No-arbitrage with multiple-priors in discrete time

Romain Blanchard, E.mail : romblanch@hotmail.com
Laurence Carassus, E.mail : laurence.carassus@devinci.fr
Léonard de Vinci Pôle Universitaire, Research Center,
92916 Paris La Défense, France and
LMR, FRE 2011 Université Reims Champagne-Ardenne.

Abstract

In a discrete time and multiple-priors setting, we propose a new characterisation of the condition of quasi-sure no-arbitrage which has become a standard assumption. This characterisation shows that it is indeed a well-chosen condition being equivalent to several previously used alternative notions of no-arbitrage and allowing the proof of important results in mathematical finance. We also revisit the so-called geometric and quantitative no-arbitrage conditions and explicit two important examples where all these concepts are illustrated.

Key words: No-arbitrage, Knightian uncertainty; multiple-priors; non-dominated model
AMS 2000 subject classification: Primary 91B70, 91B30, 28B20.

1 Introduction

The concept of no-arbitrage is fundamental in the modern theory of mathematical finance. Roughly speaking, it means that one cannot hope to make a profit without taking some risk. In a classical uni-prior setting, the Fundamental Theorem of Asset Pricing (FTAP in short) makes the link between an appropriate notion of no-arbitrage and the existence of equivalent risk-neutral probability measures. This result is essential for pricing issues, namely for the superreplication price which is for a given claim the minimum selling price needed to superreplicate it by trading in the market. The FTAP was initially formalised in [Harrison and Kreps, 1979], [Harrison and Pliska, 1981] and [Kreps, 1981] while [Dalang et al., 1990] established it in a general discrete-time setting and [Delbaen and Schachermayer, 1994] in continuous time models. The literature on the subject is huge and we refer to [Delbaen and Schachermayer, 2006] for a general overview.
However, the reliance on a single probability measure has long been questioned in the economic literature and is often referred to as Knightian uncertainty, in reference to [Knight, 1921]. In a financial context, it is called model-risk and also has a long history. The financial crisis together with the evolution of the structure and behaviour of financial markets, have made these issues even more acute for both academics and practitioners. In particular, this has motivated further research to find good notions of no-arbitrage allowing to extend the FTAP and the superreplication price characterisation while accounting for model uncertainty. A typical example of such endeavor, directly motivated by concrete situations, is to find no-arbitrage prices for some exotic derivative products (such as barrier options, lookback options, double digit options,…) using as input the prices of actively traded european options, without making any assumptions on the dynamic of the underlying. This is the so-called model-independent approach, pioneered in [Hobson, 1998]. We refer to [Hobson, 2011] for a detailed presentation including the related Skorokhod embedding problem. Importantly, [Davis and Hobson, 2007] have shown that the expected dichotomy between the existence of a suitable martingale measure and the existence of a model-independent arbitrage might not hold. [Acciaio et al., 2013] have also established a FTAP in a model-independent framework under a fairly weak notion of no-arbitrage¹¹1An arbitrage is a strategy with a strictly positive terminal payoff in all states of the world., but assuming the existence of a traded option with a super-linearly growing payoff-function.
An alternative way of modeling uncertainty is to replace the single probability measure of the classical setting with a set of priors representing all the possible models: This is the so-called quasi-sure or multiple-priors approach. As the set can vary between a singleton and all the probability measures on a given space, this formulation encompasses a wide range of settings, including the classical one. As the set of priors is not assumed to be dominated, this has raised challenging mathematical questions and has lead to the development of innovative tools such as quasi-sure stochastic analysis, non-linear expectations and G-Brownian motions. On these topics, we refer among others to [Peng, 2008, 2011], [Denis and Martini, 2006], [Denis et al., 2011], [Nutz and van Handel, 2013], [Soner et al., 2011a] and [Soner et al., 2011b].
Following this approach, [Bouchard and Nutz, 2015] have introduced in a discrete-time setting with time horizon $T$ , a no-arbitrage condition called the $NA(\mathcal{Q}^{T})$ condition (where $\mathcal{Q}^{T}$ represents all the possible models). It states that if the terminal value of a trading strategy is non-negative $\mathcal{Q}^{T}$ -quasi-surely, then it always equals $0$ $\mathcal{Q}^{T}$ -quasi-surely (see Definition 3.1). This is a natural extension of the classical uni-prior where almost sure equality and inequality are replaced with their quasi-sure pendant. [Bouchard and Nutz, 2015] established a generalisation of the FTAP together with a Superhedging Theorem. This framework has also been used to study a large range of related problems (FTAP with transaction cost, american options, worst-case optimal investment, …) and we refer among others to [Bouchard and Nutz., 2016], [Bayraktar et al., 2015], [Blanchard and Carassus, 2018] and [Bartl, 2019b].
Finally, the so-called pathwise approach is an other fruitful modeling approach: In this setting, uncertainty is introduced by describing a subset of relevant events or scenarii without references to any probability measure and without specifying their relative weight. In a discrete-time setting, [Burzoni et al., 2016c], [Burzoni et al., 2016a] introduce a set of scenarii $\mathcal{S}$ representing the agent beliefs and an Arbitrage de la Classe $\mathcal{S}$ is a trading strategy leading to a terminal value that is always non-negative for all the events in $\mathcal{S}$ and positive for a least one event in $\mathcal{S}$ . A corresponding FTAP is then obtained. Note that by choosing different sets $\mathcal{S}$ , different definitions of no-arbitrage can be considered and in particular the model independent approach previously mentioned can be recovered by choosing the whole space for $\mathcal{S}$ . Importantly, [Oblój and Wiesel, 2018] have recently unified the quasi-sure and the pathwise approaches showing that under technical assumptions both approaches are actually equivalent (see Metatheorem 1.1, see also Remark 3.34).

In this paper we follow the multiple-priors approach of [Bouchard and Nutz, 2015]. Despite its success, one might still wonder if the $NA(\mathcal{Q}^{T})$ condition is the “right” one. Indeed, at first sight at least, under this condition it is not even clear if there exists a model $P\in\mathcal{Q}^{T}$ satisfying the uni-prior no-arbitrage condition $NA(P)$ . Theorem 3.30 will prove that this is in fact possible. But as Lemmata 3.7 and 4.5 show, $\mathcal{Q}^{T}$ might still contain some models that are not arbitrage free. This means that an agent may not be able to delta-hedge a simple vanilla option using different levels of volatility in a arbitrage free way. So instead of $NA(\mathcal{Q}^{T})$ one may assume that every model is arbitrage free i.e. that the $NA(P)$ condition holds true for every model $P\in\mathcal{Q}^{T}$ . We call this $sNA(\mathcal{Q}^{T})$ for strong no-arbitrage, see Definition 3.3. This alternative condition has appeared in recent results on robust utility maximisation of unbounded functions, see for instance [Blanchard and Carassus, 2018] and [Rásonyi and Meireles-Rodrigues, 2018]. Our main result provides a characterisation of the $NA(\mathcal{Q}^{T})$ condition that gives some kind of definitive answer to these questions and confirms that the $NA(\mathcal{Q}^{T})$ condition is indeed the “right” condition in the quasi-sure setting. More precisely, Theorem 3.8 shows that the $NA(\mathcal{Q}^{T})$ condition is equivalent to the existence of a subclass of priors $\mathcal{P}^{T}\subset\mathcal{Q}^{T}$ such that $\mathcal{P}^{T}$ and $\mathcal{Q}^{T}$ have the same polar sets (roughly speaking the same relevant events) and such that the $sNA(\mathcal{P}^{T})$ hold true. In addition to enable a better economic comprehension of $NA(\mathcal{Q}^{T})$ , Theorem 3.8 also provides several interesting results. First, it allows for a short proof of a refinement of the FTAP of [Bouchard and Nutz, 2015] using the classical Dalang-Morton-Willinger Theorem (see Corollary 3.12 and [Bayraktar and Zhou, 2017, Theorem 2.1]). Then, Theorem 3.8 provides tractable theorems for the existence of solutions in the problem of robust utility maximisation. Indeed it allows to prove the equivalence between $NA(\mathcal{Q}^{T})$ and two other conditions previously used in the litterature for solving this problem. The first one is the no-arbitrage condition introduced in Bartl et al. [2019] which states that for every prior $Q\in\mathcal{Q}^{T}$ there exist a prior $P\in\mathcal{Q}^{T}$ such that $Q\ll P$ and $NA(P)$ holds true (see Corollary 3.11). The second one is the condition used by [Rásonyi and Meireles-Rodrigues, 2018] which requires the existence of a model $P^{*}\in\mathcal{Q}^{T}$ satisfying $NA(P^{*})$ and such that for this model the affine space generated by the conditional support always equals $\mathbb{R}^{d}$ (see Theorem 3.30, Remark 3.35 and also [Bayraktar and Zhou, 2017] in a one period setup). Finally, Theorem 3.8 allows to show that one may replace the set $\mathcal{Q}^{T}$ by the set $\mathcal{P}^{T}$ in the problem of maximisation of robust expected utility without changing the value function (see Lemma 3.14 and Corollary 3.17).

We then introduce local characterisations of the $NA(\mathcal{Q}^{T})$ condition called the geometric and the quantitative conditions (see Definition 3.19, 3.20 and Theorem 3.24). The geometric condition goes back in the uni-prior setup to [Jacod and Shiryaev, 1998, Theorem 3 g)] and provides some geometric intuition. Theorem 3.24 generalises the preceding result to the quasi-sure setting. The geometric condition is an important tool in the multiple-priors literature. It has been used in different setups by [Oblój and Wiesel, 2018] and by [Burzoni et al., 2016b]. It is also efficient to prove concretely that the $NA(\mathcal{Q}^{T})$ condition holds true. The quantitative no-arbitrage goes back to [Rásonyi and Stettner, 2005, Proposition 3.3] and is used to solve optimisation problems using the dynamic programming principle. For example, it provides explicit bounds on the optimal strategies in the problem of maximisation of expected utility, see Remark 3.22. Again Theorem 3.24 generalises [Rásonyi and Stettner, 2005, Proposition 3.3] to the quasi-sure setting. Together with Propositions 3.28 and 3.37, this fills a gap opened in [Blanchard and Carassus, 2018, Proposition 2.3], proving difficult measurability results and opening the possibility to solve, in the setting of [Bouchard and Nutz, 2015], the problem of multi-prior optimal investment for unbounded utility function defined on the whole real-line (see Remark 3.29).
Finally, Proposition 3.39 explicits the relation between the different notions of no-arbitrage in the dominated case while Proposition 4.1 is used to build examples of sets of probability measures $\mathcal{Q}^{T}$ which are not dominated.

The proofs follow the same idea: We first study a one-period problem with deterministic initial data where we rely on separation theorem and elementary geometric consideration in finite dimension. Then we extend the results to the multi-period setting relying on advanced measurable selections arguments. The proof of Proposition 4.1 relies also on relatively recent topological results.

Finally, these theoretical results are complemented by two concrete and useful examples. The first one proposes a multiple-priors binomial model and the second one a generic way of introducing uncertainty for the discretised dynamics of a diffusion process. In both cases, we show that the $NA(\mathcal{Q}^{T})$ conditions holds true and provide explicit expressions for the parameters introduced in the geometric and quantitative versions of the $NA(\mathcal{Q}^{T})$ condition and for the set $\mathcal{P}^{T}$ .

The paper is structured as follows: Section 2 presents the framework and notations needed in the paper. Different definitions of conditional support which are at the heart of our study are introduced and important measurability results established. Section 3 contains the different definitions of no-arbitrage together with our main result. In Section 4 we propose two detailed examples illustrating the previous results and also how to build set of probability measures which are not dominated. Finally, Section 5 collects the missing proofs.

2 The Model

This section presents our multiple-priors framework and gives introductory definitions.

2.1 Uncertainty modeling

The construction of the global probability space is based on a product of the local (between time $t$ and $t+1$ ) ones using measurable selection under Assumption 2.2 below. This is tailor made for the dynamic programming approach.

We fix a time horizon $T\in\mathbb{N}$ and introduce a sequence $\left(\Omega_{t}\right)_{1\leq t\leq T}$ of Polish spaces. Each $\Omega_{t+1}$ contains all possible scenarii between time $t$ and $t+1$ . For some $1\leq t\leq T$ , we set $\Omega^{t}:=\Omega_{1}\times\dots\times\Omega_{t}$ (with the convention that $\Omega^{0}$ is reduced to a singleton), $\mathcal{B}(\Omega^{t})$ its Borel sigma-algebra and $\mathfrak{P}(\Omega^{t})$ the set of all probability measures on $(\Omega^{t},\mathcal{B}(\Omega^{t}))$ . An element of $\Omega^{t}$ will be denoted by $\omega^{t}=(\omega_{1},\dots,\omega_{t})=(\omega^{t-1},\omega_{t})$ for $(\omega_{1},\dots,\omega_{t})\in\Omega_{1}\times\dots\times\Omega_{t}$ . We also introduce the universal sigma-algebra $\mathcal{B}_{c}(\Omega^{t})$ which is the intersection of all possible completions of $\mathcal{B}(\Omega^{t})$ .
Let $S:=\left\{S_{t},\ 0\leq t\leq T\right\}$ be a $\mathbb{R}^{d}$ -valued process where for all $0\leq t\leq T$ , $S_{t}=\left(S^{i}_{t}\right)_{1\leq i\leq d}$ represents the price of $d$ risky securities at time $t$ . We assume that there is a riskless asset whose price is constant and equals $1$ . We also make the following assumptions already stated in [Bouchard and Nutz, 2015] to which we refer for further details and motivations on the framework.

Assumption 2.1.

The process $S$ is $\left(\mathcal{B}(\Omega^{t})\right)_{0\leq t\leq T}$ -adapted.

Trading strategies are represented by $\left(\mathcal{B}_{c}(\Omega^{t-1})\right)_{1\leq t\leq T}$ -measurable and $d$ -dimensional processes $\phi:=\{\phi_{t},1\leq t\leq T\}$ where for all $1\leq t\leq T$ , $\phi_{t}=\left(\phi^{i}_{t}\right)_{1\leq i\leq d}$ represents the investor’s holdings in each of the $d$ assets at time $t$ . The set of all such trading strategies is denoted by $\Phi$ . The notation $\Delta S_{t}:=S_{t}-S_{t-1}$ will often be used. If $x,y\in\mathbb{R}^{d}$ then the concatenation $xy$ stands for their scalar product. The symbol $|\cdot|$ denotes the Euclidean norm on $\mathbb{R}^{d}$ (or on $\mathbb{R})$ . Trading is assumed to be self-financing and the value at time $t$ of a portfolio $\phi$ starting from initial capital $x\in\mathbb{R}$ is given by

V^{x,\phi}_{t}=x+\sum_{s=1}^{t}\phi_{s}\Delta S_{s}.

We construct the set $\mathcal{Q}^{T}$ of all possible priors in the market. For all $0\leq t\leq T-1$ , let $\mathcal{Q}_{t+1}:\Omega^{t}\twoheadrightarrow\mathfrak{P}(\Omega_{t+1})$ ²²2The notation $\twoheadrightarrow$ stands for set-valued mapping. where $\mathcal{Q}_{t+1}(\omega^{t})$ can be seen as the set of all possible priors for the $t$ -th period given the state $\omega^{t}$ until time $t$ .

Assumption 2.2.

For all $0\leq t\leq T-1$ , $\mathcal{Q}_{t+1}$ is a non-empty and convex valued random set such that

\displaystyle\mbox{graph}(\mathcal{Q}_{t+1})=\left\{(\omega^{t},P)\in\Omega^{t}\times\mathfrak{P}(\Omega_{t+1}),\;P\in\mathcal{Q}_{t+1}(\omega^{t})\right\}

is an analytic set.

Let $X$ be a Polish space. An analytic set of $X$ is the continuous image of some Polish space, see [Aliprantis and Border, 2006, Theorem 12.24 p447]. We denote by $\mathcal{A}(X)$ the set of analytic sets of $X$ and recall some key properties that will often be used without further reference in the rest of the paper. The projection of an analytic set is an analytic set see ([Bertsekas and Shreve, 2004, Proposition 7.39 p165]), a countable union or intersection of analytic sets is an analytic set (see [Bertsekas and Shreve, 2004, Corollary 7.35.2 p160]), the Cartesian product of analytic sets is an analytic set (see [Bertsekas and Shreve, 2004, Proposition 7.38 p165]), the image or pre-image of an analytic set is an analytic set (see [Bertsekas and Shreve, 2004, Proposition 7.40 p165]) and (see [Bertsekas and Shreve, 2004, Proposition 7.36 p161, Corollary 7.42.1 p169])

\displaystyle\mathcal{B}(X)\subset\mathcal{A}(X)\subset\mathcal{B}_{c}(X).

(1)

However the complement of an analytic set does not need to be an analytic set.
We will also use without further references a particular case of the Projection Theorem (see [Castaing and Valadier, 1977, Theorem 3.23 p75]) and of the Auman’s Theorem (see [Sainte-Beuve, 1974, Corollary 1]) which we recall for sake of completeness. Let $(X,\mathcal{T})$ be a measurable space and $Y$ be some Polish space. If $G\in\mathcal{T}\otimes\mathcal{B}(Y)$ , then the projection of $G$ on $X$ $\mbox{Proj}_{X}(G)$ belongs to $\mathcal{T}_{c}(X),$ the completion of $\mathcal{T}$ with respect to any probability measures on $(X,\mathcal{T})$ . Let $\Gamma:\,X\twoheadrightarrow Y$ be such that $\mbox{graph}(\Gamma)\in\mathcal{T}\otimes\mathcal{B}(Y).$ Then there exist a $\mathcal{T}_{c}(X)-\mathcal{B}(Y)$ measurable selector $\sigma:\,X\to Y$ such that $\sigma(x)\in\Gamma(x)$ for all $x\in\{\Gamma\neq\emptyset\}$ .

From the Jankov-von Neumann Theorem (see [Bertsekas and Shreve, 2004, Proposition 7.49 p182]) and Assumption 2.2, there exists some $\mathcal{B}_{c}(\Omega^{t})$ -measurable $q_{t+1}:\Omega^{t}\to\mathfrak{P}(\Omega_{t+1})$ such that for all $\omega^{t}\in\Omega^{t}$ , $q_{t+1}(\cdot,\omega^{t})\in\mathcal{Q}_{t+1}(\omega^{t})$ (recall that for all $\omega^{t}\in\Omega^{t}$ , $\mathcal{Q}_{t+1}(\omega^{t})\neq\emptyset$ ). For all $1\leq t\leq T$ let $\mathcal{Q}^{t}\subset\mathfrak{P}\left(\Omega^{t}\right)$ be defined by

	$\displaystyle\mathcal{Q}^{t}:=\bigl{\{}Q_{1}\otimes q_{2}\otimes\dots\otimes q_{t},$	$\displaystyle\;Q_{1}\in\mathcal{Q}_{1},\;q_{s+1}\in\mathcal{S}K_{s+1},$		(2)
		$\displaystyle q_{s+1}(\cdot,\omega^{s})\in\mathcal{Q}_{s+1}(\omega^{s}),\;\forall\,\omega^{s}\in\Omega^{s},\;\forall\,1\leq s\leq t-1\;\bigr{\}},$

where $Q^{t}:=Q_{1}\otimes q_{2}\otimes\dots\otimes q_{t}$ denotes the $t$ -fold application of Fubini’s Theorem (see [Bertsekas and Shreve, 2004, Proposition 7.45 p175]) which defines a measure on $\mathfrak{P}\left(\Omega^{t}\right)$ and $\mathcal{S}K_{t+1}$ is the set of universally-measurable stochastic kernel on $\Omega_{t+1}$ given $\Omega^{t}$ (see [Bertsekas and Shreve, 2004, Definition 7.12 p134, Lemma 7.28 p174]).

Apart from Assumption 2.2, no specific assumptions on the set of priors are made: $\mathcal{Q}^{T}$ is neither assumed to be dominated by a given probability measure nor to be weakly compact. This setting allows for various general definitions of the sets $\mathcal{Q}^{T}$ . Section 4 presents some concrete examples of non-dominated settings. We refer also to [Bartl, 2019a] for other examples.

2.2 Multiple-priors conditional supports

The following definitions are at the heart of our study.

Definition 2.3.

Let $P\in\mathfrak{P}\left(\Omega^{T}\right)$ with the fixed disintegration $P:=Q_{1}\otimes q_{2}\otimes\cdots\otimes q_{T}$ where $q_{t}\in\mathcal{SK}_{t}$ for all $1\leq t\leq T$ . For all $0\leq t\leq T-1$ , the random sets ${E}^{t+1}:\;\Omega^{t}\times\mathfrak{P}(\Omega_{t+1})\twoheadrightarrow\mathbb{R}^{d}$ , ${D}^{t+1},\ {D}_{P}^{t+1}\;:\Omega^{t}\twoheadrightarrow\mathbb{R}^{d}$ are defined for $\omega^{t}\in\Omega^{t}$ , $p\in\mathfrak{P}(\Omega_{t+1})$ by

$\displaystyle{E}^{t+1}(\omega^{t},p)$	$\displaystyle:=\bigcap\left\{A\subset\mathbb{R}^{d},\;\mbox{closed},\;p\left(\Delta S_{t+1}(\omega^{t},.)\in A\right)=1\right\},$	(3)
$\displaystyle{D}^{t+1}(\omega^{t})$	$\displaystyle:=\bigcap\left\{A\subset\mathbb{R}^{d},\;\mbox{closed},\;p\left(\Delta S_{t+1}(\omega^{t},.)\in A\right)=1,\;\forall\,p\in\mathcal{Q}_{t+1}(\omega^{t})\right\},$	(4)
$\displaystyle{D}_{P}^{t+1}(\omega^{t})$	$\displaystyle:=\bigcap\left\{A\subset\mathbb{R}^{d},\;\mbox{closed},\;q_{t+1}\left(\Delta S_{t+1}(\omega^{t},.)\in A,\omega^{t}\right)=1\right\}.$	(5)

Remark 2.4.

As $\mathbb{R}^{d}$ is second countable, $p\left(\Delta S_{t+1}(\omega^{t},\cdot)\in E^{t+1}(\omega^{t},p)\right)=1,$ see [Aliprantis and Border, 2006, Theorem 12.14] and $p\left(\Delta S_{t+1}(\omega^{t},\cdot)\in D^{t+1}(\omega^{t}\right))=1$ for all $p\in\mathcal{Q}_{t+1}(\omega^{t}),$ see [Bouchard and Nutz, 2015, Lemma 4.2].

Remark 2.5.

It is easy to verify that for all $\omega^{t}\in\Omega^{t}$ , $p\in\mathcal{Q}_{t+1}(\omega^{t})$

\displaystyle E^{t+1}(\omega^{t},p)

\displaystyle\subset D^{t+1}(\omega^{t}).

(6)

Recall that any probability $P\in\mathfrak{P}(\Omega^{T})$ can be decomposed using Borel-measurable stochastic kernel, see for instance [Bertsekas and Shreve, 2004, Corollary 7.27.2 p139]. Then for some fixed disintegration of $P\in\mathcal{Q}^{T}$ , $P:=Q_{1}\otimes q_{2}\otimes\cdots\otimes q_{T}$ , all $0\leq t\leq T-1$ and all $\omega^{t}\in\Omega^{t}$

\displaystyle{D}_{P}^{t+1}(\omega^{t})={E}^{t+1}(\omega^{t},q_{t+1}(\cdot,\omega^{t}))\subset{D}^{t+1}(\omega^{t})

(7)

as $q_{t}(\cdot,\omega^{t})\in\mathcal{Q}_{t+1}(\omega^{t})$ for all $\omega^{t}\in\Omega^{t}$ (see (2)).

The following lemma establishes some important measurability properties of the random sets previously introduced and uses the following notations. For some $R\subset\mathbb{R}^{d}$ , let

	$\displaystyle\mbox{Aff}(R)$	$\displaystyle:=\bigcap\{A\subset\mathbb{R}^{d},\;\mbox{affine subspace},\;R\subset A\},$
	$\displaystyle{{\mbox{Conv}}}(R)$	$\displaystyle:=\bigcap\{C\subset\mathbb{R}^{d},\;\mbox{ convex},\;R\subset C\},\;\quad{\overline{\mbox{Conv}}}(R):=\bigcap\{C\subset\mathbb{R}^{d},\;\mbox{closed convex},\;R\subset C\}.$

Recall that $\mbox{Conv}(R)=\left\{\sum_{i=1}^{n}\lambda_{i}p_{i},\;n\geq 1,\;p_{i}\in R,\;\sum_{i=1}^{n}\lambda_{i}=1,\lambda_{i}\geq 0\right\}$ see [Rockafellar, 1970, Theorem 2.3 p12] and that ${\overline{\mbox{Conv}}}(R)=\overline{{\mbox{Conv}}(R)}$ .
For a random set $R:\Omega\twoheadrightarrow\mathbb{R}^{d}$ , ${\overline{\mbox{Conv}}}\left(R\right)$ and $\mbox{Aff}\left(R\right)$ are the random sets defined for all $\omega\in\Omega$ by $\overline{{\mbox{Conv}}}\left(R\right)(\omega):=\overline{{\mbox{Conv}}}\left(R(\omega)\right)\;\mbox{and}\;\mbox{Aff}\left(R\right)(\omega):=\mbox{Aff}\left(R(\omega)\right).$

Lemma 2.6.

Let Assumptions 2.1 and 2.2 hold true and let $0\leq t\leq T-1$ be fixed. Let $P\in\mathcal{Q}^{T}$ with a fixed disintegration $P:=Q_{1}\otimes q_{2}\otimes\cdots\otimes q_{T}$ .

•

The random sets ${E}^{t+1}$ , $\overline{\mbox{Conv}}\left({E}^{t+1}\right)$ , $\mbox{Aff}\left(E^{t+1}\right)$ are non-empty, closed valued and $\mathcal{B}(\Omega^{t})\otimes\mathcal{B}(\mathfrak{P}\left(\Omega_{t+1})\right)$ -measurable³³3See [Rockafellar and Wets, 1998, Definition 14.1]. with graphs in $\mathcal{B}(\Omega^{t})\otimes\mathcal{B}\left(\mathfrak{P}(\Omega_{t+1})\right)\otimes\mathcal{B}(\mathbb{R}^{d})$ .
•

The random sets ${D}^{t+1}$ , ${D}_{P}^{t+1}$ , $\overline{\mbox{Conv}}\left(D^{t+1}\right)$ , $\overline{\mbox{Conv}}\left({D}_{P}^{t+1}\right)$ , $\mbox{Aff}\left(D^{t+1}\right)$ and $\mbox{Aff}\left(D_{P}^{t+1}\right)$ are non-empty, closed valued and $\mathcal{B}_{c}(\Omega^{t})$ -measurable. Furthermore their graphs belong to $\mathcal{B}_{c}(\Omega^{t})\otimes\mathcal{B}(\mathbb{R}^{d})$ .

Proof.

The measurability of ${D}^{t+1}$ follows from [Blanchard and Carassus, 2018, Lemma 2.2]. Fix some open set $O\subset\mathbb{R}^{d}$ . Assumption 2.1 and [Bertsekas and Shreve, 2004, Proposition 7.29 p144] imply that $(\omega^{t},p)\to p\left(\Delta S_{t+1}(\omega^{t},.)\in O\right)$ is $\mathcal{B}(\Omega^{t})\otimes\mathcal{B}(\mathfrak{P}(\Omega_{t+1}))$ -measurable. The measurability of ${E}^{t+1}$ and ${D}_{P}^{t+1}$ follows from

	$\displaystyle\left\{(\omega^{t},p),\;{E}^{t+1}(\omega^{t},p)\cap O\neq\emptyset\right\}$	$\displaystyle=\left\{(\omega^{t},p),\;p\left(\Delta S_{t+1}(\omega^{t},.)\in O\right)>0\right\}\in\mathcal{B}(\Omega^{t})\otimes\mathcal{B}(\mathfrak{P}(\Omega_{t+1})),$
	$\displaystyle\{\omega^{t},\;{D}_{P}^{t+1}(\omega^{t})\cap O\neq\emptyset\}$	$\displaystyle=\left\{\omega^{t},\;\exists\,q\in\mathfrak{P}(\Omega_{t+1}),\;q_{t+1}(\cdot,\omega^{t})=q,\;{E}^{t+1}(\omega^{t},q)\cap O\neq\emptyset\right\}$
		$\displaystyle=\mbox{Proj}_{\Omega^{t}}\left\{(\omega^{t},q),\;q_{t+1}(\cdot,\omega^{t})=q,\;{E}^{t+1}(\omega^{t},q)\cap O\neq\emptyset\right\}\in\mathcal{B}_{c}(\Omega^{t}),$

where we have used Assumption 2.2 and the Projection Theorem as $(\omega^{t},q)\to q_{t+1}(\cdot,\omega^{t})-q$ is $\mathcal{B}_{c}(\Omega^{t})\otimes\mathfrak{P}(\Omega_{t+1})$ -measurable.
Then, [Rockafellar and Wets, 1998, Proposition 14.2, Exercise 14.12] implies that $\overline{\mbox{Conv}}\left({E}^{t+1}\right)$ , $\mbox{Aff}\left(E^{t+1}\right)$ are $\mathcal{B}(\Omega^{t})\otimes\mathcal{B}(\mathfrak{P}(\Omega_{t+1}))$ -measurable and that $\overline{\mbox{Conv}}\left(D^{t+1}\right)$ , $\overline{\mbox{Conv}}\left({D}_{P}^{t+1}\right)$ , $\mbox{Aff}\left(D^{t+1}\right)$ and $\mbox{Aff}\left(D_{P}^{t+1}\right)$ are $\mathcal{B}_{c}(\Omega^{t})$ -measurable.
Finally, [Rockafellar and Wets, 1998, Theorem 14.8] implies that the graphs of $E^{t+1}$ , $\overline{\mbox{Conv}}\left({E}^{t+1}\right)$ and $\mbox{Aff}\left(E^{t+1}\right)$ belong to $\mathcal{B}(\Omega^{t})\otimes\mathcal{B}\left(\mathfrak{P}(\Omega_{t+1})\right)\otimes\mathcal{B}(\mathbb{R}^{d})$ while the graphs of ${D}^{t+1}$ , ${D}_{P}^{t+1}$ , $\overline{\mbox{Conv}}\left(D^{t+1}\right)$ , $\overline{\mbox{Conv}}\left({D}_{P}^{t+1}\right)$ , $\mbox{Aff}\left(D^{t+1}\right)$ , and $\mbox{Aff}\left(D_{P}^{t+1}\right)$ belong to $\mathcal{B}_{c}(\Omega^{t})\otimes\mathcal{B}(\mathbb{R}^{d}).$

3 No-arbitrage characterisations

3.1 Global no-arbitrage condition and main result

In the uni-prior case, for any $P\in\mathcal{P}^{T},$ the no-arbitrage $NA(P)$ condition holds true if $V_{T}^{0,\phi}\geq 0$ $P$ -a.s. for some $\phi\in\Phi$ implies that $V_{T}^{0,\phi}=0$ $P$ -a.s. In the multiple-priors setting, the no-arbitrage condition $NA(\mathcal{Q}^{T})$ , also referred as quasi-sure no-arbitrage, was introduced in [Bouchard and Nutz, 2015]. Our main message will be that it is indeed a good assumption. Besides being a natural extension of the classical uni-prior arbitrage condition, we will show that it is equivalent to several conditions previously used in the literature.

Definition 3.1.

The $NA(\mathcal{Q}^{T})$ condition holds true if $V_{T}^{0,\phi}\geq 0\;\mathcal{Q}^{T}\mbox{-q.s. for some $\phi\in\Phi$}$ implies that $V_{T}^{0,\phi}=0\;\mathcal{Q}^{T}\mbox{-q.s. }$

Recall that for a given $\mathcal{P}\subset\mathfrak{P}(\Omega^{T})$ , a set $N\subset\Omega^{T}$ is called a $\mathcal{P}$ -polar if for all $P\in\mathcal{P}$ , there exists some $A_{P}\in\mathcal{B}_{c}(\Omega^{T})$ such that $P(A_{P})=0$ and $N\subset A_{P}$ . A property holds true $\mathcal{P}$ -quasi-surely (q.s.), if it is true outside a $\mathcal{P}$ -polar set. Finally a set is of $\mathcal{P}$ -full measure if its complement is a $\mathcal{P}$ -polar set.

[Bouchard and Nutz, 2015] proves that Definition 3.1 allows a FTAP generalisation. The $NA(\mathcal{Q}^{T})$ is equivalent to the following: For all $Q\in\mathcal{Q}^{T}$ , there exists some $P\in\mathcal{R}^{T}$ such that $Q\ll P$ where

\displaystyle\mathcal{R}^{T}:=\{P\in\mathfrak{P}(\Omega^{T}),\;\exists\,Q^{{}^{\prime}}\in\mathcal{Q}^{T},P\ll Q^{{}^{\prime}}\;\mbox{and $P$ is a martingale measure}\}.

(8)

The next result is straightforward.

Lemma 3.2.

Let $\mathcal{P}$ and $\mathcal{M}$ be two sets of probability measures on $\mathfrak{P}(\Omega^{T})$ such that $\mathcal{P}$ and $\mathcal{M}$ have the same polar sets. Then the $NA(\mathcal{P})$ and the $NA(\mathcal{M})$ conditions are equivalent.

Nevertheless, it is not true that under the $NA(\mathcal{Q}^{T})$ condition, the $NA(P)$ condition holds true for all $P\in\mathcal{Q}^{T},$ see Lemma 3.7 below. This condition is called the “strong no-arbitrage” or $sNA(\mathcal{Q}^{T})$ .

Definition 3.3.

The $sNA(\mathcal{Q}^{T})$ condition holds true if the $NA(P)$ holds true for all $P\in\mathcal{Q}^{T}$ .

Remark 3.4.

The $sNA(\mathcal{Q}^{T})$ is a strong condition. But it is related to practical situations in finance: If it does not hold true, there exists a model $P\in\mathcal{Q}^{T}$ and a strategy $\phi\in\Phi$ such that $V_{T}^{0,\phi}\geq 0\;{P}\mbox{-a.s.}$ and $P(V_{T}^{0,\phi}>0)>0$ and an agent having sold some derivative product may not be able to use different arbitrage free models to manage the resulting position (think for instance of different volatility level to delta-hedge a simple vanilla option).
The $sNA(\mathcal{Q}^{T})$ condition is also useful to obtain tractable theorems on multiple-priors expected utility maximisation for unbounded function, see [Blanchard and Carassus, 2018, Theorem 3.6] and [Rásonyi and Meireles-Rodrigues, 2018, Theorem 3.9].
Finally, this definition seems also relevant in a continuous time setting for studying the no-arbitrage characterisation, see [Biagini et al., 2015, Definition 2.1, Theorem 3.4].

In the spirit of the model-dependent arbitrage introduced in [Davis and Hobson, 2007] (see also Remark 3.35) we introduce the notion of “weak no-arbitrage”.

Definition 3.5.

The $wNA(\mathcal{Q}^{T})$ condition holds true if there exists some $P\in\mathcal{Q}^{T}$ such that the $NA(P)$ holds true.

Remark 3.6.

The contraposition of the $wNA(\mathcal{Q}^{T})$ condition is that for all models $P\in\mathcal{Q}^{T}$ , there exists a strategy $\phi_{P}$ such that $V_{T}^{0,\phi_{P}}\geq 0\;{P}\mbox{-a.s.}$ and $P(V_{T}^{0,\phi_{P}}>0)>0$ . A concrete example of a such model-dependent arbitrage is given in [Davis and Hobson, 2007].

We illustrate now the obvious relations between the three no-arbitrage conditions introduced (see also Figure 2). The more subtle one will be addressed in Theorems 3.8 and 3.30. This last theorem shows that the $NA(\mathcal{Q}^{T})$ condition implies the $wNA(\mathcal{Q}^{T})$ one.

Lemma 3.7.

1.

Assume that $\mathcal{Q}^{T}=\{P\}$ for some $P\in\mathfrak{P}(\Omega^{T})$ . Then the $NA(\mathcal{Q}^{T}),$ $sNA(\mathcal{Q}^{T}),$ $wNA(\mathcal{Q}^{T})$ and $NA(P)$ conditions are equivalent.
2.

Assume that there exists a dominating probability measure $\widehat{P}\in\mathcal{Q}^{T}$ . Then the $NA(\mathcal{Q}^{T})$ and $NA(\widehat{P})$ conditions are equivalent.
3.

The $sNA(\mathcal{Q}^{T})$ condition implies the $wNA(\mathcal{Q}^{T})$ but the converse does not hold true.
4.

The $sNA(\mathcal{Q}^{T})$ condition implies the $NA(\mathcal{Q}^{T})$ but the converse does not true.
5.

The $wNA(\mathcal{Q}^{T})$ condition does not imply the $NA(\mathcal{Q}^{T})$ condition.

Proof.

The first item is clear. The second one follows from Lemma 3.2. The first part of item 3 is trivial and it easy to construct simple counter-example for the second part (see Example 3.36 below). We now prove item 4. If the $NA(\mathcal{Q}^{T})$ condition fails, there exists some $\phi\in\Phi$ and $P\in\mathcal{Q}^{T}$ such that $V_{T}^{0,\phi}\geq 0\;{\mathcal{Q}^{T}}\mbox{-q.s.}$ and $P(V_{T}^{0,\phi}>0)>0:$ The $sNA(\mathcal{Q}^{T})$ condition also fails. Now consider a one-period model with one risky asset $S_{0}=0$ , $S_{1}:\Omega\to\mathbb{R}$ (for some Polish space $\Omega$ ). Let $P_{1}$ such that $P_{1}\left(\pm\Delta S_{1}>0\right)>0$ and $P_{2}$ such that $P_{2}(\Delta S_{1}\geq 0)=1$ and $P_{2}(\Delta S_{1}>0)>0$ and set $\mathcal{Q}=\{\lambda P_{1}+(1-\lambda)P_{2},\;0<\lambda\leq 1\}$ . Then $NA(P_{2})$ fails while $NA(\mathcal{Q})$ holds true. Note that Lemma 4.5 provides another counter-example. Finally for item 5, consider a one period model with two risky assets $S^{1}_{0}=S^{2}_{0}=0$ and $S^{1,2}_{1}:\Omega\to\mathbb{R}$ . Let $P_{1}$ be such that $P_{1}(\Delta S_{1}^{1}\geq 0)=1$ , $P_{1}(\Delta S_{1}^{1}>0)>0$ and $P_{2}$ such that $P_{2}(\Delta S_{1}^{1}=0)=1$ , $P_{2}(\pm\Delta S_{1}^{2}>0)>0$ and set $\mathcal{Q}=\{\lambda P_{1}+(1-\lambda)P_{2},\;0<\lambda\leq 1\}$ . Then the $NA(P_{2})$ and thus the $wNA(\mathcal{Q})$ conditions are clearly verified. But the $NA(\mathcal{Q})$ condition does not hold true. Indeed, let $h=(1,0)$ . Then $h\Delta S_{1}\geq 0$ $\mathcal{Q}$ -q.s. but $P_{1}(h\Delta S_{1}>0)>0$ . Note that $\mbox{Aff}(D)=\mathbb{R}^{2}$ and $\mbox{Aff}\left(D_{P_{2}}\right)=\{0\}\times\mathbb{R}$ .

Figure 1: Relations between the no-arbitrage definitions, see Lemma3.7.

The following theorem is our main result.

Theorem 3.8.

Assume that Assumptions 2.1 and 2.2 hold true. The following conditions are equivalent.

•

The $NA(\mathcal{Q}^{T})$ condition holds true.
•

There exists some $\mathcal{P}^{T}\subset\mathcal{Q}^{T}$ such that $\mathcal{P}^{T}$ and $\mathcal{Q}^{T}$ have the same polar-sets and such that the $sNA(\mathcal{P}^{T})$ condition holds true.

Let $P^{*}$ as in Theorem 3.30 below with the fix disintegration $P^{*}:=P_{1}^{*}\otimes p_{2}^{*}\otimes\cdots\otimes p_{T}^{*}$ . The set $\mathcal{P}^{T}$ is defined recursively as follows: For all $1\leq t\leq T-1$

	$\displaystyle\begin{split}\mathcal{P}^{1}&:=\left\{\lambda P_{1}^{*}+(1-\lambda)P,\;0<\lambda\leq 1,\;P\in\mathcal{Q}^{1}\right\},\end{split}$			(9)
	$\displaystyle\begin{split}\mathcal{P}^{t+1}&:=\Bigl{\{}P\otimes\left(\lambda p^{*}_{t+1}+(1-\lambda)q_{t+1}\right),\;0<\lambda\leq 1,\\ &\quad\quad\quad\quad\quad P\in\mathcal{P}^{t},\;q_{t+1}(\cdot,\omega^{t})\in\mathcal{Q}_{t+1}(\omega^{t})\;\mbox{for all $\omega^{t}\in\Omega^{t}$}\Bigr{\}}.\end{split}$			(10)

Proof.

See Section 5.2.4.

Remark 3.9.

[Burzoni et al., 2016b, Theorem 4] delivers a similar message but in a completely different setup which does not rely on a set of priors and under the no open-arbitrage assumption. The set $\mathcal{P}^{T}$ is replaced by the set of probability measures with full support.

Remark 3.10.

In previous studies on robust pricing and hedging, it is often assumed that there exists some additional assets available only for static trading (buy and hold), see for instance [Bouchard and Nutz, 2015, Theorem 5.1]. This raises the mathematical difficulties as, roughly speaking, its breaks the dynamic consistency between time zero and future times and might prevent from obtaining a dynamic programming principle. A typical illustration of the issue arising is the so-called duality gap for American options, where the superhedging price for an American option may be strictly larger than the supremum of its expected (discounted) payoff over all stopping times and all (relevant) martingale measures (see for instance [Bayraktar et al., 2015], [Hobson and Neuberger, 2016], [Bayraktar and Zhou, 2017]).
In our setting all assets are dynamically traded and some of them may be derivatives products. Obviously the level of uncertainty regarding the behaviours of each assets might depend on its nature and this will be reflected in the set of prior $\mathcal{Q}^{T}$ . This follows the spirit of the original approach developed in [Hobson, 1998] where the prices of actively traded options is taken as input. Furthermore, from a pure practical point of view, we think that additional financial assets which provide useful informations for pricing should be traded at least on a daily basis. Hence, introducing trading constraints or transactions cost could be a better way to reflect the potential difference of liquidity between assets and derivatives. From a theoretical perspective, [Aksamit et al., 2018] shows that any setup as in [Bouchard and Nutz, 2015] can be lifted to a setup with dynamic trading in all assets in a way which does not introduce arbitrage (see [Aksamit et al., 2018, Lemma 3.1]). The idea is to assume that the options are traded dynamically and to choose a set of priors $\mathcal{Q}^{T}$ which does not impose any assumptions about their dynamics other than these resulting from no arbitrage in the initial setup. An admissible pricing measure in the original setup can be used to define dynamic options prices via conditional expectations and can thus be lifted to a martingale measure in the extended setup.

We now propose three applications of Theorem 3.8 which show how usefull it is.

The first application establishes the equivalence between the $NA(\mathcal{Q}^{T})$ condition and the no-arbitrage condition introduced by [Bartl et al., 2019] which studies the problem of robust maximisation of expected utility using medial limits.

Corollary 3.11.

Assume that Assumptions 2.1 and 2.2 hold true. The following conditions are equivalent

•

The $NA(\mathcal{Q}^{T})$ condition holds true.
•

For all $Q\in\mathcal{Q}^{T}$ , there exists some $P\in\mathcal{P}^{T}$ such that $Q\ll P$ and such that $NA(P)$ holds true.
•

For all $Q\in\mathcal{Q}^{T}$ , there exists some $P\in\mathcal{Q}^{T}$ such that $Q\ll P$ and such that $NA(P)$ holds true.

Proof.

Assume that the $NA(\mathcal{Q}^{T})$ condition holds true and choose some $Q\in\mathcal{Q}^{T}$ with the fixed disintegration $Q:=Q_{1}\otimes q_{2}\otimes\cdots\otimes q_{T}.$ Let

P:=\left(\frac{1}{2}P_{1}^{*}+\frac{1}{2}Q_{1}\right)\otimes\left(\frac{1}{2}p_{2}^{*}+\frac{1}{2}q_{2}\right)\otimes\ldots\otimes\left(\frac{1}{2}p_{T}^{*}+\frac{1}{2}q_{T}\right),

where $P^{*}$ is given in Theorem 3.30 with the fixed disintegration $P^{*}:=P_{1}^{*}\otimes p_{2}^{*}\otimes\cdots\otimes p_{T}^{*}.$ Then (10) implies that $P\in\mathcal{P}^{T}$ and obviously $Q\ll P.$ Now, Theorem 3.8 implies that the $NA(P)$ condition holds true and the second assertion is proved. As $\mathcal{P}^{T}\subset\mathcal{Q}^{T},$ the second assertion implies the third one. Assume now that the third assertion holds true and let $\phi\in\Phi$ such that $V_{T}^{0,\phi}\geq 0$ $\mathcal{Q}^{T}$ -q.s. Fix some $Q\in\mathcal{Q}^{T}$ . Then there exists $P\in\mathcal{Q}^{T}$ such that $Q\ll P$ and such that $NA(P)$ holds true. Thus $V_{T}^{0,\phi}=0$ $P$ -a.s and also $Q$ -a.s. As this is true for all $Q\in\mathcal{Q}^{T}$ , we get that $V_{T}^{0,\phi}=0$ $\mathcal{Q}^{T}$ -q.s.

The second application allows to prove the robust FTAP from the classical one. Our proof uses the one-period arguments of [Bayraktar and Zhou, 2017, Theorem 2.1] adapted to the multi-period setting. Let

\displaystyle{\mathcal{K}}^{T}:=\{P\in\mathfrak{P}(\Omega^{T}),\;\exists\,Q^{{}^{\prime}}\in\mathcal{P}^{T},P\sim Q^{{}^{\prime}}\;\mbox{and $P$ is a martingale measure}\}.

(11)

Corollary 3.12.

Assume that Assumptions 2.1 and 2.2 hold true. The following conditions are equivalent

•

The $NA(\mathcal{Q}^{T})$ condition holds true.
•

For all $Q\in\mathcal{Q}^{T}$ , there exists some $P\in{\mathcal{K}}^{T}$ such that $Q\ll P.$
•

For all $Q\in\mathcal{Q}^{T}$ , there exists some $P\in\mathcal{R}^{T}$ (see (8)) such that $Q\ll P.$

Note that this is a refinement of the version of [Bouchard and Nutz, 2015] as we have more information about the measure $P.$

Proof.

Assume that the $NA(\mathcal{Q}^{T})$ condition holds true. Corollary 3.11 implies that for all $Q\in\mathcal{Q}^{T}$ there exists some $Q^{\prime}\in\mathcal{P}^{T}$ such that $Q\ll Q^{\prime}$ and such that $NA(Q^{\prime})$ holds true. Now the classical FTAP (see [Dalang et al., 1990]) establishes the existence of some $P\sim Q^{\prime}$ such that $P$ is a martingale measure. Thus $P\in{\mathcal{K}}^{T}$ . As $Q\ll P$ , the second assertion holds true. As ${\mathcal{K}}^{T}\subset\mathcal{R}^{T}$ , the second assertion implies the third one. Assume now that the third assumption holds true and let $\phi\in\Phi$ such that $V_{T}^{0,\phi}\geq 0$ $\mathcal{Q}^{T}$ -q.s. Fix some $Q\in\mathcal{Q}^{T}$ . Then there exists $P\in\mathfrak{P}(\Omega^{T})$ and $Q^{{}^{\prime}}\in\mathcal{Q}^{T}$ such that $Q\ll P$ , $P\ll Q^{{}^{\prime}}$ and $P$ is a martingale measure. As $V_{T}^{0,\phi}\geq 0$ $Q^{\prime}$ -a.s and thus $P$ -a.s. and $E_{P}(V_{T}^{0,\phi})=0$ , we get that $V_{T}^{0,\phi}=0$ $P$ -a.s and also $Q$ -a.s. As this is true for all $Q\in\mathcal{Q}^{T}$ , we obtain that $V_{T}^{0,\phi}=0$ $\mathcal{Q}^{T}$ -q.s.

Lastly, Theorem 3.8 allows to obtain a tractable theorem on maximisation of expected utility under the $NA(\mathcal{Q}^{T})$ condition avoiding the difficult [Blanchard and Carassus, 2018, Assumption 2.1]. Note that the no-arbitrage condition is indeed related to the utility maximisation problem in the uni-prior case (see for instance [Rogers, 1994]). In the robust case, it is not clear whether a similar approach could work. This is the subject of further research.
A random utility $U$ is a function defined on $\Omega^{T}\times(0,\infty)$ taking values in $\mathbb{R}\cup\{-\infty\}$ such that for every $x\in\mathbb{R}$ , $U\left(\cdot,x\right)$ is $\mathcal{B}(\Omega^{T})$ -measurable and for every $\omega^{T}\in{\Omega}^{T}$ , $U(\omega^{T},\cdot)$ is proper⁴⁴4There exists $x\in(0,+\infty)$ such that $U(\omega^{T},x)>-\infty$ and $U(\omega^{T},x)<+\infty$ for all $x\in(0,+\infty)$ ., non-decreasing and concave on $(0,+\infty)$ . We extend $U$ by (right) continuity in $0$ and set $U(\cdot,x)=-\infty$ if $x<0$ .
Fix some $x\geq 0$ . For $P\in\mathfrak{P}(\Omega^{T})$ fixed, we denote by $\Phi(x,U,P)$ the set of all strategies $\phi\in\Phi$ such that $V_{T}^{x,\phi}(\cdot)\geq 0$ $P$ -a.s. and such that either $E_{P}U^{+}(\cdot,V_{T}^{x,\phi}(\cdot))<\infty$ or $E_{P}U^{-}(\cdot,V_{T}^{x,\phi}(\cdot))<\infty$ . Then $\Phi(x,U,\mathcal{Q}^{T}):=\bigcap_{P\in\mathcal{Q}^{T}}\Phi(x,U,P).$ The set $\Phi(x,U,\mathcal{P}^{T})$ is defined similarly changing $\mathcal{Q}^{T}$ by $\mathcal{P}^{T}$ where $\mathcal{P}^{T}$ is defined in (10). The multiple-priors portfolio problem with initial wealth $x\geq 0$ is

\displaystyle u(x):=\sup_{\phi\in\Phi(x,U,\mathcal{Q}^{T})}\inf_{P\in\mathcal{Q}^{T}}E_{P}U(\cdot,V^{x,\phi}_{T}(\cdot)).

(12)

We also define

\displaystyle u^{\mathcal{P}}(x):=\sup_{\phi\in\Phi(x,U,\mathcal{P}^{T})}\inf_{P\in\mathcal{P}^{T}}E_{P}U(\cdot,V^{x,\phi}_{T}(\cdot)).

(13)

Let for all $1\leq t\leq T$

\mathcal{W}_{t}:=\bigcap_{r>0}\left\{X:\Omega^{t}\to\mathbb{R}\cup\{\pm\infty\},\;\mbox{$\mathcal{B}(\Omega^{t})$-measurable},\;\sup_{P\in\mathcal{Q}^{t}}E_{P}|X|^{r}<\infty\right\}.

Assumption 3.13.

We have that $U^{+}(\cdot,1),U^{-}(\cdot,\frac{1}{4})\in\mathcal{W}_{T}$ and $\Delta S_{t},{1}/{\alpha^{P}_{t}}\in\mathcal{W}_{t}$ for all $1\leq t\leq T$ and $P\in\mathcal{P}^{t}$ (see Remark 3.27 for the definition of $\alpha_{t}^{P}$ ).

The first lemma shows the equality between both value functions.

Lemma 3.14.

Assume that the $NA(\mathcal{Q}^{T})$ condition and Assumptions 2.1 and 2.2 hold true. Furthermore, assume that $U$ is either bounded from above or that Assumption 3.13 holds true. Then $u(x)=u^{\mathcal{P}}(x)$ for all $x\geq 0.$

Proof.

Fix $x\geq 0$ . Theorem 3.8 will be in force. Let $P^{*}$ be given by Theorem 3.30 with the fixed disintegration $P^{*}:=P_{1}^{*}\otimes p_{2}^{*}\otimes\cdots\otimes p_{T}^{*}.$ First we show that $\Phi(x,U,\mathcal{Q}^{T})=\Phi(x,U,\mathcal{P}^{T})$ . The first inclusion follows from $\mathcal{P}^{T}\subset\mathcal{Q}^{T}.$ As $\mathcal{P}^{T}$ and $\mathcal{Q}^{T}$ have the same polar sets, $V_{T}^{x,\phi}(\cdot)\geq 0$ $\mathcal{Q}^{T}$ -q.s. and $V_{T}^{x,\phi}(\cdot)\geq 0$ $\mathcal{P}^{T}$ -q.s. are equivalent. So to prove the reverse inequality it is enough to show that for $\phi\in\Phi(x,U,\mathcal{P}^{T})$ $E_{Q}U^{+}(\cdot,V_{T}^{x,\phi}(\cdot))<\infty$ or $E_{Q}U^{-}(\cdot,V_{T}^{x,\phi}(\cdot))<\infty$ for any $Q\in\mathcal{Q}^{T}$ . It is obviously true if $U$ is bounded from above. Assume now that Assumption 3.13 holds true. Let $Q\in\mathcal{Q}^{T}$ with the fixed disintegration $Q:=P_{1}\otimes q_{2}\otimes\ldots\otimes q_{T}$ and choose

R:=\left(\frac{1}{2}P_{1}^{*}+\frac{1}{2}P_{1}\right)\otimes\left(\frac{1}{2}p_{2}^{*}+\frac{1}{2}q_{2}\right)\otimes\ldots\otimes\left(\frac{1}{2}p_{T}^{*}+\frac{1}{2}q_{T}\right).

Then $R\in\mathcal{P}^{T},$ see (10). Assume that $E_{R}U^{+}(\cdot,V_{T}^{x,\phi}(\cdot))<\infty$ (the same argument applies to the negative part). Then

\displaystyle\frac{1}{2^{T}}E_{Q}U^{+}(\cdot,V_{T}^{x,\phi}(\cdot))

\displaystyle\leq E_{R}U^{+}(\cdot,V_{T}^{x,\phi}(\cdot))<\infty.

(14)

Thus

\displaystyle u(x)=\sup_{\phi\in\Phi(x,U,\mathcal{P}^{T})}\inf_{P\in\mathcal{Q}^{T}}E_{P}U(\cdot,V^{x,\phi}_{T}(\cdot)).

(15)

Next we show that for all $x\geq 0$ and $\phi\in\Phi(x,U,\mathcal{P}^{T})$

\displaystyle u(x,\phi):=\inf_{P\in\mathcal{Q}^{T}}E_{P}U(\cdot,V^{x,\phi}_{T}(\cdot))=\inf_{P\in\mathcal{P}^{T}}E_{P}U(\cdot,V^{x,\phi}_{T}(\cdot))=:u^{\mathcal{P}}(x,\phi).

(16)

As $\mathcal{P}^{T}\subset\mathcal{Q}^{T},$ $u^{\mathcal{P}}(x,\phi)\geq u(x,\phi)$ . Let $Q\in\mathcal{Q}^{T}$ with the fixed disintegration $Q:=P_{1}\otimes q_{2}\otimes\ldots\otimes q_{T}.$ Let

P^{n}:=\left(\frac{1}{n}P_{1}^{*}+\left(1-\frac{1}{n}\right)P_{1}\right)\otimes\left(\frac{1}{n}p_{2}^{*}+\left(1-\frac{1}{n}\right)q_{2}\right)\otimes\ldots\otimes\left(\frac{1}{n}p_{T}^{*}+\left(1-\frac{1}{n}\right)q_{T}\right).

Then (10) implies that $P^{n}\in\mathcal{P}^{T},$

\displaystyle u^{\mathcal{P}}(x,\phi)\leq E_{P^{n}}U(\cdot,V^{x,\phi}_{T}(\cdot))

(17)

and the only term in $E_{P^{n}}U(\cdot,V_{T}^{x,\phi}(\cdot))$ that is not multiplied by $1/n$ is $(1-1/n)^{T}E_{Q}U(\cdot,V_{T}^{x,\phi}(\cdot)).$ Moreover, (10) implies that all the others probability measures appearing in $E_{P^{n}}U(\cdot,V_{T}^{x,\phi}(\cdot))$ belongs to $\mathcal{P}^{T}.$ Fix $R\in\mathcal{P}^{T}$ as one of this measures and note that $\phi\in\phi(x,U,R)$ . Theorem 3.8 implies that the $sNA(\mathcal{P}^{T})$ and also the $NA(R)$ conditions hold true. We first prove that $E_{R}U^{+}(\cdot,V_{T}^{x,\phi}(\cdot))<\infty.$ If $U$ is bounded from above this is immediate. Assume that Assumption 3.13 holds true. Then [Blanchard et al., 2018, Theorem 4.17] shows that for ${R}$ -almost all $\omega^{T}\in\Omega^{T}$ ,

\displaystyle|V_{T}^{x,\phi}(\omega^{T})|\leq\prod_{s=1}^{T}\left(x+\frac{|\Delta S_{s}(\omega^{s})|}{\alpha^{R}_{s-1}(\omega^{s-1})}\right)=:\frac{\lambda}{2}\in\mathcal{W}_{T}

(18)

as $\Delta S_{s},\;\frac{1}{\alpha_{s}^{R}}\in\mathcal{W}_{s}$ for all $s\geq 1$ . Suppose that $x\geq 1$ else by monotonicity of $U^{+}$ , one may replace $x$ by 1. Then [Blanchard and Carassus, 2018, Proposition 3.24] (as $\lambda\geq 1$ ) implies that

\displaystyle E_{R}U^{+}(\cdot,V_{T}^{x,\phi}(\cdot))

\displaystyle\leq 4E_{R}\left(\prod_{s=1}^{T}\left(x+\frac{|\Delta S_{s}(\cdot)|}{\alpha^{R}_{s-1}(\cdot)}\right)\left(U^{+}(\cdot,1)+U^{-}(\cdot,\frac{1}{4})\right)\right)<\infty,

as $U^{+}(\cdot,1)$ , $U^{-}(\cdot,\frac{1}{4})$ $\in$ $\mathcal{W}_{T}.$
Now if $E_{R}U^{-}(\cdot,V_{T}^{x,\phi}(\cdot))=-\infty$ , as $R\in\mathcal{P}^{T},$ we get that $u(x,\phi)\leq u^{\mathcal{P}}(x,\phi)=-\infty.$ Thus $u^{\mathcal{P}}(x,\phi)=u(x,\phi)$ . Else letting $n$ go to infinity in (17) we obtain that $u^{\mathcal{P}}(x,\phi)\leq E_{Q}U(\cdot,V^{x,\phi}_{T}(\cdot))$ and taking the infimum over all $Q\in\mathcal{Q}^{T}$ , $u^{\mathcal{P}}(x,\phi)\leq u(x,\phi)$ : (16) is proved.
Finally taking in (16) the supremum over all $\phi\in\Phi(x,U,\mathcal{P}^{T})$ , we get that $u(x)=u^{\mathcal{P}}(x).$

To state the corollary on the existence of an optimal solution for (12), we need two additional assumptions.

Assumption 3.15.

There exists some $0\leq s<\infty$ such that $-s\leq S^{i}_{t}(\omega^{t})<+\infty$ for all $1\leq i\leq d$ , $\omega^{t}\in\Omega^{t}$ and $0\leq t\leq T$ .

Assumption 3.16.

For all $r\in\mathbb{Q}$ , $r>0,$ $\sup_{P\in\mathcal{Q}^{T}}E_{P}U^{-}(\cdot,r)<+\infty.$

Corollary 3.17.

Assume that the $NA(\mathcal{Q}^{T})$ condition and Assumptions 2.1, 2.2, 3.15 and 3.16 hold true. Furthermore, assume that $U$ is either bounded from above or that Assumption 3.13 holds true. Let $x\geq 0$ . Then, there exists some optimal strategy $\phi^{*}\in\Phi(x,U,\mathcal{Q}^{T})$ such that

u(x)=\inf_{P\in\mathcal{Q}^{T}}E_{P}U(\cdot,V^{x,\phi^{*}}_{T}(\cdot))<\infty.

Proof.

Fix some $x\geq 0.$ Theorem 3.8 implies that $sNA(\mathcal{P}^{T})$ holds true. So [Blanchard and Carassus, 2018, Theorem 3.6] gives the existence of an optimal strategy for $u^{\mathcal{P}}(x)$ . Lemma 3.14 allows to conclude since $u(x)=u^{\mathcal{P}}(x)$ .

3.2 Local no-arbitrage conditions and further results

We now turn to local conditions which are at the heart of the proofs due to the structure of the model. We recall the first part of [Bouchard and Nutz, 2015, Theorem 4.5] which establishes the essential link between the global version $NA(\mathcal{Q}^{T})$ and its local version.

Theorem 3.18.

Assume that Assumptions 2.1 and 2.2 hold true. Then the following statements are equivalent.
1. The $NA(\mathcal{Q}^{T})$ condition hold true.
2. For all $0\leq t\leq T-1$ , there exists a $\mathcal{Q}^{t}$ -full measure set $\Omega^{t}_{NA}\in\mathcal{B}_{c}(\Omega^{t})$ such that for all $\omega^{t}\in\Omega^{t}_{NA}$ , $h\Delta S_{t+1}(\omega^{t},\cdot)\geq 0\;\mathcal{Q}_{t+1}(\omega^{t})\mbox{-q.s.}$ for some $h\in\mathbb{R}^{d}$ implies that $h\Delta S_{t+1}(\omega^{t},\cdot)=0\;\mathcal{Q}_{t+1}(\omega^{t})\mbox{-q.s.}$

We present two other local definitions of no-arbitrage and establish their equivalence with the $NA(\mathcal{Q}^{T})$ conditions in Theorem 3.24 which is an analogous of Theorem 3.18.
The first definition proposes a geometric view of the no-arbitrage. Theorem 3.24 extends the uni-prior result of [Jacod and Shiryaev, 1998, Theorem 3g)], see also [Kabanov and Safarian, 2010, Proposition 2.1.6]. Note that the geometric no-arbitrage has appeared in different multiple-priors contexts, see [Oblój and Wiesel, 2018, Proposition 6.4] and [Burzoni et al., 2016b, Corollary 21]. A similar idea was already exploited in [Bouchard and Nutz, 2015, Lemma 3.3]. Theorem 3.24 will also allow us to prove Proposition 3.28 and Theorem 3.30.
Recall that for a convex set $C\subset\mathbb{R}^{d}$ , the relative interior of $C$ (see [Rockafellar, 1970, Section 6]) is $\mbox{Ri}(C)=\{y\in C,\,\exists\,\varepsilon>0,\;\mbox{Aff}(C)\cap B(y,\varepsilon)\subset C\}$ where $B(y,\varepsilon)$ is the open ball in $\mathbb{R}^{d}$ centered in $y$ with radius $\varepsilon$ . Moreover for a convex-valued random set $R,$ $\mbox{Ri}\left(R\right)$ is the random set defined by $\mbox{Ri}\left(R\right)(\omega):=\mbox{Ri}\left(R(\omega)\right)$ for $\omega\in\Omega$ .

Definition 3.19.

The geometric no-arbitrage condition holds true if for all $0\leq t\leq T-1$ , there exists some $\mathcal{Q}^{t}$ -full measure set $\Omega^{t}_{gNA}\in\mathcal{B}_{c}(\Omega^{t})$ such that for all $\omega^{t}\in\Omega^{t}_{gNA}$ , $0\in\mbox{Ri}\left({\mbox{Conv}}(D^{t+1})\right)(\omega^{t})$ . In this case for all $\omega^{t}\in\Omega^{t}_{gNA}$ , there exists $\varepsilon_{t}(\omega^{t})>0$ such that

\displaystyle B(0,\varepsilon_{t}(\omega^{t}))\cap\mbox{Aff}\left(D^{t+1}\right)(\omega^{t})\subset{{\mbox{Conv}}}\left(D^{t+1}\right)(\omega^{t}).

(19)

The geometric (local) no-arbitrage condition is indeed practical: Together with Theorem 3.24 it allows to check whether the (global) $\mbox{NA}(\mathcal{Q}^{T})$ condition holds true or not. As $\mathcal{Q}^{T}$ and for all $1\leq t\leq T,$ $\Delta S_{t+1}$ are given one gets $\mbox{Ri}\left({\mbox{Conv}}(D^{t+1})\right)(\cdot)$ and it is easy to check whether $0$ is in it or not (see Section 4 for examples of such a reasoning).

Secondly, in the spirit of [Rásonyi and Stettner, 2005, Proposition 3.3] (see also [Blanchard and Carassus, 2018, Proposition 2.3]), we introduce the so-called quantitative no-arbitrage condition.

Definition 3.20.

The quantitative no-arbitrage condition holds true if for all $0\leq t\leq T-1$ , there exists some $\mathcal{Q}^{t}$ -full measure set $\Omega^{t}_{qNA}\in\mathcal{B}_{c}(\Omega^{t})$ such that for all $\omega^{t}\in\Omega^{t}_{qNA}$ , there exists $\beta_{t}(\omega^{t}),\kappa_{t}(\omega^{t})\in(0,1)$ such that for all $h\in\mbox{Aff}\left({D}^{t+1}\right)(\omega^{t})$ , $h\neq 0$ there exists $p_{h}\in\mathcal{Q}_{t+1}(\omega^{t})$ satisfying

\displaystyle p_{h}\left({h}\Delta S_{t+1}(\omega^{t},\cdot)<-\beta_{t}(\omega^{t}){|h|}\right)\geq\kappa_{t}(\omega^{t}).

(20)

In the case where there is only one risky asset and one period, (20) is interpreted as follows : There exists a prior $p^{+}$ for which the price of the risky asset increases enough and an other one $p^{-}$ for which it decreases i.e. $p^{\mp}\left(\pm\Delta S(\cdot)<-\beta\right)\geq\kappa$ where $\beta,\kappa\in(0,1)$ . The number $\kappa$ serves as a measure of the gain/loss probability and the number $\beta$ of their size.

Remark 3.21.

Definition 3.20 is the direct adaptation to the multiple-priors set-up of [Rásonyi and Stettner, 2005, Proposition 3.3]: The probability measure depends of the strategy. For an agent buying or selling some quantity of risky assets, there is always a prior in which she is exposed to a potential loss. Proposition 3.37 will show that one can in fact choose a comment prior for all strategies in Definition 3.20.

Remark 3.22.

Theorem 3.8 and Proposition 3.37 are precious for solving the problem of maximisation of expected utility. For example when the utility function $U$ is defined on $(0,\infty)$ they provide natural bounds for the one step strategies or for $U(V_{T}^{x,\Phi})$ , see (18) and [Blanchard and Carassus, 2018, Lemma 3.11 and (44)]. This is used to prove the existence of the optimal strategy but it could also be used to compute it numerically. We propose in Section 4 explicit values for $\beta_{t}$ and $\kappa_{t}$ .

Remark 3.23.

In (20), $\beta_{t}(\omega^{t})$ provides information on $D^{t+1}(\omega^{t})$ while $\kappa_{t}(\omega^{t})$ provides information on $\mathcal{Q}_{t+1}(\omega^{t})$ . Moreover, Definition 3.20 can equivalently be formulated as follow: For all $0\leq t\leq T-1$ , there exists some $\mathcal{Q}^{t}$ -full measure set $\Omega^{t}_{qNA}\in\mathcal{B}_{c}(\Omega^{t})$ such that for all $\omega^{t}\in\Omega^{t}_{qNA}$ , there exists $\alpha_{t}(\omega^{t})\in(0,1)$ such that for all $h\in\mbox{Aff}\left({D}^{t+1}\right)(\omega^{t})$ , $h\neq 0$ there exists $p_{h}\in\mathcal{Q}_{t+1}(\omega^{t})$ satisfying

\displaystyle p_{h}\left({h}\Delta S_{t+1}(\omega^{t},\cdot)<-\alpha_{t}(\omega^{t}){|h|}\right)\geq\alpha_{t}(\omega^{t}).

(21)

Indeed, (21) implies (20) and assuming (20), (21) is true with $\alpha_{t}(\omega^{t})=\min(\kappa_{t}(\omega^{t}),\beta_{t}(\omega^{t}))\in(0,1)$ .

Theorem 3.24.

Assume that Assumptions 2.1 and 2.2 hold true. Then the $NA(\mathcal{Q}^{T})$ condition (see Definition 3.1), the geometric no-arbitrage (see Definition 3.19) and the quantitative no-arbitrage (see Definition 3.20) are equivalent and one can choose $\Omega^{t}_{NA}=\Omega^{t}_{qNA}=\Omega^{t}_{gNA}$ for all $0\leq t\leq T-1$ . Furthermore, one can choose $\beta_{t}={\varepsilon_{t}}/{2}$ in (20) (for $\varepsilon_{t}$ introduced in (19)).

Proof.

See Section 5.2.2.

Remark 3.25.

Under Assumptions 2.1 and 2.2 and any of the no-arbitrage condition, $0\in\mbox{Conv}\left(D^{t+1}\right)(\omega^{t})$ and $\mbox{Aff}\left(D^{t+1}\right)(\omega^{t})$ is a vector space for all $\omega^{t}\in\Omega^{t}_{NA}$ .

The next proposition is [Jacod and Shiryaev, 1998, Theorem 3] but could also be obtained as a direct application of Theorem 3.24 together with [Bertsekas and Shreve, 2004, Lemma 7.28 p174] and [Aliprantis and Border, 2006, Theorem 12.28] in the specific setting where $\mathcal{Q}^{T}=\{P_{1}\otimes p_{2}\otimes\dots\otimes p_{T}\}$ . Indeed, Theorem 3.24 does not apply directly as $\mbox{graph}(p_{t})$ belongs a priory to $\mathcal{B}_{c}\left(\Omega^{t}\times\mathfrak{P}(\Omega_{t+1})\right)$ and not to $\mathcal{A}\left(\Omega^{t}\times\mathfrak{P}(\Omega_{t+1})\right)$ , and one needs to build some Borel-measurable version of $p_{t}$ . Proposition 3.26 will be used in the sequel to prove that the $NA(P)$ condition holds true.

Proposition 3.26.

Assume that Assumption 2.1 holds true and let $P\in\mathfrak{P}(\Omega^{T})$ with the fixed disintegration $P:=P_{1}\otimes p_{2}\otimes\dots\otimes p_{T}$ where $p_{t}\in\mathcal{S}K_{t}$ for all $1\leq t\leq T$ . Then the $NA(P)$ condition holds true if and only if $0\in\mbox{Ri}\left(\mbox{Conv}\left(D_{P}^{t+1}\right)\right)(\cdot)$ $P^{t}$ -a.s. for all $0\leq t\leq T-1$ .

Remark 3.27.

Similarly, under the assumption of Proposition 3.26, one can show that the $NA(P)$ condition holds true if and only if the quantitative no-arbitrage holds true for $\mathcal{Q}^{T}=\{P\}$ which is exactly [Rásonyi and Stettner, 2005, Proposition 3.3]. In this case, we denote $\alpha_{t}$ in (21) by $\alpha^{P}_{t}$ .

We now establish some tricky measurability properties.

Proposition 3.28.

Assume that Assumptions 2.1 and 2.2 hold true. Under one of the no-arbitrage conditions (see Definitions 3.1, 3.19 and 3.20) one can choose a $\mathcal{B}_{c}(\Omega^{t})$ -measurable version of $\varepsilon_{t}$ (in (19)) and $\beta_{t}$ (in (20)).

Proof.

See Section 5.2.2.

Remark 3.29.

The measurability of $\kappa_{t}$ cannot be directly inferred from the one of $\varepsilon_{t}$ but will be obtained in Proposition 3.37 as a consequence of Theorem 3.8. The measurability of $\kappa_{t}$ is useful to solve the problem of multi-priors optimal investment for unbounded utility function defined on the whole real-line since the bounds on the optimal strategies depends on $\kappa_{t}$ see for instance [Rásonyi and Stettner, 2005, (17)] in a non-robust setting and [Rásonyi and Meireles-Rodrigues, 2018, Proof of Lemma 3.3] in the robust context.

The next theorem is crucial. It is a first step towards Theorem 3.8: It gives the existence of the measure $P^{*}$ which allows to build recursively the set $\mathcal{P}^{T}$ (see (10)). But it is also of own interest since it gives the equivalence between $NA(\mathcal{Q}^{T})$ and a stronger form of $wNA(\mathcal{Q}^{T})$ .

Theorem 3.30.

Assume that Assumptions 2.1 and 2.2 hold true. The $NA(\mathcal{Q}^{T})$ condition holds true if and only if there exists some $P^{*}\in\mathcal{Q}^{T}$ such that $\mbox{Aff}\left(D_{P^{*}}^{t+1}\right)(\omega^{t})=\mbox{Aff}\left(D^{t+1}\right)(\omega^{t})$ and $0\in\mbox{Ri}\left({\mbox{Conv}}(D_{P^{*}}^{t+1})\right)(\omega^{t})$ for all $0\leq t\leq T-1$ , $\omega^{t}\in\Omega_{NA}^{t}$ ⁵⁵5The set $\Omega^{t}_{NA}$ was introduced in Theorem 3.18, see also (45)..

Proof.

See Section 5.2.3.

Remark 3.31.

Theorem 3.30 was proved in a one period setting in [Bayraktar and Zhou, 2017, Lemma 2.2].

Remark 3.32.

The probability measure $P^{*}$ of Theorem 3.30 is not unique. In fact, under $NA(\mathcal{Q}^{T})$ , all $P\in\mathcal{P}^{T}$ satisfy $\mbox{Aff}\left(D_{P}^{t+1}\right)(\omega^{t})=\mbox{Aff}\left(D^{t+1}\right)(\omega^{t})$ and $0\in\mbox{Ri}\left({\mbox{Conv}}(D_{P}^{t+1})\right)(\omega^{t})$ for all $0\leq t\leq T-1$ , $\omega^{t}\in\Omega_{NA}^{t},$ see proof of Theorem 3.8 step 2 $iii).$

Remark 3.33.

The main (and difficult) point in Theorem 3.30 is that $P^{*}\in\mathcal{Q}^{T}$ . Thus any $\mathcal{Q}^{t}$ -null set is also a $P^{*}$ -null set and in particular $\Omega^{t}_{NA}$ is of $P^{*}$ -full measure (see Theorem 3.18). So $0\in\mbox{Ri}\left({\mbox{Conv}}(D_{P^{*}}^{t+1})\right)(\omega^{t})$ for $\omega^{t}\in\Omega^{t}_{NA}$ and the $NA(P^{*})$ condition holds true (see Proposition 3.26). We have actually more since $\Omega^{t}_{NA}$ is of $\mathcal{Q}^{t}$ -full measure. We will provide in Section 4 explicit form of $P^{*}$ .

Remark 3.34.

Theorem 3.30 is related and complements [Oblój and Wiesel, 2018, Theorem 3.1]. Indeed, in both cases the main issue is to find some $p_{t+1}^{*}(\cdot,\omega^{t})\in\mathcal{Q}_{t+1}(\omega^{t})$ such that $0\in\mbox{Ri}\left({\mbox{Conv}}(D_{P^{*}}^{t+1})\right)(\omega^{t})\subset\mbox{Ri}\left({\mbox{Conv}}(D^{t+1})\right)(\omega^{t})$ (recall (7)). This is used in [Oblój and Wiesel, 2018] to make the link with the quasi-sure setting and in our case to establish Theorem 3.8.

Remark 3.35.

[Rásonyi and Meireles-Rodrigues, 2018, Assumption 2.1] asserts that there exists at least one arbitrage free model (in the uni-prior sense) and that for this model the affine space generated by the conditional support always equals $\mathbb{R}^{d}$ . Those are the conditions verified by $P^{*}$ in Theorem 3.30 and thus [Rásonyi and Meireles-Rodrigues, 2018, Theorem 3.7] which shows the existence in the problem of maximisation of expected utility for bounded function defined on the whole real line works under $NA(\mathcal{Q}^{T}).$

Example 3.36.

The probability measure $P^{*}\in\mathcal{Q}^{T}$ of Theorem 3.30 provides a kind of stronger $NA(P^{*})$ . The counter example of the last item in Lemma 3.7 illustrates why the condition $\mbox{Aff}\left(D_{P^{*}}^{t+1}\right)(\cdot)=\mbox{Aff}\left(D^{t+1}\right)(\cdot)$ $\mathcal{Q}^{t}$ -q.s. is needed in Theorem 3.30. However this is not enough to obtain equivalence with the $NA(\mathcal{Q}^{T})$ condition and the following counterexample illustrates why $0\in\mbox{Ri}\left({\mbox{Conv}}(D_{P^{*}}^{t+1})\right)(\cdot)$ $\mathcal{Q}^{t}$ -q.s. is needed and why $0\in\mbox{Ri}\left({\mbox{Conv}}(D_{P^{*}}^{t+1})\right)(\cdot)$ ${P}_{t}^{*}$ -p.s. is not enough.
Let $T=2$ , $d=1$ , $\Omega_{1}:=\Omega_{2}:=\{-1,0,1\}$ , $S_{0}:=2$ , $S_{1}(\omega_{1}):=2+\omega_{1}$ , $S_{2}(\omega_{1},\omega_{2}):=2+\omega_{1}+\omega_{2}$ . Let $P_{na}:=\frac{1}{2}(\delta_{-1}+\delta_{1})$ , $P_{0}:=\delta_{0}$ and $P_{1}:=\delta_{1}$ be three probability measures on $\mathfrak{P}(\Omega_{1})$ . Set $\mathcal{Q}_{1}:=\mbox{Conv}\left(P_{0},P_{na}\right)$ and define $\mathcal{Q}_{2}(\cdot)$ as follow: $\mathcal{Q}_{2}(\pm 1)=\{P_{na}\}$ and $\mathcal{Q}_{2}(0)=\{P_{1}\}$ . This is illustrated in Figure 2.

Figure 2: Left-hand side: The model. Right-hand side: In green

P^{*}

and in orange

\overline{Q}

It is clear that Assumptions 2.1 and 2.2 hold true.
Let $p_{2}(\cdot)\in\mathcal{Q}_{2}(\cdot)$ and set $P^{*}:=P_{na}\otimes p_{2}\in\mathcal{Q}^{2}$ (see Figure 2). It is immediate that the $NA(P^{*})$ and thus the $wNA(\mathcal{Q}^{2})$ conditions hold true. Furthermore $D_{P^{*}}^{2}(\pm 1)=D^{2}(\pm 1)=\{-1,1\}$ and $D_{P^{*}}^{2}(0)=D^{2}(0)=\{1\}$ . Thus for all $\omega_{1}$ , $\mbox{Aff}\left(D_{P^{*}}^{2}\right)(\omega_{1})=\mbox{Aff}\left(D^{2}\right)(\omega_{1})=\mathbb{R}$ . As $0\in\mbox{Ri}\left(\mbox{Conv}\left(D_{P^{*}}^{2}\right)\right)(\pm 1)$ and $P_{1}^{*}\left(\{\pm 1\}\right)=1$ , $0\in\mbox{Ri}\left(\mbox{Conv}\left(D_{P^{*}}^{2}\right)\right)(\cdot)$ $P_{1}^{*}$ -a.s. Now let $\bar{Q}:=P_{0}\otimes p_{2}\in\mathcal{Q}^{2}$ (see Figure 2). Then $\bar{Q}^{1}(\{0\})=1$ and $0\notin\mbox{Ri}\left(\mbox{Conv}\left(D_{P^{*}}^{2}\right)\right)(0)$ implies that $0\in\mbox{Ri}\left(\mbox{Conv}\left(D_{P^{*}}^{2}\right)\right)(\cdot)$ $\bar{Q}^{1}$ -p.s. and thus $0\in\mbox{Ri}\left({\mbox{Conv}}(D_{P^{*}}^{2})\right)(\cdot)$ $\mathcal{Q}^{1}$ -q.s. are not verified.
Let us check that the $NA(\mathcal{Q}^{2})$ condition does not hold true. Choose $\phi\in\Phi$ such that $\phi_{1}=0$ and $\phi_{2}(\omega_{1})=1_{0}(\omega_{1})$ and use again $\bar{Q}=P_{0}\otimes p_{2}\in\mathcal{Q}^{2}$ . Then $V_{2}^{0,\phi}\geq 0$ $\mathcal{Q}^{2}$ -q.s. and $\bar{Q}\left(\{V_{2}^{0,\phi}>0\}\right)=\delta_{1}(\{\omega_{2}>0\})=1$ .
Now replace $\mathcal{Q}_{2}$ by $\widetilde{\mathcal{Q}}_{2}(\cdot):=\mbox{Conv}\left(P_{na},P_{1}\right)$ while keeping $\widetilde{\mathcal{Q}}_{1}=\mathcal{Q}_{1}$ as before and set $\widetilde{P}^{*}:=P_{na}\otimes\widetilde{p}_{2}$ , where $\widetilde{p}_{2}(\cdot,\omega_{1}):=P_{na}(\cdot)$ for all $\omega_{1}$ . Then $D_{\widetilde{P}^{*}}^{2}(\omega_{1})=\{-1,1\}$ , $\mbox{Aff}(D_{\widetilde{P}^{*}}^{2})(\omega_{1})=\mbox{Aff}\left(D^{2}\right)(\omega_{1})=\mathbb{R}$ and $0\in\mbox{Ri}(\mbox{Conv}(D_{\widetilde{P}^{*}}^{2}))(\omega_{1})$ for all $\omega_{1}$ . One can directly check that the $NA(\widetilde{\mathcal{Q}}^{2})$ condition holds true.
Finally, one may build $\widetilde{\mathcal{P}}^{2}$ using (10) and $\widetilde{P}^{*}$ . It is clear that $\widetilde{\mathcal{P}}^{2}$ is strictly included in $\widetilde{\mathcal{Q}}^{2}$ since it does not contain $\{P_{0}\otimes q_{2},q_{2}(\cdot,\omega_{2})\in\widetilde{\mathcal{Q}}_{2}(\omega_{2})\}$ .

The following result provides an answer to the measurability issue raised in Remark 3.29 and also provides a commun prior for all strategies.

Proposition 3.37.

Assume that Assumptions 2.1 and 2.2 as well as the $NA(\mathcal{Q}^{T})$ condition hold true. Then for all $0\leq t\leq T-1$ there exists some $\mathcal{B}_{c}(\Omega^{t})$ -measurable random variables $\beta_{t}(\cdot),\kappa_{t}(\cdot)\in(0,1)$ such that for all $\omega^{t}\in\Omega^{t}_{NA}$ and $h\in\mbox{Aff}\left({D}^{t+1}\right)(\omega^{t})$ , $h\neq 0$

\displaystyle p^{*}_{t+1}\left({h}\Delta S_{t+1}(\omega^{t},\cdot)<-\beta_{t}(\omega^{t}){|h|},\omega^{t}\right)\geq\kappa_{t}(\omega^{t}),

(22)

where $p_{t+1}^{*}(\cdot,\omega^{t})$ is defined in Theorem 3.30 with the fix disintegration $P^{*}:=P_{1}^{*}\otimes p_{2}^{*}\otimes\cdots\otimes p_{T}^{*}$ .

Remark 3.38.

We have that $\beta_{t}(\omega^{t})=\kappa_{t}(\omega^{t})=1$ only if $D^{t+1}_{P^{*}}(\omega^{t})=\{0\}$ . Indeed if $\beta_{t}(\omega^{t})=\kappa_{t}(\omega^{t})=1$ and $D^{t+1}_{P^{*}}(\omega^{t})\neq\{0\}$ , then for all $h\in\mbox{Aff}\left({D}^{t+1}\right)(\omega^{t})$ with $|h|=1$ $p^{*}_{t+1}\left({h}\Delta S_{t+1}(\omega^{t},\cdot)<-1,\omega^{t}\right)=1.$ Fix such a $h$ and let $F_{h}:=\{y\in\mathbb{R}^{d},\;hy\leq-1\}.$ Then $p^{*}_{t+1}\left(\Delta S_{t+1}(\omega^{t},\cdot)\in F_{\pm h},\omega^{t}\right)=1$ and $D^{t+1}_{P^{*}}(\omega^{t})=E^{t+1}(\omega^{t},p_{t+1}^{*}(\cdot,\omega^{t}))\subset F_{-h}\cap F_{+h}=\emptyset,$ see Remark 2.5. Note that it is not easy to obtain this result for Theorem 3.24 as the prior in (20) depends on $h$ .

Proof.

See Section 5.2.5.

Finally, if there exists a dominating probability measure $\widehat{P}\in\mathcal{Q}^{T}$ , the following result holds true.

Proposition 3.39.

Assume that Assumptions 2.1 and 2.2 hold true. Assume furthermore that there exists some dominating measure $\widehat{P}\in\mathcal{Q}^{T}$ . Then the $NA(\widehat{P})$ and the $NA(\mathcal{Q}^{T})$ conditions are equivalent. In this case, for all $0\leq t\leq T-1,$

\displaystyle D_{\widehat{P}}^{t+1}(\cdot)=D^{t+1}(\cdot)\mbox{ and }0\in\mbox{Ri}\left({\mbox{Conv}}(D_{\widehat{P}}^{t+1})\right)(\cdot)\;\;\mathcal{Q}^{t}\mbox{-q.s.}

(23)

Remark 3.40.

One can choose $P^{*}=\widehat{P}$ in Proposition 3.37 changing $\Omega^{t}_{NA}$ by the full-measure set where (23) holds true. Moreover, $\mathcal{P}^{T}$ (see (10)) in Theorem 3.8 can be constructed from $\widehat{P}$ .

Proof.

See Section 5.2.6.

4 Examples

This section proposes concrete examples of multiple-priors setting illustrating our results. We also use these examples to present how to build sets of probability measures which are not dominated. This relies on the following result.

Proposition 4.1.

Assume that Assumption 2.2 holds true and that there exists some $\widetilde{P}\in\mathcal{Q}^{T}$ , some $0\leq t\leq T-1$ and some $\Omega^{t}_{N}\in\mathcal{B}_{c}(\Omega^{t})$ such that $\widetilde{P}^{t}(\Omega^{t}_{N})>0$ and such that the set $\mathcal{Q}_{t+1}(\omega^{t})$ is not dominated for all $\omega^{t}\in\Omega^{t}_{N}$ . Then $\mathcal{Q}^{T}$ is not dominated.

Proof.

See Section 5.3.

4.1 Robust Binomial model

Suppose that $T\geq 1$ , $d=1$ and $\Omega_{t}=\mathbb{R}$ (or $(0,\infty)$ ) for all $1\leq t\leq T$ . The risky asset $\left(S_{t}\right)_{0\leq t\leq T}$ is such that $S_{0}=1$ and $S_{t+1}=S_{t}Y_{t+1}$ where $Y_{t+1}$ is a real-valued and $\mathcal{B}(\Omega_{t+1})$ -measurable random variable such that $Y_{t+1}(\Omega_{t+1})=(0,\infty)$ for all $0\leq t\leq T-1$ (if $\Omega_{t}=(0,\infty)$ you can think of $Y_{t}=\omega_{t}$ ). The positivity of $Y_{t}$ implies that $S_{t}(\omega^{t})>0$ for all $\omega^{t}\in\Omega^{t}$ . It is clear that Assumption 2.1 is verified. Then, for $0\leq t\leq T-1$ let

\displaystyle\mathcal{B}_{t+1}(\omega^{t}):=\{\pi\delta_{u}+(1-\pi)\delta_{d},\;\pi_{t}(\omega^{t})\leq\pi\leq\Pi_{t}(\omega^{t}),

\displaystyle\;u_{t}(\omega^{t})\leq u\leq U_{t}(\omega^{t}),\;d_{t}(\omega^{t})\leq d\leq D_{t}(\omega^{t})\},

where $\pi_{t},\Pi_{t},u_{t},U_{t},d_{t},D_{t}$ are real-valued $\mathcal{B}(\Omega^{t})$ -measurable random variables such that $0\leq\pi_{t}(\omega^{t})\leq\Pi_{t}(\omega^{t})\leq 1$ , $u_{t}(\omega^{t})\leq U_{t}(\omega^{t})$ and $d_{t}(\omega^{t})\leq D_{t}(\omega^{t})$ for all $\omega^{t}\in\Omega^{t}$ .⁶⁶6This could be generalised by setting $\mathcal{B}_{t+1}(\omega^{t}):=\{\pi\delta_{u}+(1-\pi)\delta_{d},\;\pi\in\mathcal{S}_{t}(\omega^{t}),\;u\in\mathcal{U}_{t}(\omega^{t}),\;d\in\mathcal{D}_{t}(\omega^{t})\}$ , where $\mathcal{S}_{t}$ , $\mathcal{U}_{t}$ , $\mathcal{D}_{t}$ are Borel-measurable random sets $\Omega^{t}\twoheadrightarrow\mathbb{R}.$

Assumption 4.2.

We have that $\pi_{t}(\omega^{t})<1$ , $\Pi_{t}(\omega^{t})>0$ and $0<d_{t}(\omega^{t})<1<U_{t}(\omega^{t})$ for all $0\leq t\leq T-1$ and $\omega^{t}\in\Omega^{t}$ .

For all $0\leq t\leq T-1$ and $\omega^{t}\in\Omega^{t},$ let

\displaystyle\widetilde{\mathcal{Q}}_{t+1}(\omega^{t})

\displaystyle:=\left\{q\in\mathfrak{P}(\Omega_{t+1}),\;q\left(Y_{t+1}\in\cdot\right)\in\mathcal{B}_{t+1}(\omega^{t})\right\}\mbox{ and }\mathcal{Q}_{t+1}(\omega^{t}):=\mbox{Conv}\left(\widetilde{\mathcal{Q}}_{t+1}(\omega^{t})\right),

(24)

where $q\left(Y_{t+1}\in\cdot\right)$ is the law of $Y_{t+1}$ under $q$ . In words, at each step, the risky asset can go up or down and there is uncertainty not only on the probability of the jumps but also on their sizes.

Remark 4.3.

The usual binomial model (see [Cox et al., 1979]) corresponds to $\pi_{t}=\Pi_{t}=\pi$ , $u_{t}=U_{t}=u$ and $d_{t}=D_{T}=d$ where $0<\pi<1$ , $d<1<u$ .

Lemma 4.4.

Under Assumption 4.2, Assumption 2.2 holds true.

Proof.

First, $\mathcal{Q}_{t+1}$ is convex valued by definition. Since $Y_{t+1}(\Omega_{t+1})=(0,\infty)$ , $\widetilde{\mathcal{Q}}_{t+1}(\omega^{t})\neq\emptyset$ , hence ${\mathcal{Q}}_{t+1}(\omega^{t})\neq\emptyset$ for all $\omega^{t}\in\Omega^{t}$ . We show successively that $\mbox{graph}\left(\mathcal{B}_{t+1}\right)$ , $\mbox{graph}\left(\widetilde{\mathcal{Q}}_{t+1}\right)$ and $\mbox{graph}\left({\mathcal{Q}}_{t+1}\right)$ are analytic sets. For $\omega^{t}\in\Omega^{t},$ let

	$\displaystyle E(\omega^{t})$	$\displaystyle:=\{(u,d,\pi)\in\mathbb{R}^{3},\;\pi_{t}(\omega^{t})\leq\pi\leq\Pi_{t}(\omega^{t}),\;u_{t}(\omega^{t})\leq u\leq U_{t}(\omega^{t}),\;d_{t}(\omega^{t})\leq d\leq D_{t}(\omega^{t})\},$
	$\displaystyle F(\omega^{t},$	$\displaystyle u,d,\pi):=(\omega^{t},\pi\delta_{u}+(1-\pi)\delta_{d})\quad\mbox{for $(\omega^{t},u,d,\pi)\in\Omega^{t}\times\mathbb{R}^{3}$}.$

Then $F$ is Borel-measurable (see [Bertsekas and Shreve, 2004, Corollary 7.21.1 p130]), $\mbox{graph}(E)\in{\cal B}_{c}(\Omega^{t})\otimes{\cal B}(\mathbb{R}^{3})$ as $\pi_{t},\,\Pi_{t},\,u_{t},\,U_{t},\,d_{t}$ and $D_{t}$ are Borel-measurable. We conclude that $\mbox{graph}\left(\mathcal{B}_{t+1}\right)$ $=F\left(\mbox{graph}(E)\right)$ is analytic. Let $\Phi:\mathfrak{P}(\Omega_{t+1})\to\mathfrak{P}(\mathbb{R})$ be defined by $\Phi(q):=q\left(Y_{t+1}\in\cdot\right)$ . Using [Bertsekas and Shreve, 2004, Propositions 7.29 p144 and 7.26 p134], $\Phi$ is a Borel-measurable stochastic kernel on $\mathbb{R}$ given $\mathfrak{P}(\Omega_{t+1})$ . So $\hat{\Phi}(\omega^{t},q):=(\omega^{t},\Phi(q))$ is also Borel-measurable and $\mbox{graph}(\widetilde{\mathcal{Q}}_{t+1})=\hat{\Phi}^{-1}\left(\mbox{graph}\left(\mathcal{B}_{t+1}\right)\right)$ is analytic. Then one can show as in [Bartl, 2019b, Proofs for Section 2.3] that $\mbox{graph}\left({\mathcal{Q}}_{t+1}\right)$ is analytic since ${\mathcal{Q}}_{t+1}$ is the convex hull of $\widetilde{\mathcal{Q}}_{t+1}$ .

Lemma 4.5.

Under Assumption 4.2, the $NA(\mathcal{Q}^{T})$ condition holds true and the $sNA(\mathcal{Q}^{T})$ condition might fails.

Proof.

It is clear that for all $0\leq t\leq T-1$ , all $\omega^{t}\in\Omega^{t}$ ,

\mbox{Conv}\left(D^{t+1}\right)(\omega^{t})=[S_{t}(\omega^{t})(d_{t}(\omega^{t})-1),S_{t}(\omega^{t})(U_{t}(\omega^{t})-1)].

So the $NA(\mathcal{Q}^{T})$ condition holds true as $0\in\mbox{Ri}\left(\mbox{Conv}\left(D^{t+1}\right)\right)(\omega^{t})$ for all $\omega^{t}\in\Omega^{t}$ (see Theorem 3.24). Under Assumption 4.2, one may have that $u_{t}(\omega^{t})<1$ for all $\omega^{t}\in\Omega^{t}$ , $0\leq t\leq T-1$ and find some $a_{t}(\omega^{t})\in[u_{t}(\omega^{t}),1)$ . For all $0\leq t\leq T-1$ and $\omega^{t}\in\Omega^{t}$ , let

q_{t+1}(Y_{t+1}\in\cdot,\omega^{t}):=r_{t}(\omega^{t})\delta_{a_{t}(\omega^{t})}(\cdot)+\left(1-r_{t}(\omega^{t})\right)\delta_{d_{t}(\omega^{t})}(\cdot),

where $r_{t}(\omega^{t})\in[\pi_{t}(\omega^{t}),\Pi_{t}(\omega^{t})]$ . Set $Q:=Q_{1}\otimes q_{2}\otimes\cdots\otimes q_{T}\in\mathcal{Q}^{T}$ . As

\mbox{Conv}\left(D_{Q}^{t+1}\right)(\omega^{t})=\left[S_{t}(\omega^{t})(d_{t}(\omega^{t})-1),S_{t}(\omega^{t})(a_{t}(\omega^{t})-1)\right],

$0\notin\mbox{Conv}\left(D_{Q}^{t+1}\right)(\omega^{t})$ for all $\omega^{t}\in\Omega^{t}$ and Proposition 3.26 implies that $NA({Q})$ and thus $sNA(\mathcal{Q}^{T})$ fail.

We now provide some explicit expressions for $\varepsilon_{t}$ , $\beta_{t}$ and $\kappa_{t}$ of (19) and (20) and exhibit a candidate for the measure $P^{*}$ of Theorem 3.30.

Lemma 4.6.

Assume that Assumption 4.2 holds true. For all $0\leq t\leq T-1$ , all $\omega^{t}\in\Omega^{t}$ let

	$\displaystyle\bar{\pi}_{t}(\omega^{t})$	$\displaystyle:=\frac{\pi_{t}(\omega^{t})+\Pi_{t}(\omega^{t})}{2}\in(0,1)$
	$\displaystyle\frac{\varepsilon_{t}(\omega^{t})}{2}$	$\displaystyle=\beta_{t}(\omega^{t}):=\frac{S_{t}(\omega{{}^{t}})}{N}\min\left(\frac{U_{t}(\omega^{t})-1}{2},\frac{1-d_{t}(\omega^{t})}{2}\right)>0,$
	$\displaystyle\kappa_{t}(\omega^{t})$	$\displaystyle:=\frac{1}{M}\min\left(\bar{\pi}_{t}(\omega^{t}),1-\bar{\pi}_{t}(\omega^{t})\right)>0,$
	$\displaystyle a_{t}^{+}(\omega^{t})$	$\displaystyle:=U_{t}(\omega^{t})>1,\;\;\;\;b_{t}^{+}(\omega^{t}):=\min\left(D_{t}(\omega^{t}),\frac{d_{t}(\omega^{t})+1}{2}\right)<1,$
	$\displaystyle a_{t}^{-}(\omega^{t})$	$\displaystyle:=\max\left(u_{t}(\omega^{t}),\frac{U_{t}(\omega^{t})+1}{2}\right)>1,\;\;\;\;b_{t}^{-}(\omega^{t}):=d_{t}(\omega^{t})<1,$
	$\displaystyle r_{t+1}^{\pm}(\cdot,\omega^{t})$	$\displaystyle:=\bar{\pi}_{t}(\omega^{t})\delta_{a_{t}^{\pm}(\omega^{t})}(\cdot)+(1-\bar{\pi}_{t}(\omega^{t}))\delta_{b_{t}^{\pm}(\omega^{t})}(\cdot)\in\mathcal{B}_{t+1}(\omega^{t}),$
	$\displaystyle r_{t+1}^{*}(\cdot,\omega^{t})$	$\displaystyle:=\frac{1}{2}\left(r_{t+1}^{+}(\cdot,\omega^{t})+r_{t+1}^{-}(\cdot,\omega^{t})\right)\in\mathcal{B}_{t+1}(\omega^{t}),\;\;p_{t+1}^{}(Y_{t+1}\in\cdot,\omega^{t}):=r_{t+1}^{}(\cdot,\omega^{t})\in\mathcal{Q}_{t+1}(\omega^{t})$

where $N>1$ and $M>1$ are fixed and allows to get sharper bound for $\varepsilon_{t}(\omega^{t}),\;\beta_{t}(\omega^{t})$ and $\kappa_{t}(\omega^{t})$ . Then

\displaystyle p_{t+1}^{*}\left(\pm\Delta S_{t+1}(\omega^{t},\cdot)<-\beta_{t}(\omega^{t}),\omega^{t}\right)\geq\kappa_{t}(\omega^{t}),

(25)

and (20) is satisfied; (19) also holds true.
Moreover, for $P^{*}:=P_{0}^{*}\otimes p^{*}_{1}\cdots\otimes p^{*}_{T}\in\mathcal{Q}^{T},$ $0\in\mbox{Ri}\left({\mbox{Conv}}(D_{P^{*}}^{t+1})\right)(\omega^{t})$ and $\mbox{Aff}\left(D_{P^{*}}^{t+1}\right)(\omega^{t})=\mbox{Aff}\left(D^{t+1}\right)(\omega^{t})=\mathbb{R}$ for all $\omega^{t}\in\Omega^{t}$ .
Finally, assume that for some $0\leq t\leq T-1$ and some $\omega^{t}\in\Omega^{t}$ , $u_{t}(\omega^{t})<U_{t}(\omega^{t})$ or $d_{t}(\omega^{t})<D_{t}(\omega^{t}).$ Then the set $\mathcal{Q}_{t+1}(\omega^{t})$ is not dominated and one can construct sets $\mathcal{Q}^{T}$ which are not dominated.

Remark 4.7.

Note that $P^{*}$ is not unique. The (Borel) measurability of $\varepsilon_{t},\beta_{t}$ and $\kappa_{t}$ are clear. Similarly they will inherit any integrability conditions imposed on $S_{t}$ , $\pi_{t}$ , $\Pi_{t}$ , $d_{t}$ , $D_{t},$ $u_{t}$ and $U_{t}$ . For instance if they belong to $\mathcal{W}_{t}$ for all $1\leq t\leq T$ so do $\varepsilon_{t},\beta_{t}$ and $\kappa_{t}$ .

Proof.

Fix some $0\leq t\leq T-1$ , $\omega^{t}\in\Omega^{t}$ . Let $q_{t+1}^{\pm}(Y_{t+1}\in\cdot,\omega^{t}):=r_{t+1}^{\pm}(\cdot,\omega^{t})\in\mathcal{Q}_{t+1}(\omega^{t})$ . Then

	$\displaystyle q_{t+1}^{+}\left(\Delta S_{t+1}(\omega^{t},\cdot)<-\beta_{t}(\omega^{t}),\omega^{t}\right)$	$\displaystyle\geq q_{t+1}^{+}\left(Y_{t+1}(\cdot)<\frac{d_{t}(\omega^{t})+1}{2},\omega^{t}\right)\geq 1-\bar{\pi}_{t}(\omega^{t})\geq\kappa_{t}(\omega^{t})$		(26)
	$\displaystyle q_{t+1}^{-}\left(\Delta S_{t+1}(\omega^{t},\cdot)>\beta_{t}(\omega^{t}),\omega^{t}\right)$	$\displaystyle\geq q_{t+1}^{-}\left(Y_{t+1}(\cdot)>\frac{U_{t}(\omega^{t})+1}{2},\omega^{t}\right)\geq\bar{\pi}_{t}(\omega^{t})\geq\kappa_{t}(\omega^{t})$		(27)

and (25) follows while (19) follows from Theorem 3.24.
As $p_{t+1}^{*}\in\mathcal{SK}_{t+1}$ , $P^{*}\in\mathcal{Q}^{T}$ . From (25), the quantitative no-arbitrage (20) holds true for all $\omega^{t}\in\Omega^{t}$ with $p_{h}=p_{t+1}^{*}(\cdot,\omega^{t})$ for all possible strategy $h$ . Therefore the $NA(P^{*})$ condition holds true (see Remark 3.27). Theorem 3.24 implies also that $0\in\mbox{Ri}\left({\mbox{Conv}}(D_{P^{*}}^{t+1})\right)(\omega^{t})$ . Moreover $\mbox{Aff}\left(D_{P^{*}}^{t+1}\right)(\omega^{t})=\mbox{Aff}\left(D^{t+1}\right)(\omega^{t})=\mathbb{R}$ for all $\omega^{t}$ .
For the last item, assume that for some $0\leq t\leq T-1$ and some $\omega^{t}\in\Omega^{t}$ , $u_{t}(\omega^{t})<U_{t}(\omega^{t})$ and that the set $\mathcal{Q}_{t+1}(\omega^{t})$ is dominated by some measure $\widehat{p}$ . For $x\in(0,\infty)$ let $A_{x}:=\{Y_{t+1}^{-1}(\{x\})\}\neq\emptyset$ as $Y_{t+1}(\Omega^{t})=(0,\infty)$ . Fix $x(\omega^{t})\in(\min(1,u_{t}(\omega^{t})),U_{t}(\omega^{t}))$ and choose $a(\omega^{t})\in A_{x(\omega^{t})}$ and $b(\omega^{t})\in A_{d_{t}(\omega^{t})}$ . Let $r_{x}(.,\omega^{t}):=\Pi_{t}(\omega^{t})\delta_{a(\omega^{t})}+(1-\Pi_{t}(\omega^{t}))\delta_{b(\omega^{t})}\in\mathcal{B}_{t+1}(\omega^{t})$ and $p_{x}(Y_{t+1}\in\cdot,\omega^{t}):=r_{x}(.,\omega^{t})\in\mathcal{Q}_{t+1}(\omega^{t}).$ As $r_{x}(\{a(\omega^{t})\},\omega^{t})=\Pi_{t}(\omega^{t})>0$ , $\widehat{p}(\{a(\omega^{t})\})>0$ , which leads to an uncountable number of atoms for $\widehat{p}$ .
Then, Proposition 4.1 allows to build examples of sets $\mathcal{Q}^{T}$ which are not dominated.

4.2 Discretized $d$ -dimensional diffusion

We provide now an example for the discretized dynamics of a multi-dimensional diffusion process in the spirit of [Carassus and Rásonyi, 2015, Example 8.2].
Fix a period $T\geq 1$ and $n\geq d$ . Denote by $M_{n}$ the set of real-valued matrix with $n$ rows and $n$ columns. Choose some constant $Y_{0}\in\mathbb{R}^{n}$ and let $Y_{t+1}$ be defined by the following difference equation for all $0\leq t\leq T-1$ , $(\omega^{t},\omega_{t+1})\in\Omega^{t}\times\Omega_{t+1}$

\displaystyle Y_{t+1}(\omega^{t},\omega_{t+1})-Y_{t}(\omega^{t})=\mu_{t+1}\left(Y_{t}(\omega^{t}),\omega^{t},\omega_{t+1}\right)+\nu_{t+1}\left(Y_{t}(\omega^{t}),\omega^{t}\right)Z_{t+1}(\omega^{t},\omega_{t+1})

(28)

where $\mu_{t+1}:\mathbb{R}^{n}\times\Omega^{t}\times\Omega_{t+1}\to\mathbb{R}^{n}$ , $\nu_{t+1}:\mathbb{R}^{n}\times\Omega^{t}\to M_{n}$ , $Z_{t+1}:\Omega^{t}\times\Omega_{t+1}\to\mathbb{R}^{n}$ are assumed to be Borel-measurable.
Two cases will be studied: $S^{i}_{t}=Y^{i}_{t}$ and $S^{i}_{t}=e^{Y^{i}_{t}}$ for all $1\leq i\leq d$ . In a uni-prior setting if the law of $Z_{t+1}$ is assumed to be normal, this corresponds to the popular normal and lognormal dynamic for the underlying assets. Note that in both cases if $d<n$ we may think that $Y^{i}_{t}$ for $i>d$ represents some non-traded assets or the evolution of some economic factors that will influence the market.
Assume that some $P^{0}\in\mathfrak{P}(\Omega^{T})$ is given with fixed disintegration $P^{0}:=P^{0}_{1}\otimes p_{2}^{0}\otimes\cdots\otimes p_{T}^{0}$ , where $p^{0}_{t+1}\in\mathcal{SK}_{t+1}$ for all $0\leq t\leq T-1$ : $P^{0}$ could be an initial guess or estimate for the prior. For all $0\leq t\leq T-1$ , let $r_{t}$ and $q_{t}$ be functions from $\Omega^{t}$ to $(0,\infty)$ : $r_{t}$ will be the bound on the drift while $q_{t}$ guarantees that the diffusion is non-degenerated (in dimension one it is a lower bound on the volatility). We make the following assumptions on the dynamic of $Y$ .

Assumption 4.8.

For all $0\leq t\leq T-1$ , $r_{t}$ is $\mathcal{B}(\Omega^{t})$ -measurable. For all $\omega^{t}\in\Omega^{t},\;x\in\mathbb{R}^{n}$ ,

•

$\nu_{t+1}(x,\omega^{t})\in M_{n}^{{q_{t}(\omega^{t})}}$ where $M^{\delta}_{n}:=\left\{M\in M_{n},\;\forall\,h\in\mathbb{R}^{n},\;h^{t}MM^{t}h\geq{\delta}h^{t}h\right\}$ for $\delta>0$ .
•

$Z_{t+1}(\omega^{t},\cdot)$ and $\mu_{t+1}(\omega^{t},\cdot)$ are independent under $p^{0}_{t+1}(\cdot,\omega^{t})$ .
•

$p^{0}_{t+1}\left(\mu_{t+1}(Y_{t}(\omega^{t}),\omega^{t},\cdot)\in[-r_{t}(\omega^{t}),r_{t}(\omega^{t})]^{n},\omega^{t}\right)=1$
•

$D_{Z_{t+1}}^{t+1}(\omega^{t})=\mathbb{R}^{n}$ , where $D_{Z_{t+1}}^{t+1}(\omega^{t})$ is the support of $Z_{t+1}(\omega^{t},\cdot)$ under $p_{t+1}^{0}(\cdot,\omega^{t}),$ see (5).

The model uncertainty on the laws of $\mu_{t+1}$ and $Z_{t+1}$ is given by the folowing sets.

$\displaystyle\mathcal{Q}^{1}_{t+1}(\omega^{t})$	$\displaystyle:=\left\{p\in\mathfrak{P}(\Omega_{t+1}),\;p\left(\mu_{t+1}(Y_{t}(\omega^{t}),\omega^{t},\cdot)\in[-r_{t}(\omega^{t}),r_{t}(\omega^{t})]^{n}\right)=1\right\},$	(29)
$\displaystyle\mathcal{Q}^{2}_{t+1}(\omega^{t})$	$\displaystyle:=\left\{p\in\mathfrak{P}(\Omega_{t+1}),\;F_{t}(p,\omega^{t})=0\right\},$	(30)
$\displaystyle\mathcal{Q}_{t+1}(\omega^{t})$	$\displaystyle:=\mathcal{Q}^{1}_{t+1}(\omega^{t})\bigcap\mathcal{Q}^{2}_{t+1}(\omega^{t}),$

where for some $k\geq 1$ , $F_{t}:\mathfrak{P}(\Omega_{t+1})\times\Omega^{t}\to\mathbb{R}^{k}$ is a Borel-measurable function such that $F_{t}(p_{t+1}^{0}(\cdot,\omega^{t}),\omega^{t})=0$ for all $0\leq t\leq T-1$ , $\omega^{t}\in\Omega^{t}$ . By assumption $p_{t+1}^{0}(\cdot,\omega^{t})\in\mathcal{Q}_{t+1}(\omega^{t})$ for all $\omega^{t}\in\Omega^{t}$ and thus $P^{0}\in\mathcal{Q}^{T}$ . Note that for a given $p\in\mathcal{Q}_{t+1}(\omega^{t})$ the law of $Z_{t+1}(\omega^{t},\cdot)$ and $\mu_{t+1}(\omega^{t},\cdot)$ under $p$ are not necessarily independent.

The financial interpretation is the following. The set $\mathcal{Q}^{1}_{t+1}(\omega^{t})$ allows the drift of the diffusion to be not only stochastic but with an unknown distribution. It is only assumed to be bounded. If $F_{t}(p,\omega^{t})=1_{\mbox{dist}_{t}\left(p\;,\;p^{0}_{t+1}(\cdot,\omega^{t})\right)\leq b_{t}(\omega^{t})}-1$ with $b_{t}(\omega^{t})>0$ and $\mbox{dist}_{t}$ some kind of distance function between probability measures, the set $\mathcal{Q}^{2}_{t+1}(\omega^{t})$ contains models which are close enough from $p_{t+1}^{0}(\cdot,\omega^{t})$ . This could happen if the physical measure is not known but estimated from data at each step. A popular choice for the $\mbox{dist}_{t}$ function is the Wasserstein distance. But one may also choose for the coordinate $i$ of $F(p,\omega^{t})$ (with $1\leq i\leq k$ ) the difference between the moments of order $i$ of $Z_{t+1}(\omega^{t},\cdot)$ under $p$ and under $p_{t+1}^{0}(\cdot,\omega^{t})$ and incorporate all the models $p$ such that the moments of $Z_{t+1}(\omega^{t},\cdot)$ under $p$ are equals to the ones of $Z_{t+1}(\omega^{t},\cdot)$ under $p^{0}_{t+1}(\cdot,\omega^{t})$ up to order $k$ .

Lemma 4.9.

Under Assumption 4.8, Assumptions 2.1 and 2.2 are satisfied.

Proof.

Assumption 2.1 follows from the Borel measurability of $\mu_{t+1}$ , $\nu_{t+1}$ , $Z_{t+1}$ and thus of $Y_{t+1}$ . As the function $(\omega^{t},p)\to p\left(\mu_{t+1}(Y_{t}(\omega^{t}),\omega^{t},\cdot)\in[-r_{t}(\omega^{t}),r_{t}(\omega^{t})]^{n}\right)$ is Borel-measurable (see [Bertsekas and Shreve, 2004, Proposition 7.29 p144]), $\mbox{graph}\left(\mathcal{Q}^{1}_{t+1}\right)$ is analytic. The Borel-measurability of $F_{t}$ implies that $\mbox{graph}\left(\mathcal{Q}^{2}_{t+1}\right)$ is an analytic set and so is $\mbox{graph}\left(\mathcal{Q}_{t+1}\right)$ . It is clear that $\mathcal{Q}^{1}_{t+1}$ is convex valued. If $F_{t}(\cdot,\omega^{t})$ is convex for all $\omega^{t}\in\Omega^{t}$ , then $\mathcal{Q}^{2}_{t+1}$ is convex valued. Else one may consider the convex hull of $\mathcal{Q}^{2}_{t+1}$ whose analyticity can be established as in the proof of Lemma 4.4. Assumption 2.2 is proved.

Now we give explicit values for $\beta_{t}$ and $\kappa_{t}$ in (20) with $p_{h}=p_{t+1}^{0}(\cdot,\omega^{t})$ and prove $NA(\mathcal{Q}^{T})$ .

Lemma 4.10.

Assume that Assumption 4.8 is satisfied and that $S^{i}_{t}=Y^{i}_{t}$ for all $1\leq i\leq d$ and all $1\leq t\leq T$ . Then $D^{t+1}(\omega^{t})=\mathbb{R}^{d}$ for all $\omega^{t}\in\Omega^{t}$ and $1\leq t\leq T-1$ and $NA(\mathcal{Q}^{T})$ condition holds true. Let

\displaystyle\kappa_{t}(\omega^{t}):=\min_{k\in K}\left(p^{0}_{t+1}\left(G_{k}(\omega^{t}),\omega^{t}\right)\right)>0\;\mbox{and}\;\beta_{t}(\omega^{t}):=\frac{\mbox{ln 2}}{\sqrt{n}}>0,

(31)

where $K$ is the (finite) set of functions from $\{1,\cdots,d\}$ to $\{-1,1\}$ and for some $k\in K$

\displaystyle G_{k}(\omega^{t}):=\left\{k(i)\Delta Y^{i}_{t+1}(\omega^{t},\cdot)<-\mbox{ln 2},\;1\leq i\leq d\right\}.

(32)

Then, for all $h\in\mathbb{R}^{d}$ with $|h|=1$

\displaystyle p_{t+1}^{0}\left(h\Delta S_{t+1}(\omega^{t},\cdot)<-\beta_{t}(\omega^{t}),\omega^{t}\right)\geq\kappa_{t}.

(33)

Proof.

First, we show that for all $\omega^{t}\in\Omega^{t},$ $D^{t+1}(\omega^{t})=\mathbb{R}^{d}.$ To do that we prove that

\displaystyle D^{t+1}_{Y}(\omega^{t}):=\bigcap\left\{A\subset\mathbb{R}^{n},\;\mbox{closed},\;p\left(\Delta Y_{t+1}(\omega^{t},\cdot)\in A,\omega^{t}\right)=1\;\forall p\in\mathcal{Q}_{t+1}(\omega^{t})\right\}=\mathbb{R}^{n}.

(34)

Let $D_{Y,P_{0}}^{t+1}(\omega^{t})$ be the support of $Y(\omega^{t},\cdot)$ under $p_{t+1}^{0}(\cdot,\omega^{t}),$ see (5). Using (7) $D^{t+1}_{Y,P_{0}}(\omega^{t})\subset D_{Y}^{t+1}(\omega^{t})$ and it is enough to prove that $D^{t+1}_{Y,P_{0}}(\omega^{t})=\mathbb{R}^{n}.$ Fix some $\omega^{t}\in\Omega^{t}$ . For ease of reading, we adopt the following notations. Let $\Delta Y(\cdot)=\Delta Y_{t+1}(\omega^{t},\cdot)$ , $R(\cdot)=\mu_{t+1}(Y_{t}(\omega^{t}),\omega^{t},\cdot)$ , $X(\cdot)=\Delta Y(\cdot)-R(\cdot)$ , $M=\nu_{t+1}(Y_{t}(\omega^{t}),\omega^{t})$ , $Z(\cdot)=Z_{t+1}(\omega^{t},\cdot)$ and $p^{0}(\cdot)=p^{0}_{t+1}(\cdot,\omega^{t})$ . As $X(\cdot)=MZ(\cdot)$ (see (28)) and $Z$ and $R$ are independent under $p^{0}$ , $X$ and $R$ are also independent under $p^{0}$ .
Fix some $x_{0}\in\mathbb{R}^{n}$ , $\varepsilon>0$ . By assumption $M$ is an invertible matrix: There exists some $y_{0}\in\mathbb{R}^{n}$ , $\alpha>0$ , such that $B(y_{0},\alpha)\subset M^{-1}\left(B(x_{0},\varepsilon)\right)$ ⁷⁷7 $M^{-1}\left(B(x_{0},\varepsilon)\right)$ is open in $\mathbb{R}^{n}$ and is not empty because $M$ is a bijective function on $\mathbb{R}^{n}$ .. The forth item of Assumption 4.8 together with Lemma 5.2 imply that⁸⁸8With the notation $p^{0}_{R}(A)=p^{0}(R\in A)$ for all $A\in{\cal B}(\mathbb{R}^{n})$ .

	$\displaystyle p^{0}\left(X(\cdot)\in B(x_{0},\varepsilon)\right)$	$\displaystyle=p^{0}\left(Z(\cdot)\in M^{-1}\left(B(x_{0},\varepsilon)\right)\right)\geq p^{0}\left(Z(\cdot)\in B(y_{0},\alpha)\right)>0$
	$\displaystyle p^{0}\left(\Delta Y(\cdot)\in B(x_{0},\varepsilon)\right)$	$\displaystyle=p^{0}\left(X(\cdot)+R(\cdot)\in B(x_{0},\varepsilon)\right)=\int_{\mathbb{R}}p^{0}\left(X(\cdot)\in B(x_{0}-u,\varepsilon)\right)p^{0}_{R}(du)>0,$

as $X$ and $R$ are independent under $p^{0}$ . Lemma 5.2 implies that the supports of $X$ and of $\Delta Y$ under $p^{0}$ are equal to $\mathbb{R}^{n}$ .

For all $0\leq t\leq T-1$ , $\omega^{t}\in\Omega^{t}$ , $D^{t+1}(\omega^{t})=\mathbb{R}^{d}$ and $0\in\mbox{Ri}\left(\mbox{Aff}\left({D}^{t+1}\right)(\omega^{t})\right)$ . Theorem 3.24 implies that the $NA(\mathcal{Q}^{T})$ condition is verified.
Fix now some $\omega^{t}\in\Omega^{t}$ and $h\in\mathbb{R}^{d}$ with $|h|=1$ . First, $D_{Y,P_{0}}^{t+1}(\omega^{t})=\mathbb{R}^{n}$ implies that for all $k\in K$ , $\omega^{t}\in\Omega^{t}$

\displaystyle p_{t+1}^{0}\left(G_{k}(\omega^{t}),\omega^{t}\right)=p^{0}_{t+1}(\Delta Y_{t+1}(\omega^{t},\cdot)\in\mathcal{O}_{h},\omega^{t})>0,

(35)

where $\mathcal{O}_{h}:=\{z\in\mathbb{R}^{n},\;k(i)z_{i}<-\mbox{ln 2},\;\forall\,1\leq i\leq d\}$ is an open set of $\mathbb{R}^{n}$ . Set $k^{*}(i):=\mbox{sign}(h_{i})$ for all $1\leq i\leq d$ , then $k^{*}\in K$ . Let $\omega_{t+1}\in G_{k^{*}}(\omega^{t})$ as (35) implies that $G_{k^{*}}(\omega^{t})$ is not empty. For all $1\leq i\leq d$ ,

h_{i}\Delta S^{i}_{t+1}(\omega^{t},\omega_{t+1})=|h_{i}|k^{*}(i)\Delta Y^{i}_{t+1}(\omega^{t},\omega_{t+1})\leq-\mbox{ln 2}|h_{i}|\leq 0.

As $|h|=1$ there exists $1\leq i^{*}\leq d$ such that $\frac{1}{\sqrt{n}}\leq\frac{1}{\sqrt{d}}\leq|h_{i^{*}}|\leq 1$ and

\displaystyle h\Delta S_{t+1}(\omega^{t},\omega_{t+1})<-\frac{\mbox{ln 2}}{\sqrt{n}}+\sum_{i\neq i^{*}}h_{i}\Delta Y^{i}_{t+1}(\omega^{t},\omega_{t+1})\leq-\frac{\mbox{ln 2}}{\sqrt{n}}.

Therefore $p_{t+1}^{0}\left(h\Delta S_{t+1}(\omega^{t},\cdot)<-{\mbox{ln 2}}/{\sqrt{n}},\omega^{t}\right)\geq\min_{k\in K}\left(p_{t+1}^{0}\left(G_{k}(\omega^{t}),\omega^{t}\right)\right)$ . Recalling (35), (33) is satisfied.

We now treat the log-normal case.

Lemma 4.11.

Assume that Assumption 4.8 is satisfied and that $S^{i}_{t}=e^{Y^{i}_{t}}$ for all $1\leq i\leq d$ and all $1\leq t\leq T$ . Then $D^{t+1}(\omega^{t})=\mathbb{R}^{d}$ for all $\omega^{t}\in\Omega^{t}$ and $1\leq t\leq T-1$ and $NA(\mathcal{Q}^{T})$ condition holds true. Let

\displaystyle\kappa_{t}(\omega^{t}):=

\displaystyle\min_{k\in K}\left(p_{t+1}^{0}\left(G_{k}(\omega^{t}),\omega^{t}\right)\right)>0\;\;\;\beta_{t}(\omega^{t}):=\frac{1}{2}\min\left({1},\frac{\min_{1\leq i\leq d}S^{i}_{t}(\omega^{t})}{\sqrt{n}}\right)>0,

(36)

recall (32) for the definition of $G_{k}(\omega^{t}).$ Then, for all $h\in\mathbb{R}^{d}$ with $|h|=1$

\displaystyle p_{t+1}^{0}\left(h\Delta S_{t+1}(\omega^{t},\cdot)<-\beta_{t}(\omega^{t}),\omega^{t}\right)\geq\kappa_{t}.

(37)

Proof.

Let $0\leq t\leq T-1$ and fix $\omega^{t}\in\Omega^{t}$ . Using (7) $D^{t+1}_{P_{0}}(\omega^{t})\subset D^{t+1}(\omega^{t})$ and it is enough to prove that $D^{t+1}_{P_{0}}(\omega^{t})=\mathbb{R}^{d}.$ This will follow from Lemma 5.2 if for any open set $O$ of $\mathbb{R}^{d},$ $p^{0}\left(\Delta S_{t+1}(\cdot,\omega^{t})\in O,\omega^{t}\right)>0.$ Fix an open set $O$ of $\mathbb{R}^{d}$ and let $F_{\omega^{t}}:\mathbb{R}^{n}\to\mathbb{R}^{d}$ be defined by $F_{\omega^{t}}(x_{1},\cdots,x_{n})=(e^{Y_{t}^{1}(\omega^{t})}(e^{x_{1}}-1),\cdots,e^{Y_{t}^{d}(\omega^{t})}(e^{x_{d}}-1))$ . As $F_{\omega^{t}}$ is continuous $F^{-1}_{\omega^{t}}(O)$ is an open set of $\mathbb{R}^{n}$ . Then

	$\displaystyle p^{0}\left(e^{Y_{t+1}(\cdot,\omega^{t})}-e^{Y_{t}(\omega^{t})}\in O,\omega^{t}\right)$
	$\displaystyle=p^{0}\left(\left(e^{Y^{1}_{t}(\omega^{t})}\left(e^{\Delta Y^{1}_{t+1}(\cdot,\omega^{t})}-1\right),\cdots,e^{Y^{d}_{t}(\omega^{t})}\left(e^{\Delta Y^{d}_{t+1}(\cdot,\omega^{t})}-1\right)\right)\in O,\omega^{t}\right)$
	$\displaystyle=p^{0}\left(\Delta Y_{t+1}(\cdot,\omega^{t})\in F^{-1}_{\omega^{t}}(O),\omega^{t}\right)>0,$

using (34) and Lemma 5.2 again. Thus for all $0\leq t\leq T-1$ , $\omega^{t}\in\Omega^{t}$ , $D^{t+1}(\omega^{t})=\mathbb{R}^{d}$ and $0\in\mbox{Ri}\left(\mbox{Aff}\left({D}^{t+1}\right)(\omega^{t})\right)$ . Theorem 3.24 implies that the $NA(\mathcal{Q}^{T})$ condition is verified.
Fix a $\omega^{t}\in\Omega^{t}$ , $h\in\mathbb{R}^{d}$ with $|h|=1$ . Then

\displaystyle h\Delta S_{t+1}(\omega^{t},\omega_{t+1})

\displaystyle=\sum_{i=1}^{d}h_{i}S^{i}_{t}(\omega^{t})\left(e^{\Delta Y^{i}_{t+1}(\omega^{t},\omega_{t+1})}-1\right).

(38)

Let $k^{*}\in K$ as in the proof of the preceding lemma and let $\omega_{t+1}\in G_{k^{*}}(\omega^{t})$ . First, for all $1\leq i\leq d$ ,

\displaystyle h_{i}S^{i}_{t}(\omega^{t})\left(e^{\Delta Y^{i}_{t+1}(\omega^{t},\omega_{t+1})}-1\right)<\begin{cases}-\frac{|h_{i}|S^{i}_{t}(\omega^{t})}{2}\;\mbox{if $k^{*}({i})=1$}\\ -|h_{i}|{S^{i}_{t}(\omega^{t})}\mbox{ if $k^{*}({i})=-1$}\end{cases}\leq 0.

(39)

As $|h|=1$ there is a component $h_{i^{*}}$ such that $\frac{1}{\sqrt{n}}\leq\frac{1}{\sqrt{d}}\leq|h_{i^{*}}|\leq 1$ and as $S^{i^{*}}_{t}(\omega^{t})>0$ , (38) implies that

h\Delta S_{t+1}(\omega^{t},\omega_{t+1})<-\frac{S^{i^{*}}_{t}(\omega^{t})}{2\sqrt{n}}+\sum_{i\neq i^{*}}h_{i}S^{i}_{t}(\omega^{t})\left(e^{\Delta Y^{i}_{t+1}(\omega^{t},\omega_{t+1})}-1\right)\leq-\frac{\min_{1\leq i\leq d}S^{i}_{t}(\omega^{t})}{2\sqrt{n}}.

So,

p_{t+1}^{0}\left(h\Delta S_{t+1}(\omega^{t},\cdot)<-\frac{\min_{1\leq i\leq d}S^{i}_{t}(\omega^{t})}{2\sqrt{n}},\omega^{t}\right)\geq\min_{k\in K}p_{t+1}^{0}\left(G_{k}(\omega^{t}),\omega^{t}\right),

and using (35), (37) is satisfied.

Remark 4.12.

Note that in both cases ( $S^{i}_{t}=Y^{i}_{t}$ and $S^{i}_{t}=e^{Y^{i}_{t}}$ ), we can choose $P^{*}=P^{0}$ in Theorem 3.30.

We now give a one dimension illustration of the previous setting where $\mathcal{Q}^{T}$ is not dominated. Take $n=d=1$ and $\Omega_{t}:=\Omega$ for some Polish space $\Omega$ . Let $Z$ be some real-valued random variable defined on $\Omega$ and $p_{0}\in\mathfrak{P}(\Omega)$ be such that under $p_{0}$ , $Z$ is normally distributed with mean $0$ and standard deviation $1$ . Set $P^{0}:=p_{0}\otimes\cdots\otimes p_{0}$ and $Z_{t+1}(\omega^{t},\omega_{t+1}):=Z(\omega_{t+1})$ for all $0\leq t\leq T-1$ and $\omega^{t}\in\Omega^{t}$ . Define $F:\mathfrak{P}(\Omega)\to\mathbb{R}^{2}$ by $F(p):=\left(E_{p}(Z),E_{p}\left(Z-E_{p}(Z)\right)^{2}-1\right)$ and $F(\omega^{t},\omega_{t+1}):=F(\omega_{t+1})$ for all $0\leq t\leq T-1$ and $\omega^{t}\in\Omega^{t}$ . Finally, set $\mathcal{Q}_{t+1}(\omega^{t}):=\{p\in\mathfrak{P}(\Omega),F(p)=0\}=:\mathcal{Q}$ for all $0\leq t\leq T-1$ , $\omega^{t}\in\Omega^{t}$ . For each $\omega^{t}$ , the law of the driving process $Z$ for the next period is centered with variance $1$ but not necessarily normally distributed.
Assumption 4.8 on the dynamic of $Y$ are verified if we choose $Y_{0}:=1$ and for all $0\leq t\leq T-1$ , $x\in\mathbb{R}$ , $(\omega^{t},\omega_{t+1})\in\Omega^{t}\times\Omega$

\mu_{t+1}(x,\omega^{t},\omega_{t+1}):=r_{t}(\omega^{t}):=r,\;\;\;\nu_{t+1}(x,\omega^{t}):=\sigma,\;\;\;q_{t}(\omega^{t}):=\sigma^{2},

for some $r\in\mathbb{R}$ and $\sigma>0$ fixed.
As $\Delta Y_{t}=r+\sigma Z$ and $Z$ is normally distributed with mean $0$ and standard deviation $1$ under $p_{0}$ , (31) (or (36)) implies that

\kappa_{t}=\kappa=\min\left(\Phi\left(-\frac{\mbox{ln 2}+r}{\sigma}\right),1-\Phi\left(\frac{\mbox{ln 2}-r}{\sigma}\right)\right)

where $\Phi$ is the cumulative distribution function of some normal law with mean $0$ and standard deviation $1$ . We have already seen that $\beta_{t}(\omega^{t})=\beta={(\mbox{ln 2}})/{\sqrt{n}}$ when $S_{t}={Y_{t}}$ . In the other case, $S_{t}(\omega^{t})=\mbox{exp}\left({Y_{t}(\omega^{t})}\right)=\mbox{exp}\left(1+rt+\sigma\sum_{i=1}^{t}Z(\omega_{i})\right)$ and $\beta_{t}(\omega^{t})=(1/2)\min\left(1,S_{t}(\omega^{t})\right)$ (see (36)).
Finally, the set $\mathcal{Q}^{T}$ is not dominated. Indeed, we show that $\mathcal{Q}$ is not dominated and conclude using Proposition 4.1. Assume that there is some $\widehat{p}\in\mathfrak{P}(\Omega)$ which dominates $\mathcal{Q}$ . For $x\neq 0$ , let $q_{x}\in\mathfrak{P}(\Omega)$ such that

q_{x}(Z=x)=\frac{1}{2x^{2}},\,\,q(Z=-x)=\frac{1}{2x^{2}},\,\,q(Z=0)=1-\frac{1}{x^{2}}.

Then $q_{x}\in\mathcal{Q}$ and $\{x\in\mathbb{R},\;\widehat{p}(\{Z=x\})>0\}=\mathbb{R}\backslash\{0\}$ , a contradiction.

5 Proofs

The first section presents the one-period version of our problems with deterministic initial data. We will study the different notions of arbitrage and their equivalence (see Proposition 5.7). We also prove Proposition 5.8 that will be used in the proof of Theorem 3.30. In the second section the multi-period results are proved relying on the one-period results together with measurable selections technics. Finally, the third section presents the proof of Proposition 4.1.

5.1 One-period model

Let $(\overline{\Omega},{\cal G})$ be a measured space, $\mathfrak{P}(\overline{\Omega})$ the set of all probability measures defined on $\mathcal{G}$ and $\mathcal{Q}$ a non-empty convex subset of $\mathfrak{P}(\overline{\Omega})$ . For $P\in\mathcal{Q}$ fixed, $E_{P}$ denotes the expectation under $P$ . Let $Y$ be a ${\cal G}$ -measurable $\mathbb{R}^{d}$ -valued random variable.
The following sets are the pendant in the one-period case of the ones introduced in Definition 2.3. Let $P\in\mathcal{Q}$

	$\displaystyle{E}(P)$	$\displaystyle:=\bigcap\left\{A\subset\mathbb{R}^{d},\;\mbox{closed},\;P\left(Y(.)\in A\right)=1\right\},$		(40)
	$\displaystyle{D}$	$\displaystyle:=\bigcap\left\{A\subset\mathbb{R}^{d},\;\mbox{closed},\;P\left(Y(\cdot)\in A\right)=1,\;\forall P\in\mathcal{Q}\right\}.$		(41)

The next lemma will be used in the proof of Proposition 3.28.

Lemma 5.1.

Let $C$ be a convex set of $\mathbb{R}^{d}$ and fix some $\varepsilon>0$ . Then $B(0,\varepsilon)\cap\mbox{Aff}(C)\subset\overline{C}$ if and only if $B(0,\varepsilon)\cap\mbox{Aff}(C)\subset{C}$ .

Proof.

The reverse implication is trivial. Assume that $B(0,\varepsilon)\cap\mbox{Aff}(C)\subset\overline{C}$ and let $x\in B(0,\varepsilon)\cap\mbox{Aff}(C)$ . As $|x|<\varepsilon$ , there exists some $\delta>0$ such that $B(x,\delta)\cap\mbox{Aff}(C)\subset B(0,\varepsilon)\cap\mbox{Aff}(C)\subset\overline{C}.$ Hence $x\in\mbox{Ri}(\overline{C})=\mbox{Ri}(C)\subset C$ (see [Rockafellar, 1970, Theorem 6.3 p46]).

This lemma allows an easy characterisation of the support and was used several time in the paper.

Lemma 5.2.

Let $h\in\mathbb{R}^{d}$ and $P\in\mathfrak{P}(\overline{\Omega})$ be fixed. Then, $h\in{E}(P)$ if and only if for all $n\geq 1$ , $P\left(Y(\cdot)\in B\left(h,{1}/{n}\right)\right)>0$ . Similarly, $h\in{D}$ if and only if for all $n\geq 1$ , there exists some $P^{n}\in\mathcal{Q}$ , such that $P^{n}\left(Y(\cdot)\in B\left(h,{1}/{n}\right)\right)>0$ .

Proof.

Fix some $h\in\mathbb{R}^{d}$ . By definition $h\notin{E}(P)$ if and only if there exists an open set $O\subset\mathbb{R}^{d}$ such that $h\in O$ and $P(Y(\cdot)\in O)=0$ and the first item follows. Similarly, $h\notin{D}$ if and only if there exists an open set $O\subset\mathbb{R}^{d}$ such that $h\in O$ and $P(Y(\cdot)\in O)=0$ for all $P\in\mathcal{Q}$ and the second item follows.

Now, we introduce the definitions of no-arbitrage in this one period setting. The first one is the one-period pendant of the $NA(\mathcal{Q}^{T})$ condition while the two others are the pendant of Definitions 3.19 and 3.20.

Definition 5.3.

The one-period no-arbitrage condition holds true if $hY(\cdot)\geq 0$ $\mathcal{Q}$ -q.s. for some $h\in\mathbb{R}^{d}$ implies that $hY(\cdot)=0$ $\mathcal{Q}$ -q.s.

Definition 5.4.

The one-period geometric no-arbitrage condition holds true if $0\in\mbox{Ri}\left(\mbox{Conv}(D)\right).$ This is equivalent to $0\in\mbox{Conv}(D)$ and there exists some $\varepsilon>0$ such that $B(0,\varepsilon)\cap{\mbox{Aff}(D)}\subset{\mbox{Conv}}(D).$

Definition 5.5.

The one-period quantitative no-arbitrage condition holds true if there exists some constants $\beta,\kappa\in(0,1]$ such that for all $h\in\mbox{Aff}(D)$ , $h\neq 0$ there exists $P_{h}\in\mathcal{Q}$ satisfying

\displaystyle P_{h}(hY(\cdot)<-\beta|h|)\geq\kappa.

(42)

Remark 5.6.

We recall that if $0\notin\mbox{Ri}\left({\mbox{Conv}}(D)\right)$ there exists some $h^{*}\in\mbox{Aff}(D)$ , $h^{*}\neq 0$ such that $h^{*}Y(\cdot)\geq 0$ $\mathcal{Q}$ -q.s. This is a classical exercise relying on separation arguments in $\mathbb{R}^{d}$ , see [Rockafellar, 1970, Theorems 11.1, 11.3 p97] or [Föllmer and Schied, 2002, Proposition A.1].

Proposition 5.7 establishes that these three preceding conditions are actually equivalent.

Proposition 5.7.

Definitions 5.3, 5.4 and 5.5 are equivalent. Moreover, one can choose $\beta={\varepsilon}/{2}$ in (42) where $\varepsilon>0$ is such that $B(0,\varepsilon)\cap{\mbox{Aff}(D)}\subset{\mbox{Conv}}(D)$ in Definition 5.4.

Proof.

Step 1 : Definition 5.3 implies Definitions 5.4 and 5.5.
First we show by contradiction that for all $h\in\mbox{Aff}(D)$

\displaystyle hY(\cdot)\geq 0\;\mathcal{Q}\mbox{-q.s.}\Rightarrow h=0.

(43)

Assume that there exists some $h\in\mbox{Aff}(D)$ , $h\neq 0$ such that $hY(\cdot)\geq 0\;\mathcal{Q}$ -q.s. Definition 5.3 implies that $hY(\cdot)=0\;\mathcal{Q}\mbox{-q.s.}$ and⁹⁹9 ${X}^{\perp}$ stands for the orthogonal space of some set $X.$

h\in\{h\in\mathbb{R}^{d},hy=0\;\mbox{for all $y\in{D}$}\}={D}^{\perp}=\left(\mbox{Aff}(D)\right)^{\perp},

see for instance [Nutz, 2016, Proof of Lemma 2.6]. This implies that $h\in\mbox{Aff}(D)\cap\left(\mbox{Aff}(D)\right)^{\perp}\subset\{0\}$ , a contradiction.
Now we show that Definition 5.4 holds true. If $0\notin\mbox{Ri}\left({\mbox{Conv}}(D)\right)$ , Remark 5.6 implies that there exists some $h^{*}\in\mbox{Aff}(D)$ , $h^{*}\neq 0$ such that $h^{*}Y(\cdot)\geq 0$ $\mathcal{Q}$ -q.s. which contradicts (43). Then, we prove that Definition 5.5 holds also true. For all $n\geq 1,$ let

\displaystyle A_{n}:=\left\{h\in\mbox{Aff}(D),\;|h|=1,\;P\left(hY(\cdot)<-\frac{1}{n}\right)<\frac{1}{n}\;\forall P\in\mathcal{Q}\right\}\;\;n_{0}:=\inf\{n\geq 1,A_{n}=\emptyset\}

(44)

with the convention that $\inf\emptyset=+\infty$ . We have seen that Definition 5.4 holds true: $0\in\mbox{Ri}\left({\mbox{Conv}}(D)\right)\subset\mbox{Aff}(D)$ and $\mbox{Aff}(D)$ is a vector space. If $\mbox{Aff}(D)=\{0\}$ , then $n_{0}=1<\infty$ . Assume now that $\mbox{Aff}(D)\neq\{0\}$ . We prove by contradiction that $n_{0}<\infty$ . Assume that $n_{0}=\infty$ . For all $n\geq 1$ , there exists some $h_{n}\in A_{n}.$ By passing to a sub-sequence we can assume that $h_{n}$ tends to some $h^{*}\in\mbox{Aff}(D)$ with $|h^{*}|=1$ . Let $B_{n}:=\left\{h_{n}Y(\cdot)<-{1}/{n}\right\}$ . Then $\{h^{*}Y(\cdot)<0\}\subset\liminf_{n}B_{n}$ and Fatou’s Lemma implies that for any $P\in\mathcal{Q}$

\displaystyle P\left(h^{*}Y(\cdot)<0\right)

\displaystyle\leq\int_{\overline{\Omega}}1_{\liminf_{n}B_{n}}(\omega)P(d\omega)\leq\liminf_{n}\int_{\overline{\Omega}}1_{B_{n}}(\omega)P(d\omega)=0.

So $h^{*}Y(\cdot)\geq 0$ $\mathcal{Q}$ -q.s. and (43) implies that $h^{*}=0$ which contradicts $|h^{*}|=1$ . Thus $n_{0}<\infty$ and we can set $\beta=\kappa={1}/{n_{0}}.$ It is clear that $\beta,\kappa\in(0,1]$ and by definition of $A_{n_{0}}$ , (42) holds true.

Step 2 : Definition 5.5 implies Definition 5.4.
Else, Remark 5.6 implies that there exists some $h^{*}\in\mbox{Aff}(D)$ , $h^{*}\neq 0$ such that $h^{*}Y(\cdot)\geq 0$ $\mathcal{Q}$ -q.s.: A contradiction with (42).

Step 3 : Definition 5.4 implies Definition 5.3.
Fix some $h\in\mathbb{R}^{d}$ such that $hY(\cdot)\geq 0$ $\mathcal{Q}$ -q.s. Let $p(h)$ be the orthogonal projection of $h$ on $\mbox{Aff}(D)$ (recall that $\mbox{Aff}(D)$ is a vector space since $0\in\mbox{Ri}({\mbox{Conv}}(D))\subset\mbox{Aff}(D)$ ). Assume for a moment that $p(h)=0$ . Remark 2.4 shows that $P(\{Y(\cdot)\in{D}\})=1$ for all $P\in\mathcal{Q},$ $hY(\cdot)=p(h)Y(\cdot)=0$ $\mathcal{Q}$ -q.s. and Definition 5.3 is verified.
Next we show that $hy\geq 0$ for all $y\in{D}$ and by convex combinations for all $y\in{\mbox{Conv}}(D)$ . Indeed if there exists $y_{0}\in D$ such that $hy_{0}<0$ , then there exists some $\delta>0$ such that $hy<0$ for all $y\in B(y_{0},\delta)$ . But Lemma 5.2 implies the existence of some $P\in\mathcal{Q}$ such that $P(Y(\cdot)\in B(y_{0},\delta))>0$ , a contradiction. Now, if $p(h)\neq 0$ , as $0\in\mbox{Ri}({\mbox{Conv}}(D))$ , there exists some $\varepsilon>0$ such that $B(0,\varepsilon)\cap\mbox{Aff}(D)\subset{\mbox{Conv}}(D),$ $-\varepsilon{p(h)}/{|p(h)|}\in{\mbox{Conv}}(D)$ and

-\varepsilon\frac{p(h)}{|p(h)|}h=-\varepsilon\frac{p(h)}{|p(h)|}p(h)<0,

a contradiction.

Step 4: If $B(0,\varepsilon)\cap\mbox{Aff}(D)\subset{\mbox{Conv}}(D)$ one can choose $\beta={\varepsilon}/{2}$ in (42).
This is similar to the proof of Definition 5.4 implies Definition 5.5. The set $A_{n}$ is modified by setting

A_{n}:=\left\{h\in\mbox{Aff}(D),\;|h|=1,P\left(hY(\cdot)<-\frac{\varepsilon}{2}\right)<\frac{1}{n},\;\forall P\in\mathcal{Q}\right\}.

The same arguments as before apply and if $n_{0}=\infty$ there exists some $h^{*}\in\mbox{Aff}(D)$ , $|h^{*}|=1$ such that $h^{*}Y\geq-{\varepsilon}/{2}$ $\mathcal{Q}^{T}$ -q.s. We also get that $h^{*}y\geq-{\varepsilon}/{2}$ for all $y\in{\mbox{Conv}}(D)$ . Choosing $y=-(2/3)\varepsilon h^{*}\in B(0,\varepsilon)\cap\mbox{Aff}(D)\subset{\mbox{Conv}}(D)$ , we obtain a contradiction. So, (42) holds true with $\beta=\varepsilon/{2}$ and $\kappa={1}/{n_{0}}$ ¹⁰¹⁰10The same argument shows that one can set $\kappa=\inf_{h\in\scriptsize{\mbox{Aff}}(D),\;|h|=1}\ \sup_{P\in\mathcal{Q}}P(hY(\cdot)<-\frac{\varepsilon}{2})>0$ illustrating why the measurability of $\kappa$ cannot be directly obtained, see Remark 3.29..

The next proposition follows from [Bayraktar and Zhou, 2017, Lemma 2.2] and will be used in the proof of Theorem 3.30.

Proposition 5.8.

Assume that the one-period no-arbitrage condition (see Definition 5.3) holds true. Then there exists some $P^{*}\in\mathcal{Q}$ such that $0\in\mbox{Ri}\left({\mbox{Conv}}(E(P^{*}))\right)$ and $\mbox{Aff}(E(P^{*}))=\mbox{Aff}(D)$ .

Proof.

[Bayraktar and Zhou, 2017, Lemma 2.2] gives the existence of some $P^{*}\in\mathcal{Q}$ such that $NA(P^{*})$ holds true and $\mbox{Aff}(E(P^{*}))=\mbox{Aff}(D)$ . Note that the proof of [Bayraktar and Zhou, 2017, Lemma 2.2] relies on the convexity of $\mathcal{Q}$ . Now Proposition 3.26 (for $T=1$ ) shows that $0\in\mbox{Ri}\left({\mbox{Conv}}(E(P^{*}))\right)$ .

5.2 Multi-period model

First we define the distance of a point $x\in\mathbb{R}^{d}$ to a set $F\subset\mathbb{R}^{d}$ by $d(x,F):=\inf\{|x-f|,\;f\in F\}$ and the Hausdorff distance between two sets $F,G\subset\mathbb{R}^{d}$ by $d(F,G)=\sup_{x\in\mathbb{R}^{d}}|d(x,F)-d(x,G)|.$

5.2.1 Proof of Theorem 3.24

Proof.

Fix some $0\leq t\leq T-1$ and $\omega^{t}\in\Omega^{t}.$ We say that the $NA(\mathcal{Q}_{t+1}(\omega^{t}))$ condition holds true if $h\Delta S_{t+1}(\omega^{t},\cdot)\geq 0$ $\mathcal{Q}_{t+1}(\omega^{t})$ -q.s. for some $h\in\mathbb{R}^{d}$ implies that $h\Delta S_{t+1}(\omega^{t},\cdot)=0$ $\mathcal{Q}_{t+1}(\omega^{t})$ -q.s. Proposition 5.7 implies that the $NA(\mathcal{Q}_{t+1}(\omega^{t}))$ condition is equivalent to (19) and (20) for any $\omega^{t}\in\Omega^{t}$ . Then Theorem 3.18 shows that Definition 3.1 is equivalent to the fact that

\displaystyle\Omega^{t}_{NA}=\{\omega^{t}\in\Omega^{t},\;\;NA(\mathcal{Q}_{t+1}(\omega^{t}))\mbox{ holds true}\}

(45)

is a $\mathcal{Q}^{t}$ -full measure set and belongs to $\mathcal{B}_{c}(\Omega^{t})$ for all $0\leq t\leq T-1$ . Thus, for all $0\leq t\leq T-1,$ one may choose $\Omega^{t}_{NA}=\Omega^{t}_{qNA}=\Omega^{t}_{gNA}.$ Furthermore, Proposition 5.7 shows that one can take $\beta_{t}(\omega^{t})={\varepsilon_{t}}(\omega^{t})/{2}$ for $\omega^{t}\in\Omega^{t}_{NA}$ .

5.2.2 Proof of Proposition 3.28

Proof of Proposition 3.28

Proof.

Fix some $0\leq t\leq T-1$ . We set $\Gamma^{t+1}(\omega^{t})=\emptyset$ for $\omega^{t}\notin\Omega_{NA}^{t}$ and for all $\omega^{t}\in\Omega_{NA}^{t}$

	$\displaystyle\Gamma^{t+1}(\omega^{t})$	$\displaystyle:=\left\{\varepsilon\in\mathbb{Q},\;\varepsilon>0,\;B(0,\varepsilon)\cap\mbox{Aff}\left(D^{t+1}\right)(\omega^{t})\subset{\mbox{Conv}}\left({D}^{t+1}\right)(\omega^{t})\right\}$
		$\displaystyle=\left\{\varepsilon\in\mathbb{Q},\;\varepsilon>0,\;B(0,\varepsilon)\cap\mbox{Aff}\left(D^{t+1}\right)(\omega^{t})\subset\overline{\mbox{Conv}}\left({D}^{t+1}\right)(\omega^{t})\right\},$

where the equality comes from Lemma 5.1. Assume for a moment that $\mbox{graph}\;\Gamma^{t+1}\in\mathcal{B}_{c}(\Omega^{t})\otimes\mathcal{B}(\mathbb{R}^{d})$ has been proved. The Aumann Theorem implies the existence of a $\ \mathcal{B}_{c}(\Omega^{t})$ -measurable selector ${\varepsilon}_{t}:\{\Gamma^{t+1}\neq\emptyset\}\to\mathbb{R}$ such that ${\varepsilon}_{t}(\omega^{t})\in\Gamma^{t+1}(\omega^{t})$ for every $\omega^{t}\in\{\Gamma^{t+1}\neq\emptyset\}$ . Now, Theorem 3.24 and (19) imply that $\Omega^{t}_{NA}=\left\{\Gamma^{t+1}(\omega^{t})\neq\emptyset\right\}$ (recall that $\Gamma^{t+1}(\omega^{t})=\emptyset$ outside $\Omega_{NA}^{t}$ ). Setting $\varepsilon_{t}=1$ outside $\Omega^{t}_{NA},$ $\varepsilon_{t}$ is $\mathcal{B}_{c}(\Omega^{t})$ -measurable and Proposition 3.28 is proved as we can choose $\beta_{t}={\varepsilon_{t}}/2$ (see Theorem 3.24).
It remains to show that $\mbox{graph}\;\Gamma^{t+1}\in\mathcal{B}_{c}(\Omega^{t})\otimes\mathcal{B}(\mathbb{R}^{d})$ . For all $\varepsilon>0$ , $\varepsilon\in\mathbb{Q}$ , let

A_{\varepsilon}:=\left\{\omega^{t}\in\Omega_{NA}^{t},\;B(0,\varepsilon)\cap\mbox{Aff}\left(D^{t+1}\right)(\omega^{t})\subset\overline{\mbox{Conv}}\left({D}^{t+1}\right)(\omega^{t})\right\}.

As $\mbox{graph}\;\Gamma^{t+1}=\bigcup_{\varepsilon\in\mathbb{Q},\;\varepsilon>0}A_{\varepsilon}\times\{\varepsilon\},$ it is enough to prove that $A_{\varepsilon}\in\mathcal{B}_{c}(\Omega^{t})$ . Let $h:\mathbb{R}^{d}\times\Omega^{t}$ be defined by

h(x,\omega^{t}):=d\left(x,B(0,\varepsilon)\cap\mbox{Aff}\left(D^{t+1}\right)(\omega^{t})\right)-d\left(x,\overline{\mbox{Conv}}\left({D}^{t+1}\right)(\omega^{t})\right).

Then [Aliprantis and Border, 2006, Theorem 18.5 p595] and Lemma 2.6 show that for all $x\in\mathbb{R}^{d}$ $h(x,\cdot)$ is $\mathcal{B}_{c}(\Omega^{t})$ -measurable and that $h(\cdot,\omega^{t})$ is continuous for all $\omega^{t}\in\Omega^{t}$ . As $\overline{\mbox{Conv}}\left({D}^{t+1}\right)(\omega^{t})$ is closed-valued,

A_{\varepsilon}=\left\{\omega^{t}\in\Omega_{NA}^{t},\;h(x,\omega^{t})\geq 0,\,\forall x\in\mathbb{R}^{d}\right\}=\cap_{q\in\mathbb{Q}^{d}}\left\{\omega^{t}\in\Omega_{NA}^{t},\;h(q,\omega^{t})\geq 0\right\}\in\mathcal{B}_{c}(\Omega^{t}).

5.2.3 Proof of Theorem 3.30

As mentioned in Remark 3.34, our proof uses similar ideas as the one used in the proof of [Oblój and Wiesel, 2018, Theorem 3.1] and relies crucially on the measurability and convexity of Graph( $\mathcal{Q}_{t+1}(\omega^{t}))$ (see Assumption 2.2).

Proof.

Reverse implication.
Fix some $0\leq t\leq T-1$ and $\omega^{t}\in\Omega^{t}_{NA}.$ As $P^{*}\in\mathcal{Q}^{T},$ Remark 2.5 implies that ${D}_{P^{*}}^{t+1}(\omega^{t})\subset{D}^{t+1}(\omega^{t}).$ As $\mbox{Aff}\left(D^{t+1}\right)(\omega^{t})=\mbox{Aff}\left(D_{P^{*}}^{t+1}\right)(\omega^{t})$ and $0\in\mbox{Ri}\left({\mbox{Conv}}(D_{P^{*}}^{t+1})\right)(\omega^{t})$ , there exists some $\varepsilon>0$ such that

\displaystyle B(0,\varepsilon)\bigcap\mbox{Aff}\left(D^{t+1}\right)(\omega^{t})

\displaystyle=B(0,\varepsilon)\bigcap\mbox{Aff}\left(D_{P^{*}}^{t+1}\right)(\omega^{t})\subset{\mbox{Conv}}(D_{P^{*}}^{t+1})(\omega^{t})\subset{\mbox{Conv}}(D^{t+1})(\omega^{t})

and NA $(\mathcal{Q}^{T})$ follows from Theorem 3.24.

Direct implication.
For all $0\leq t\leq T-1$ , let $\mathcal{E}_{t+1}:\Omega^{t}\twoheadrightarrow\mathfrak{P}(\Omega_{t+1})$ be defined by $\mathcal{E}_{t+1}(\omega^{t})=\emptyset$ if $\omega^{t}\in\Omega^{t}_{NA}$ and if $\omega^{t}\in\Omega^{t}_{NA}$

\displaystyle\mathcal{E}_{t+1}(\omega^{t}):=\{p\in\mathcal{Q}_{t+1}(\omega^{t}),\;0\in\mbox{Ri}\left({\mbox{Conv}}(E^{t+1})\right)(\omega^{t},p)\mbox{ and }

\displaystyle\mbox{Aff}\left(E^{t+1}\right)(\omega^{t},p)=\mbox{Aff}\left(D^{t+1}\right)(\omega^{t})\}.

Theorem 3.18 and Proposition 5.8 show that $\Omega^{t}_{NA}=\{\mathcal{E}_{t+1}\neq\emptyset\}$ . Assume for a moment that we have proved the existence of $p_{t+1}^{*}\in\mathcal{SK}_{t+1}$ such that $p_{t+1}^{*}(\cdot,\omega^{t})\in\mathcal{E}_{t+1}(\omega^{t})$ for all $\omega^{t}\in\Omega^{t}_{NA}$ . Let $P^{*}:=p^{*}_{1}\otimes\cdots\otimes p^{*}_{T}.$ Then, $P^{*}\in\mathcal{Q}^{T}$ (see (2)), (7) implies that

\mbox{Aff}\left(D_{P^{*}}^{t+1}\right)(\omega^{t})=\mbox{Aff}\left(E^{t+1}\right)(\omega^{t},p^{*}_{t+1}(\cdot,\omega^{t}))\subset\mbox{Aff}\left(D^{t+1}\right)(\omega^{t})=\mbox{Aff}\left(E^{t+1}\right)(\omega^{t},p^{*}_{t+1}(\cdot,\omega^{t}))

and $0\in\mbox{Ri}\left({\mbox{Conv}}(D_{P^{*}}^{t+1})\right)(\omega^{t})$ for all $\omega^{t}\in\Omega_{NA}^{t}$ .
So it remains to prove the existence of $p^{*}_{t+1}$ . Fix some $0\leq t\leq T-1$ and let

	$\displaystyle B$	$\displaystyle:=\{(\omega^{t},p)\in\Omega^{t}\times\mathfrak{P}(\Omega_{t+1}),\;\mbox{Ri}\left(\overline{\mbox{Conv}}(E^{t+1})\right)(\omega^{t},p)\cap\{0\}\neq\emptyset\},$
	$\displaystyle C$	$\displaystyle:=\{(\omega^{t},p)\in\Omega^{t}\times\mathfrak{P}(\Omega_{t+1}),\;\mbox{Aff}\left(E^{t+1}\right)(\omega^{t},p)=\mbox{Aff}\left(D^{t+1}\right)(\omega^{t})\}.$

[Artstein, 1972, Lemma 5.6] and Lemma 2.6 show that $\mbox{Ri}\left(\overline{\mbox{Conv}}(E^{t+1})\right)$ is $\mathcal{B}(\Omega^{t})\otimes\mathcal{B}(\mathfrak{P}(\Omega_{t+1}))$ -measurable and $B\in\mathcal{B}(\Omega^{t})\otimes\mathcal{B}(\mathfrak{P}(\Omega_{t+1}))$ follows. Let $h$ be defined by

h(\omega^{t},p):=d\left(\mbox{Aff}\left(E^{t+1}\right)(\omega^{t},p),\mbox{Aff}\left(D^{t+1}\right)(\omega^{t})\right).

Note that $C=\{h^{-1}(0)\}$ . Then [Aliprantis and Border, 2006, Theorem 18.5 p595] and Lemma 2.6 show that $x\in\mathbb{R}^{d}$ $(\omega^{t},p)\to d\left(x,\mbox{Aff}\left(E^{t+1}\right)(\omega^{t},p)\right)$ is $\mathcal{B}(\Omega^{t})\otimes\mathcal{B}(\mathfrak{P}(\Omega_{t+1}))$ measurable and $\omega^{t}\to d\left(x,\mbox{Aff}\left(D^{t+1}\right)(\omega^{t})\right)$ is $\mathcal{B}_{c}(\Omega^{t})$ measurable. They also show
$x\to|d\left(x,\mbox{Aff}\left(E^{t+1}\right)(\omega^{t},p)\right)-d\left(x,\mbox{Aff}\left(D^{t+1}\right)(\omega^{t})\right)|$ is continuous. Thus

\displaystyle h(\omega^{t},p)=\sup_{x\in\mathbb{Q}^{d}}|d\left(x,\mbox{Aff}\left(E^{t+1}\right)(\omega^{t},p)\right)-d\left(x,\mbox{Aff}\left(D^{t+1}\right)(\omega^{t})\right)|

(46)

and $h$ is $\mathcal{B}_{c}(\Omega^{t})\otimes\mathcal{B}(\mathfrak{P}(\Omega_{t+1}))$ measurable. It follows that $C\in\mathcal{B}_{c}(\Omega^{t})\otimes\mathcal{B}(\mathfrak{P}(\Omega_{t+1})).$ [Rockafellar, 1970, Theorem 6.3 p46], Assumption 2.2 and Lemma 5.9 show that

\mbox{graph}\left(\mathcal{E}_{t+1}\right)=\mbox{graph}\left(\mathcal{Q}_{t+1}\right)\cap B\cap C\in\mathfrak{A}\left(\mathcal{B}_{c}(\Omega^{t})\otimes\mathfrak{P}(\Omega_{t+1})\right),

where for some Polish space $X$ and some paving $\mathcal{J}$ (i.e. a non-empty collection of subsets of $X$ containing the empty set), $\mathfrak{A}(\mathcal{J})$ denotes the set of all nuclei of Suslin Scheme on $\mathcal{J}$ (see [Bertsekas and Shreve, 2004, Definition 7.15 p157]). Now [Bouchard and Nutz, 2015, Lemma 4.11] (which relies on [Leese, 1978]) gives the existence of $p_{t+1}^{*}\in\mathcal{SK}_{t+1}$ such that $p_{t+1}^{*}(\cdot,\omega^{t})\in\mathcal{E}_{t+1}(\omega^{t})$ for all $\omega^{t}\in\Omega^{t}_{NA}=\{\mathcal{E}_{t+1}\neq\emptyset\}$ . The proof is complete.

The following lemma was used in the previous proof.

Lemma 5.9.

Let $X,Y$ be two Polish spaces. Let $\Gamma_{1}\in\mathcal{A}(X\times Y)$ and $\Gamma_{2}\in\mathcal{B}_{c}(X)\otimes\mathcal{B}(Y)$ . Then $\Gamma_{1}\cap\Gamma_{2}\in\mathfrak{A}\left(\mathcal{B}_{c}(X)\otimes\mathcal{B}(Y)\right)$ .

Proof.

[Bertsekas and Shreve, 2004, Proposition 7.35 p158, Proposition 7.41 p166] imply that $\Gamma_{1}\in\mathcal{A}(X\times Y)=\mathfrak{A}(\mathcal{B}(X)\otimes\mathcal{B}(Y))\subset\mathfrak{A}\left(\mathcal{B}_{c}(X)\otimes\mathcal{B}(Y)\right)$ and $\Gamma_{2}\in\mathcal{B}_{c}(X)\otimes\mathcal{B}(Y)\subset\mathfrak{A}\left(\mathcal{B}_{c}(X)\otimes\mathcal{B}(Y)\right)$ and thus $\Gamma_{1}\cap\Gamma_{2}\in\mathfrak{A}\left(\mathcal{B}_{c}(X)\otimes\mathcal{B}(Y)\right)$ .

5.2.4 Proof of Theorem 3.8

Proof.

Step 1: Reverse implication.
Lemma 3.7 implies that the $NA(\mathcal{P}^{T})$ condition holds true and Lemma 3.2 shows that the $NA(\mathcal{Q}^{T})$ is satisfied.

Step 2: Direct implication.
Theorem 3.30 implies that there exists some $P^{*}\in\mathcal{Q}^{T}$ with the fixed disintegration $P^{*}:=P_{1}^{*}\otimes p_{2}^{*}\otimes\cdots\otimes p_{T}^{*}$ such that $\mbox{Aff}\left(D_{P^{*}}^{t+1}\right)(\omega^{t})=\mbox{Aff}\left(D^{t+1}\right)(\omega^{t})$ and $0\in\mbox{Ri}\left({\mbox{Conv}}\left(D_{P^{*}}^{t+1}\right)\right)(\omega^{t})$ for all $\omega^{t}\in\Omega^{t}_{NA}$ and all $0\leq t\leq T-1$ . The direct implication holds true if i), ii) and iii) below are proved¹¹¹¹11Note that $i$ ) and $ii)$ are true if we only assume that $P^{*}\in\mathcal{Q}^{T}$ ..
i) $\mathcal{P}^{t}\subset\mathcal{Q}^{t}$ for all $1\leq t\leq T$ .
This follows by induction from (10), $p^{*}_{t+1}(\cdot,\omega^{t})\in\mathcal{Q}_{t+1}(\omega^{t})$ and the convexity of $\mathcal{Q}_{t+1}(\omega^{t})$ .
ii) $\mathcal{Q}^{t}$ and $\mathcal{P}^{t}$ have the same polar-sets for all $1\leq t\leq T$ .
Fix some $1\leq t\leq T$ . As $\mathcal{P}^{t}\subset\mathcal{Q}^{t}$ , it is clear that a $\mathcal{Q}^{t}$ -polar set is also a $\mathcal{P}^{t}$ -polar set. To establish the other inclusion, we prove by induction that for all $1\leq t\leq T$ and all $Q^{t}\in\mathcal{Q}^{t}$ , there exists some $(\lambda_{1}^{t},\cdots,\lambda_{2t}^{t})\in(0,1]^{2t}$ such that $\sum_{i=1}^{2t}\lambda_{i}^{t}=1$ and some $\left(R_{i}^{t}\right)_{3\leq i\leq 2t}\subset\mathcal{Q}^{t}$ (if $t\geq 2$ ) such that

\displaystyle Q^{t}<<P^{t}:=\lambda_{1}^{t}P^{*t}+\lambda_{2}^{t}Q^{t}+\sum_{i=3}^{2t}\lambda_{i}^{t}R_{i}^{t}\in\mathcal{P}^{t}.

(47)

For $t=1$ , let $Q_{1}\in\mathcal{Q}^{1}$ and $P_{1}:=(P_{1}^{*}+Q_{1})/2$ . Then, $Q_{1}<<P_{1}$ and $P_{1}\in\mathcal{P}^{1},$ see (10). Now assume that the property is true for some $t\geq 1.$ Let $Q^{t+1}\in\mathcal{Q}^{t+1}$ with the fixed disintegration $Q^{t+1}:=Q^{t}\otimes q_{t+1}$ where $Q^{t}\in\mathcal{Q}^{t}$ and $q_{t+1}(\cdot,\omega^{t})\in\mathcal{Q}_{t+1}(\omega^{t})$ for all $\omega^{t}\in\Omega^{t}$ . Then, there exists some $\left(R_{i}^{t}\right)_{3\leq i\leq 2t}\subset\mathcal{Q}^{t}$ , $(\lambda_{1}^{t},\cdots,\lambda_{2t}^{t})\in(0,1]^{2t}$ such that (47) holds true. Let

$\displaystyle P^{t+1}$	$\displaystyle:=$	$\displaystyle P^{t}\otimes\frac{1}{2}(p_{t+1}^{*}+q_{t+1})$
$\displaystyle R_{i}^{t+1}$	$\displaystyle:=$	$\displaystyle R_{i}^{t}\otimes\frac{1}{2}(p^{*}_{t+1}+q_{t+1})\quad\quad\lambda_{i}^{t+1}:=\lambda_{i}^{t}\quad\quad\forall 3\leq i\leq 2t$
$\displaystyle R_{2t+1}^{t+1}$	$\displaystyle:=$	$\displaystyle Q^{t}\otimes p^{}_{t+1}\quad\quad R_{2t+2}^{t+1}:=P^{t}\otimes q_{t+1}$
$\displaystyle\lambda_{1}^{t+1}$	$\displaystyle:=$	$\displaystyle\frac{\lambda_{1}^{t}}{2}\quad\quad\lambda_{2}^{t+1}:=\frac{\lambda_{2}^{t}}{2}\quad\quad\lambda_{2t+1}^{t+1}:=\frac{\lambda_{2}^{t}}{2}\quad\quad\lambda_{2t+2}^{t+1}:=\frac{\lambda_{1}^{t}}{2}.$

Then $P^{t+1}\in\mathcal{P}^{t+1}$ (see (10)), $(R_{i}^{t+1})_{3\leq i\leq 2(t+1)}\subset\mathcal{Q}^{t+1},$ $\sum_{i=1}^{2(t+1)}\lambda_{i}^{t+1}=1$ and

\displaystyle P^{t+1}=\lambda_{1}^{t+1}P^{*t+1}+\lambda_{2}^{t+1}Q^{t+1}+\sum_{i=3}^{2(t+1)}\lambda_{i}^{t+1}R_{i}^{t+1}.

As $Q^{t+1}<<P^{t+1},$ the induction is proven.

iii) The $sNA(\mathcal{P}^{T})$ condition holds true.
Fix some $P\in\mathcal{P}^{T}\subset\mathcal{Q}^{T}$ , some $0\leq t\leq T-1$ and $\omega^{t}\in\Omega_{NA}^{t}$ . We establish that $0\in\mbox{Ri}\left({\mbox{Conv}}\left(D_{P}^{t+1}\right)\right)(\omega^{t})$ . Then $P^{t}\left(\Omega^{t}_{NA}\right)=1$ and Proposition 3.26 shows that $NA(P)$ holds true and $iii)$ follows. Remark 2.5 and (10) imply that ${D}_{P^{*}}^{t+1}(\omega^{t})\subset{D}_{P}^{t+1}(\omega^{t})\subset{D}^{t+1}(\omega^{t})$ . Thus, $0\in{\mbox{Conv}}(D_{P^{*}}^{t+1})(\omega^{t})\subset{\mbox{Conv}}(D_{P}^{t+1})(\omega^{t})$ . We have that

\mbox{Aff}\left(D^{t+1}\right)(\omega^{t})=\mbox{Aff}\left(D_{P^{*}}^{t+1}\right)(\omega^{t})\subset\mbox{Aff}\left(D_{P}^{t+1}\right)(\omega^{t})\subset\mbox{Aff}\left(D^{t+1}\right)(\omega^{t}).

As $0\in\mbox{Ri}\left({\mbox{Conv}}(D_{P^{*}}^{t+1})\right)(\omega^{t})$ , there exists some $\varepsilon>0$ such that

\displaystyle B(0,\varepsilon)\bigcap\mbox{Aff}\left(D_{P}^{t+1}\right)(\omega^{t})

\displaystyle=B(0,\varepsilon)\bigcap\mbox{Aff}\left(D_{P^{*}}^{t+1}\right)(\omega^{t})\subset{\mbox{Conv}}(D_{P^{*}}^{t+1})(\omega^{t})\subset{\mbox{Conv}}(D_{P}^{t+1})(\omega^{t}).

5.2.5 Proof of Proposition 3.37

Proof.

Theorem 3.30 implies that there exists some $P^{*}\in\mathcal{Q}^{T}$ with the fixed disintegration $P^{*}=P_{1}^{*}\otimes p_{2}^{*}\otimes\cdots\otimes p_{T}^{*}$ such that $\mbox{Aff}\left(D_{P^{*}}^{t+1}\right)(\omega^{t})=\mbox{Aff}\left(D^{t+1}\right)(\omega^{t})$ and $0\in\mbox{Ri}\left({\mbox{Conv}}(D_{P^{*}}^{t+1})\right)(\omega^{t})$ for all $\omega^{t}\in\Omega^{t}_{NA}$ and all $0\leq t\leq T-1$ . To find a $\mathcal{B}_{c}(\Omega^{t})$ -measurable version of $\beta_{t}$ and ${\kappa}_{t}$ in (20) we follow the same idea as in [Blanchard et al., 2018, Proposition 3.7]. Fix some $0\leq t\leq T-1$ . Set $n_{t}(\omega^{t}):=\inf\{n\geq 1,\;A^{P^{*}}_{n}(\omega^{t})=\emptyset\}$ where for all $n\geq 1$ $A^{P^{*}}_{n}(\omega^{t})=\emptyset$ if $\omega^{t}\notin\Omega^{t}_{NA}$ and if $\omega^{t}\in\Omega^{t}_{NA}$ ,

\displaystyle A^{P^{*}}_{n}(\omega^{t}):=\left\{h\in\mbox{Aff}\left(D_{P^{*}}^{t+1}\right)(\omega^{t}),\;|h|=1,\,p_{t+1}^{*}\left(h\Delta S_{t+1}(\omega^{t},\cdot)<-\frac{1}{n},\omega^{t}\right)<\frac{1}{n}\right\}.

(48)

For all $\omega^{t}\in\Omega^{t}$ , as in the proof of Proposition 5.7, $n_{t}(\omega^{t})<\infty$ and one may set $\kappa_{t}(\omega^{t})={\beta}_{t}(\omega^{t}):={1}/{n_{t}(\omega^{t})}\in(0,1)$ . Then, by definition of $A^{P^{*}}_{n}$ , (20) is true with $P_{h}(\cdot)=p^{*}_{t+1}(\cdot,\omega^{t})\in\mathcal{Q}_{t+1}(\omega^{t})$ since $\mbox{Aff}\left(D_{P^{*}}^{t+1}\right)(\omega^{t})=\mbox{Aff}\left(D^{t+1}\right)(\omega^{t})$ for all $\omega^{t}\in\Omega^{t}_{NA}$ .
To prove that $\kappa_{t}={\beta}_{t}$ is $\mathcal{B}_{c}(\Omega^{t})$ -measurable, we show that $\{A^{P^{*}}_{n}\neq\emptyset\}\in\mathcal{B}_{c}(\Omega^{t})$ since for all $k\geq 1$ ,

\{n_{t}\geq k\}=\Omega^{t}_{NA}\cap\left(\bigcap_{1\leq j\leq k-1}\{A^{P^{*}}_{j}=\emptyset\}\right).

Fix some $n\geq 1.$ As $p_{t+1}^{*}$ is only universally-measurable, we use Lemma 5.10 to prove that $\{A^{P^{*}}_{n}\neq\emptyset\}\in\mathcal{B}_{c}(\Omega^{t})$ . Fix $P\in\mathfrak{P}(\Omega^{t})$ . First, applying [Bertsekas and Shreve, 2004, Lemma 7.28 p173], there exists $p_{t+1}^{P}$ a Borel-measurable stochastic kernel on $\Omega_{t+1}$ given $\Omega^{t}$ and $\Omega^{t}_{P}\in\mathcal{B}(\Omega^{t})$ such that $P(\Omega^{t}_{P})=1$ and $p_{t+1}^{P}(\cdot,\omega^{t})=p_{t+1}^{*}(\cdot,\omega^{t})$ for all $\omega^{t}\in\Omega^{t}_{P}$ . Set $A^{P}_{n}$ as in (48) replacing $p^{*}_{t+1}$ with $p^{P}_{t+1}$ if $\omega^{t}\in\Omega^{t}_{NA}$ (and $A^{P}_{n}(\omega^{t})=\emptyset$ if $\omega^{t}\notin\Omega^{t}_{NA}$ ). Then

\{A^{P^{*}}_{n}\neq\emptyset\}\cap\Omega^{t}_{P}=\{A^{P}_{n}\neq\emptyset\}\cap\Omega^{t}_{P}

and it remains to establish that $\{A^{P}_{n}\neq\emptyset\}\in\mathcal{B}_{c}(\Omega^{t})$ . Remark that

\mbox{graph}\left(A^{P}_{n}\right)=\mbox{graph}\left(\mbox{Aff}\left(D_{P^{*}}^{t+1}\right)\right)\bigcap\left\{(\omega^{t},h),\;|h|=1,\;p_{t+1}^{P}\left(h\Delta S_{t+1}(\omega^{t},\cdot)<-\frac{1}{n},\omega^{t}\right)<\frac{1}{n}\right\}.

Lemma 2.6 implies that $\mbox{graph}\left(\mbox{Aff}\left(D_{P^{*}}^{t+1}\right)\right)\in\mathcal{B}_{c}(\Omega^{t})\otimes\mathcal{B}(\mathbb{R}^{d}).$ As $(\omega^{t},h,\omega_{t+1})\to h\Delta S_{t+1}(\omega^{t},\omega_{t+1})$ and $p_{t+1}^{P}$ are Borel-measurable, [Bertsekas and Shreve, 2004, Proposition 7.29 p144] implies that $(\omega^{t},h)\to p_{t+1}^{P}\left(h\Delta S_{t+1}(\omega^{t},\cdot)<-{1}/{n},\omega^{t}\right)$ is $\mathcal{B}(\Omega^{t})\otimes\mathcal{B}(\mathbb{R}^{d})$ -measurable. Thus, applying the Projection Theorem, $\mbox{Proj}_{\Omega^{t}}\left(\mbox{graph}\left(A^{P}_{n}\right)\right)=\{A^{P}_{n}\neq\emptyset\}\in\mathcal{B}_{c}(\Omega^{t})$ and the proof is complete.

Lemma 5.10.

Let $X$ be a Polish space. Let $A\subset X$ . Assume that for all $P\in\mathfrak{P}(X)$ there exists some $A_{P}\in\mathcal{B}_{c}(X)$ and some $P$ -full measure set $X_{P}\in\mathcal{B}(X)$ such that $A\cap X_{P}=A_{P}\cap X_{P}$ . Then $A\in\mathcal{B}_{c}(X)$ .

Proof.

Fix some $P\in\mathfrak{P}(X).$ We show that $A\in\mathcal{B}_{P}(X),$ the completion of $\mathcal{B}(X)$ with respect to $P.$ As this is true for all $P\in\mathfrak{P}(X)$ , $A\in\mathcal{B}_{c}(X)$ will follow.
There exists $A_{P}\in\mathcal{B}_{c}(X)$ and $X_{P}\in\mathcal{B}(X)$ such that $P(X_{P})=1$ and $A\cap X_{P}=A_{P}\cap X_{P}$ . As $A_{P}\cap X_{P}\in\mathcal{B}_{c}(\Omega^{t})\subset\mathcal{B}_{P}(X)$ there exists a $P$ -negligible set $N_{P}$ and $\tilde{A}_{P}\in\mathcal{B}(X)$ such that $A_{P}\cap X_{P}=\tilde{A}_{P}\cup N_{P}$ . Now, let $M_{P}:=A\cap\left(X\backslash{X_{P}}\right)\subset X\backslash{X_{P}}$ . As $X\backslash{X_{P}}\in\mathcal{B}(X)$ and $P(X\backslash{X_{P}})=0$ , $M_{P}$ is a $P$ -negligible set and

A=\left(A\cap X_{P}\right)\cup\left(A\cap\left(X\backslash{X_{P}}\right)\right)=\tilde{A}_{P}\cup N_{P}\cup M_{P}\in\mathcal{B}_{P}(X).

5.2.6 Proof of Proposition 3.39

Proof.

Lemma 3.7 implies that $NA(\widehat{P})$ and $NA(\mathcal{Q}^{T})$ are equivalent. Fix some disintegration of $\widehat{P}\in\mathcal{Q}^{T},$ $\widehat{P}:=\widehat{P}_{1}\otimes\widehat{p}_{2}\otimes\cdots\otimes\widehat{p}_{T}$ and some $1\leq t\leq T$ . As $\widehat{P}^{t}$ dominates $\mathcal{Q}^{t}$ Proposition 3.26 implies that

0\in\mbox{Ri}\left(\mbox{Conv}\left(D_{\widehat{P}}^{t+1}\right)\right)(\cdot)\;\;\mathcal{Q}^{t}\mbox{-q.s.}

Lemma 5.12 below provides a $\mathcal{Q}^{t}$ -full measure set $\Omega^{t}\backslash{\Omega^{t}_{nd}}$ such that $\widehat{p}_{t+1}(\cdot,\omega^{t})$ dominates $\mathcal{Q}_{t+1}(\omega^{t})$ for all $\omega^{t}\in\Omega^{t}\backslash{\Omega^{t}_{nd}}$ . Thus ${D}^{t+1}(\omega^{t})\subset{D}_{\widehat{P}}^{t+1}(\omega^{t})$ and the equality follows from (7) as $\widehat{P}\in\mathcal{Q}^{T}.$

5.3 Proof of Proposition 4.1

The proof of Proposition 4.1 follows directly from Lemma 5.12. Indeed assume that the set $\mathcal{Q}^{T}$ is dominated. As $\Omega^{t}_{N}\subset\Omega^{t}_{nd}$ , $\Omega^{t}_{N}$ is a $\mathcal{Q}^{t}$ -polar set which contradicts $\widetilde{P}^{t}(\Omega^{t}_{N})>0$ .

The proof of Lemma 5.12 is fairly technical and needs the introduction of the Wijsman topology as well as Lemma 5.11. Note that the reverse implication in Proposition 4.1 seems intuitive but raises challenging technical issues.
Let $(X,d)$ be a Polish space and $\mathcal{F}$ be the set of non-empty closed subsets of $X$ . The Wijsman topology on $\mathcal{F}$ denoted by $\mathcal{T}_{W}$ is such that

F_{n}\underset{n\rightarrow+\infty}{\overset{\tau_{w}}{\longrightarrow}}F\iff d(x,F_{n})\underset{n\rightarrow+\infty}{\longrightarrow}d(x,F)\;\mbox{for all $x\in X$,}

where $d(x,F):=\inf\{d(x,f),\;f\in F\}$ . Note that $\mathcal{F}$ endowed with $\mathcal{T}_{W}$ is a Polish space (see [Beer, 1991]).

Lemma 5.11.

The function $(F,x)\in\mathcal{F}\times X\to 1_{F}(x)$ is $\mathcal{B}(\mathcal{F})\otimes\mathcal{B}(X)$ -measurable.

Proof.

The function $d:(x,F)\in X\times\mathcal{F}\to d(x,F)$ is separately continuous. Indeed for all fixed $x\in X$ , $d(x,\cdot)$ is continuous by definition of $\mathcal{T}_{W}$ and [Aliprantis and Border, 2006, Theorem 3.16] implies that $d(\cdot,F)$ is continuous for all fixed $F\in\mathcal{F}$ . Using [Aliprantis and Border, 2006, Lemma 4.51 p153] $d$ is $\mathcal{B}(X)\otimes\mathcal{B}(\mathcal{F})$ -measurable. We conclude since $x\in F$ if and only if $d(x,F)=0$ .

Lemma 5.12.

Assume that Assumption 2.2 holds true and that $\mathcal{Q}^{T}$ is dominated by $\widehat{P}\in\mathfrak{P}(\Omega^{T})$ with the fix disintegration $\widehat{P}:=\widehat{P}_{0}\otimes\widehat{p}_{1}\otimes\cdots\otimes\widehat{p}_{T}$ where $\hat{p}_{t}\in\mathcal{SK}_{t}$ for all $1\leq t\leq T$ . Then

\Omega^{t}_{nd}:=\left\{\omega^{t}\in\Omega^{t},\;\mbox{$\mathcal{Q}_{t+1}(\omega^{t})$ is not dominated by $\widehat{p}_{t+1}(\cdot,\omega^{t})$}\right\}\in\mathcal{B}_{c}(\Omega^{t})

and is a $\mathcal{Q}^{t}$ -polar set for all $0\leq t\leq T-1$ .

Proof.

Fix some $0\leq t\leq T-1$ . We proceed in two steps.
Step 1: $\Omega^{t}_{nd}\in\mathcal{B}_{c}(\Omega^{t}).$
To prove Step $1$ , we use Lemma 5.10 and fix $R\in\mathfrak{P}(\Omega^{t})$ . Applying [Bertsekas and Shreve, 2004, Lemma 7.28 p174], there exists $p^{R}_{t+1}$ a Borel-measurable stochastic kernel on $\Omega_{t+1}$ given $\Omega^{t}$ and a $R$ -full-measure set $\Omega^{t}_{R}\in\mathcal{B}(\Omega^{t})$ such that

\displaystyle p^{R}_{t+1}(\cdot,\omega^{t})=\widehat{p}_{t+1}(\cdot,\omega^{t})\;\mbox{ for all $\omega^{t}\in\Omega^{t}_{R}$.}

(49)

Let $\mathcal{F}_{t+1}$ be the set of non-empty and closed subsets of $\Omega_{t+1}$ and let $N^{R}_{t}:\Omega^{t}\twoheadrightarrow\mathfrak{P}(\Omega_{t+1})\times\mathcal{F}_{t+1}$ be defined for all $\omega^{t}\in\Omega^{t}$ by

\displaystyle N^{R}_{t}(\omega^{t}):=\left\{(q,F)\in\mathfrak{P}(\Omega_{t+1})\times\mathcal{F}_{t+1},q\in\mathcal{Q}_{t+1}(\omega^{t}),\;p^{R}_{t+1}\left(F,\omega^{t}\right)=0,\;q\left(F\right)>0\right\}.

(50)

We first claim that

\displaystyle\Omega^{t}_{nd}\cap\Omega^{t}_{R}=\{N^{R}_{t}\neq\emptyset\}\cap\Omega^{t}_{R}.

(51)

Let $\omega^{t}\in\Omega^{t}_{nd}\cap\Omega^{t}_{R}$ . As $\mathcal{Q}_{t+1}(\omega^{t})$ is not dominated by $\widehat{p}_{t+1}(\cdot,\omega^{t})=p^{R}_{t+1}(\cdot,\omega^{t})$ , there exists some $q\in\mathcal{Q}_{t+1}(\omega^{t})$ and some $A\in\mathcal{B}(\Omega^{t})$ such that $p^{R}_{t+1}(A,\omega^{t})=0$ and $q(A)>0$ . As $q\in\mathfrak{P}(\Omega_{t+1})$ is inner-regular (see [Aliprantis and Border, 2006, Definition 12.2 p435, Theorem 12.7 p438, Lemma 12.3 p435]), there exists some $F\in\mathcal{F}_{t+1}$ , $F\subset A$ such that $q(F)>0$ and $(q,F)\in N_{t}^{R}(\omega^{t})$ follows. The reverse inclusion is clear.
Thus Lemma 5.10 applies and Step 1 is completed if $\{N^{R}_{t}\neq\emptyset\}=\mbox{Proj}_{\Omega^{t}}\left(\mbox{graph}\left(N^{R}_{t}\right)\right)\in\mathcal{B}_{c}(\Omega^{t}).$ This will follows from Jankov-von Neumann Theorem (see [Bertsekas and Shreve, 2004, Proposition 7.49 p182]) if

\displaystyle\mbox{graph}(N^{R}_{t})\in\mathcal{A}\left(\Omega^{t}\times\mathfrak{P}(\Omega_{t+1})\times\mathcal{F}_{t+1}\right).

(52)

This follows from $\mbox{graph}(N^{R}_{t})=A\cap B\cap C$ where

	$\displaystyle A$	$\displaystyle:=\mbox{graph}(\mathcal{Q}_{t+1})\times\mathcal{F}_{t+1}\in\mathcal{A}\left(\Omega^{t}\times\mathfrak{P}(\Omega_{t+1})\times\mathcal{F}_{t+1}\right),$
	$\displaystyle B$	$\displaystyle:=\{(\omega^{t},q,F),\;p^{R}_{t+1}(F,\omega^{t})=0\}\in\mathcal{B}(\Omega^{t})\otimes\mathcal{B}(\mathfrak{P}(\Omega_{t+1})\otimes\mathcal{B}(\mathcal{F}_{t+1}),$
	$\displaystyle C$	$\displaystyle:=\{(\omega^{t},q,F),\;q(F)>0\}\in\mathcal{B}(\Omega^{t})\otimes\mathcal{B}(\mathfrak{P}(\Omega_{t+1})\otimes\mathcal{B}(\mathcal{F}_{t+1}),$

see Assumption 2.2 for the measurability of $A$ . For $B$ and $C$ , Lemma 5.11 together with [Bertsekas and Shreve, 2004, Proposition 7.29 p144] imply that $(\omega^{t},q,F)\to p^{R}_{t+1}(F,\omega^{t})$ and $(\omega^{t},q,F)\to q(F)$ are Borel-measurables (recall that $p_{t+1}^{R}(d\omega_{t+1}|\omega^{t},q,F)=p^{R}_{t+1}(d\omega_{t+1},\omega^{t})$ and $q(d\omega_{t+1}|\omega^{t},q,F)=q(d\omega_{t+1})$ are Borel-measurable stochastic kernels).

Step 2: $\Omega^{t}_{nd}$ is a $\mathcal{Q}^{t}$ -polar set.
We proceed by contradiction and assume that there exists some $\overline{P}\in\mathcal{Q}^{T}$ such that $\overline{P}^{t}(\Omega^{t}_{nd})>0$ . We choose $R=\widehat{P}^{t}$ in (49) and (50) and we denote by

\Omega^{t}_{nd1}:=\Omega^{t}_{nd}\cap\Omega^{t}_{\widehat{P}^{t}}=\{N^{\widehat{P}^{t}}_{t}\neq\emptyset\}\cap\Omega^{t}_{\widehat{P}^{t}}\in\mathcal{B}_{c}(\Omega^{t}),

see (51) and Step 1. The Jankov-von Neumann Theorem and (52) also give the existence of ${q}^{\widehat{P}}_{t+1}$ a universally-measurable stochastic kernel on $\Omega_{t+1}$ given $\Omega^{t}$ and a universally measurable function ${F}^{\widehat{P}}_{t+1}:\Omega^{t}\to\mathcal{F}_{t+1}$ such that $({q}^{\widehat{P}}_{t+1}(\cdot,\omega^{t}),F^{\widehat{P}}_{t+1}(\omega^{t}))\in N^{\widehat{P}^{t}}_{t}(\omega^{t})$ for all $\omega^{t}\in\Omega^{t}_{nd1}$ . For $\omega^{t}\notin\ \Omega^{t}_{nd1}$ we set $F^{\widehat{P}}_{t+1}(\omega^{t})=\emptyset$ and ${q}^{\widehat{P}}_{t+1}(\cdot,\omega^{t})=q_{t+1}(\cdot,\omega^{t})$ where $q_{t+1}$ is a given universally-measurable selector of $\mathcal{Q}_{t+1}$ .
Note that as $\widehat{P}^{t}$ dominates $\mathcal{Q}^{t}$ , $1=\widehat{P}^{t}(\Omega^{t}_{\widehat{P}^{t}})=\overline{P}^{t}(\Omega^{t}_{\widehat{P}^{t}})$ and $\overline{P}^{t}(\Omega^{t}_{nd1})>0$ .
We now build some $\widehat{Q}\in\mathcal{Q}^{T}$ , $E\in\mathcal{B}_{c}(\Omega^{t+1})$ such that $\widehat{P}^{t+1}(E)=0$ but $\widehat{Q}^{t+1}(E)>0$ which contradicts the fact that $\widehat{P}$ dominates $\mathcal{Q}^{T}$ . Let

	$\displaystyle\widehat{Q}$	$\displaystyle:=\overline{P}^{t}\otimes{q}^{\widehat{P}}_{t+1}\otimes\overline{p}_{t+2}\otimes\cdots\otimes\overline{p}_{T}\in\mathcal{Q}^{T},$
	$\displaystyle E$	$\displaystyle:=\left\{(\omega^{t},\omega_{t+1})\in\Omega^{t}\times\Omega_{t+1},\;\omega^{t}\in\Omega^{t}_{nd1},\;\omega_{t+1}\in F^{\widehat{P}}_{t+1}(\omega^{t})\right\}=\varphi^{-1}(\{1\})\cap\left(\Omega^{t}_{nd1}\times\Omega_{t+1}\right),$
	$\displaystyle\varphi(\omega^{t},\omega_{t+1})$	$\displaystyle:=1_{F^{\widehat{P}}_{t+1}(\omega^{t})}(\omega_{t+1}).$

Lemma 5.11 implies that $(F,\omega_{t+1})\to 1_{F}(\omega_{t+1})$ is $\mathcal{B}(\mathcal{F}_{t+1})\otimes\mathcal{B}(\Omega_{t+1})$ -measurable and as $(\omega^{t},\omega_{t+1})\to(F^{\widehat{P}}_{t+1}(\omega^{t}),\omega_{t+1})$ is $\mathcal{B}_{c}(\Omega^{t+1})$ -measurable, $\varphi$ is $\mathcal{B}_{c}(\Omega^{t+1})$ -measurable by composition. Thus $E$ belong to $\mathcal{B}_{c}(\Omega^{t+1})$ . Let $\left(E\right)_{\omega^{t}}:=\{\omega_{t+1}\in\Omega_{t+1},\;(\omega^{t},\omega_{t+1})\in E\}$ , then

\displaystyle\widehat{P}^{t+1}(E)=\int_{\Omega_{nd1}^{t}}\widehat{p}_{t+1}\left(\left(E\right)_{\omega^{t}},\omega^{t}\right)\widehat{P}^{t}(d\omega^{t})

\displaystyle=\int_{\Omega_{nd1}^{t}}\widehat{p}_{t+1}\left(F^{\widehat{P}}_{t+1}(\omega^{t}),\omega^{t}\right)\widehat{P}^{t}(d\omega^{t})=0

where we have used that for $\omega^{t}\notin\Omega_{nd1}^{t}$ $\left(E\right)_{\omega^{t}}=\emptyset$ and for $\omega^{t}\in\Omega^{t}_{nd1}$ $\left(E\right)_{\omega^{t}}=F^{\widehat{P}}_{t+1}(\omega^{t})$ and that $\widehat{p}_{t+1}\left(F^{\widehat{P}}_{t+1}(\omega^{t}),\omega^{t}\right)=p^{\widehat{P}^{t}}_{t+1}\left(F^{\widehat{P}}_{t+1}(\omega^{t}),\omega^{t}\right)=0$ . But

\displaystyle\widehat{Q}^{t+1}(E)

\displaystyle=\int_{\Omega_{nd1}^{t}}{q}^{\widehat{P}}_{t+1}\left(\left(E\right)_{\omega^{t}},\omega^{t}\right)\overline{P}^{t}(d\omega^{t})=\int_{\Omega_{nd1}^{t}}{q}^{\widehat{P}}_{t+1}\left(F^{\widehat{P}}_{t+1}(\omega^{t}),\omega^{t}\right)\overline{P}^{t}(d\omega^{t})>0

since $\overline{P}^{t}\left(\Omega^{t}_{nd1}\right)>0$ and ${q}^{\widehat{P}}_{t+1}\left(F^{\widehat{P}}_{t+1}(\omega^{t}),\omega^{t}\right)>0$ for all $\omega^{t}\in\Omega^{t}_{nd1}$ . This concludes the proof.

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