HJB equation for maximization of wealth under insider trading

The theory of enlargement of a filtration was initiated in 1976 by Itô [6]. This author has pointed out that one way to extend the domain of the stochastic integral (in the Itô sense) with respect to an $\mathbf{F}$-martingale $Y$ is to enlarge the filtration $\mathbf{F}$ to another filtration $\mathbf{G}$ in such a way that $Y$ remains a semimartingale with respect to the new and bigger filtration $\mathbf{G}$. In this way, we can now integrate processes that are $\mathbf{G}$-adapted, which include processes that are not necessarily adapted to the underlying filtration $\mathbf{F}$. In particular, Itô [6] shows that if $\mathbf{G}_{1}$ and $\mathbf{G}_{2}$ are two filtrations such that $\mathbf{G}_{1}\subset\mathbf{G}_{2}$ and $Y$ is semimartingale with respect to both filtrations, then the stochastic integrals with respect to the $\mathbf{G}_{1}$ and $\mathbf{G}_{2}$ semimartingale $Y$ are the same in the intersection of the domains of both integrals. But, this problem have not been considered in [6] when $\mathbf{G}_{1}\not\subset\mathbf{G}_{2}$ and $\mathbf{G}_{2}\not\subset\mathbf{G}_{1}$. This problem has been solved by Russo and Vallois [17] using the forward integral. The forward integral is a limit in probability and agrees with the Itô integral if the integrator is a semimartingale (see Section 4 and Remark 4.2), which answer the problem that Itô did not address. So, the forward integral is an anticipating integral, that is, it allows us to integrate processes that are not adapted to the underlying filtration with respect to other processes that are not necessarily semimartingales, therefore the forward integral coincides with the Itô’s integral if this last one is well-defined for the filtration $\mathbf{G}$. In consequence, the forward integral becomes an appropriate tool to deal with problems that involve processes that are no adapted to the underlying filtration. Now, we have another anticipating integrals such as the divergence operator in the Malliavin calculus as it is defined in Nualart [14], or the Stratonovich integral introduced in [17] (see also León[9]). But these integrals do not agree with the Itô integral when we apply the enlargement of filtrations. Examples where we can apply anticipating integrals, together with the Malliavin calculus, are the study of stability of solutions to stochastic differential equations with a random variable as initial condition (León et al. [10]), optimal portfolio of an investor with extra information from the beginning (see, for instance, Biagini and Øksendal [3], León et al. [11] and references therein, and Pikovsky and Karatzas [15]), the study of stochastic differential equations driven by fractional Brownian motion, which is not a semimartingale (see, for example, Alòs et al. [1], or Garzón et al. [5]), the study of short-time behaviour of the implied volatility investigated by Alòs et al. [2], etc. The last problem contains only adapted processes to the underlying filtration but employs the future volatility as a main tool, which is a process that is not adapted (i.e., it is an anticipating process).

The use of the forward integral in financial markets was first introduced by León et al. [11] to figure an optimal portfolio out of an insider to maximize the expected logarithmic utility from terminal wealth. An insider is an investor that possesses extra information of the development of the market from the beginning, which is represented by a random variable $L$. In this way, we obtain an approach based on the Malliavin calculus to analyse the dynamics of the wealth equation of this insider since the forward integral is related to the divergence and derivative operators, as it is shown in Nualart [14, equality (3.14)], and in Russo and Vallois [17, Remark 2.5].

It is well-know that the wealth equation is a controlled stochastic differential equation. So, the problem of calculating an optimal portfolio to maximize the utility from terminal wealth is nothing else than a problem of stochastic control. That is, we must compute an optimal control that maximize/minimize a cost functional. A main tool in stochastic control theory is the verification theorem, which involves an optimal control and the so called Hamilton-Jacobi-Bellman equation (for short HJB-equation). The version of the classical verification theorem considered in this paper is the one given in the book by Korn and Korn [8]. Therefore, in this verification theorem, it is natural to consider controls that are adapted to a bigger filtration than the underlying one in the HJB-equation, as it is done in Theorem 3.2 below. Thus, the first goal of this paper is to study an extension of the verification theorem that is based on a classical controlled stochastic differential equation and on a classical cost function, but with controls adapted to the filtration generated by the underlying filtration and a random variable $L$ that stands for a certain extra information of the problem (see the filtration $\mathbf{G}$ defined in (1)).

Since the forward integral allows us to integrate with respect to stochastic processes that are not semimartingales, we could think that we can deal with a forward controlled stochastic differential equation driven by a process that is a martingale with respect to the underlying filtration, but with controls adapted to a filtration bigger than the one generated by this martingale. However, we show that if we can find an optimal control in this case, then the driven process is still a semimartingale with respect to the bigger filtration. This is the second goal in this paper.

The paper is organized as follows. In Section 2, we establish the framework that we use in the remaining of this article. Section 3 is devoted to state the extended verification theorem. In Section 4, we analyse an inverse type result for the extended verification theorem. Namely, we show that if there exists an optimal control with respect to certain cost function and certain filtration $\tilde{\mathbf{G}}$, then the given Brownian motion is still a $\tilde{\mathbf{G}}$-semimartingale. Finally, as an example, we provide two application of our extended verification theorem, which appear in financial markets.

Let $B=\{B_{t}:t\in[0,T]\}$ be a Brownian motion defined on a complete probability space $(\Omega,P,\mathcal{F})$ and $\mathbf{F}=\{\mathcal{F}_{t}\}_{t\in[0,T]}$ the filtration generated by $B$ augmented with the null sets. It is well-known that $\{\mathcal{F}_{t}\}_{t\geq 0}$ satisfies the usual conditions. We know that every $\sigma$-algebra $\mathcal{F}_{t}$ in $\mathbf{F}$ contains the events for which is possible determine their occurrence or not only from the history of the process $B$ until time $t$. If we assume the arrival of new information from a random variable $L$ this leads up to consider a new filtration $\mathbf{G}=\{\mathcal{G}_{t}\}_{t\in[0,T]}$ given by

which also satisfies the usual conditions. Under suitable assumptions on $L$ (see, for example, Yor and Mansuy [12, Section 1.3], León [11, Section 3] or Protter [16, Section 6]), $B$ is still a special $\mathbf{G}$-semimartingale with decomposition

where $\tilde{B}$ is a $\mathbf{G}$-Brownian motion and the information drift $\alpha=\{\alpha_{s}(x):s\in[0,T],\ \mbox{and}\ x\in\mathbb{R}\}$ is an $\mathbf{F}$-adapted random field such that $\alpha(L)\in L^{p}(0,t)$ w.p.1, for each $t\in[0,T]$ and some $p>1$.

In the financial framework, the initial enlargement of filtrations can be interpreted in this fashion: Consider a classical financial market with one bond and one risky asset. Then, by Karatzas [7], the wealth $X$ of an honest investor follows the dynamics of the Itô’s stochastic differential equation

Here, $u$ stands for the amount that the investor invest in the stock (i.e., the risky asset), and the processes $r$, $\tilde{r}$ and $\sigma$ are $\mathbf{F}$-adapted stochastic processes that represent the rate of the bound, the rate of the stock and the volatility of the market, respectively. Now suppose that this investor is an insider. That is, he/she has from the beginning some extra knowledge of the future development of the market given by the random variable $L$. So, this insider can use strategies of the form $u(L)$ to invest in the stock to make profit, where $u=\{u_{s}(x):s\in[0,T],\ \mbox{and}\ x\in\mathbb{R}\}$ is an $\mathbf{F}$-adapted random field (see León et al. [11] or Navarro [13], and Pikovsky and Karatzas [15]). In this case, from (2) and (3), the wealth equation of the insider is

Actually, equations (3) and (4) are equivalent (i.e., they have the same solutions). Also, in this case, we have that equation (3) is a controlled stochastic differential equation driven by the $\mathbf{G}$-semimartingale $B$ that involves controls that are $\mathbf{G}$-adapted. Hence, to take advantage of the extra information $L$, we can figure out a $\mathbf{G}$-adapted optimal control with respect to a certain cost function, unlike the classical stochastic control problem, where the controls are $\mathbf{F}$-adapted processes. This is extended as follows.

Let $U$ and $\mathcal{O}$ be a closed and an open subsets of $\mathbb{R}$, respectively. For $t_{0}\in[0,T)$, we will denote $Q=(t_{0},T)\times\mathcal{O}$ and $\bar{Q}=[t_{0},T]\times\bar{\mathcal{O}}$. Throughout this work, we will assume that the extra information is modeled by a random variable $L$. Now, consider two measurable functions $b,\sigma:\bar{Q}\times U\to\mathbb{R}$ satisfying suitable conditions that are given in Section 4 and the controlled stochastic differential equation for the filtration $\mathbf{G}$

Here, $u:[0,T]\times\Omega\rightarrow U$ has the form $u_{s}=u_{s}(L)$ as in equation (4). In consequence, under assumption (2), this last equation is also written as

That is, the solution $Y:\Omega\times[0,T]\to\mathcal{O}$ to these two equations is an Itô process adapted to the filtration $\mathbf{G}$. Therefore, $Y$ would be only controlled whenever it remains in the open set $\mathcal{O}$. Thus, it is necessary to introduce the $\mathbf{G}$-stopping time

Remember that $\tau$ is a stopping time since the filtration $\mathbf{G}$ satisfies the usual conditions, as it is established in Protter [16]. Moreover, by definition, $\tau\leq T$.

The main task in stochastic control consists in determining a control $u^{*}$ which is optimal with respect to a certain cost function. In this paper, the cost function has the form

where the deterministic functions $L:Q\times U\to\mathbb{R}$ and $\psi:\bar{Q}\to\mathbb{R}$ are the initial and final cost functions, respectively. Furthermore, the expectation $E_{t,x}$ indicates that the solution $Y$ to the controlled equation (5) has initial condition $x$ at time $t$. The classical tool to solve this optimization problem is the so called Hamilton-Jacobi-Bellman equation (HJB-equation), which is related with the value function (see (11) below), through the verification theorem. Consequently, in this paper we are interested in establishing an extension of the verification theorem that allows us to deal with controls adapted to a bigger filtration than the underlying filtration, for which the Brownian motion $B$ is a semimartingale. This is done in Section 3. Conversely, in Section 4, we show that if we can find an $\mathbf{G}$-adapted optimal control (with respect to a certain cost function), where the filtration $\mathbf{G}$ is bigger than the one generated by $B$, then $B$ is an $\mathbf{G}$-semimartingale. Finally, we provide two examples where we apply our extended verification theorem in Section 5.

The goal of this section is to state a verification theorem for the initial enlargement of filtrations. We first introduce the general assumptions and notation that we use throughout this section.

$(\Omega,P,\mathcal{F})$ is a complete probability space where it is defined a Brownian motion $B=\{B_{t}:t\in[0,T]\}$ and $L:\Omega\rightarrow\mathbb{R}$ is a random variable such that there are a $\mathbf{G}$-Brownian motion $\tilde{B}=\{\tilde{B}_{t}:t\in[0,T]\}$ and an $\mathbf{F}$-random field $\alpha=\{\alpha_{s}(x):s\in[0,T],\ \mbox{and}\ x\in\mathbb{R}\}$ satisfying equality (2), for all $t\in[0,T]$, w.p.1. Here, $\mathbf{F}$ is defined in Section 2 and $\mathbf{G}$ is the filtration introduced in (1). In this paper, we do not necessarily have that $\mathcal{F}$ is the $\sigma$-algebra $\mathcal{F}_{T}$. That is, we could have $\mathcal{F}_{T}\subset\mathcal{F}$.

In this section, we deal with equation (5). That is, the controlled stochastic differential equation

Here, $t_{0}\in[0,T)$, the coefficients $b,\sigma$ and the control $u$ satisfy the following hypothesis and definition, respectively. Remember that $Q$ and $U$ were introduced in Section 2.

• ($\mathbf{H}$)

  The coefficients $b,\sigma:Q\times U\to\mathbb{R}$ are measurable and satisfy the following conditions:

  • i)

    $b(t,\cdot,u),\sigma(t,\cdot,u)\in C^{1}(\mathcal{O})$, for all $(t,u)\in(t_{0},T)\times U$.

  • ii)

    There exists a constant $C>0$ such that, for all $(t,x,u)\in Q\times U$,

    $$|\partial_{x}b|\leq C,\quad|\partial_{x}\sigma|\leq C,\quad\mbox{and}\quad|b(t,x,u)|+|\sigma(t,x,u)|\leq C(1+|x|+|u|).$$

Observe that equation (5) can only have a solution $X$ up to the first time it exploits and consequently, it will be only controlled as long as it remains in the set $\mathcal{O}$. In this case, it means that equation (5) has a solution $t\mapsto X_{t}$ up to either it reaches the boundary $\partial\mathcal{O}$ of the set $\mathcal{O}$, or $t=T$.

Now, we are ready to defined the admissible strategies.

Note that if $u\in\mathcal{A}(t_{0},x)$ and $x\in\mathcal{O}$, then equation (5) has a unique solution such that $X_{t_{0}}=x$ because of the definition of admissible control and Hypothesis ($\mathbf{H}$).ii), which implies that the coefficients $b$ and $\sigma$ are Lipschitz on any interval contained in $\mathcal{O}$, uniformly on $[0,T]\times U$.

The main task in stochastic control consists in determining a control $u^{*}$ which is optimal with respect to a certain cost functional. For our purposes, the cost functional has the form as in equality (8) where the deterministic functions $L:Q\times U\to\mathbb{R}$ and $\psi:\bar{Q}\to\mathbb{R}$ verify

for some $k\in\mathbb{N}$. Remember that the notation $\mathbb{E}_{t,x}$ corresponds to the expectation of functionals of the solution $X$ to equation (5) with an initial condition $x$ at time $t$.

Before stating the control problem for this work, we need to introduce some extra definitions and conventions.

The control problem that we consider here is to compute $u^{*}\in\mathcal{A}(t,x)$, which minimizes the cost functional (8). That is, a control $u^{*}$ in $\mathcal{A}(t,x)$ satisfying

Note that the function $V:[0,T]\times\mathcal{O}\rightarrow\mathbb{R}$ describes the evolution of the minimal costs as a function of $(t,x)$. This function is called the value function.

In analogy with the adapted case, where $\alpha\equiv 0$, we use the convention

for $G\in C^{1,2}(Q)\cap C(\bar{Q})$ and $(t,x,u)\in Q\times U$. We observe that we need to deal with the extra term $(t,x,u)\mapsto\alpha_{t}(L)\sigma(t,x,u)\partial_{x}G(t,x)$ since equation (5) is equivalent to equation (6) due to condition (2). Remember that equation (6) is a controlled stochastic differential equation driven by the $\mathbf{G}$-Brownian motion $\tilde{B}$. So, in the remaining of this section, we assume that $\alpha(L)$ defined in (2) belongs to $L^{p}([0,T]\times\Omega)$, for some $p>1$.

Now we are in position to enunciate the main result of this section, where we use the $\mathbf{G}$-stopping time $\tau$ given in (7). Note that $\tau\equiv T$ in the case that $\mathcal{O}=\mathbb{R}$.

The purpose of this section is to give a converse type result of the verification theorem proved in Section 3. Towards this end, the main tool in this section is the forward integral with respect to the Brownian motion $B$. Remember that $\mathbf{F}$ stands for the filtration generated by $B$ augmented with the $P$-null sets.

Using Definition 4.1 the involve stochastic equation for the wealth process of an investor is the following controlled stochastic process (see equation (3))

Here, the coefficients satisfy the following condition:

Throughout this section, we assume that we have a filtration $\tilde{\mathbf{G}}$ bigger than $\mathbf{F}$. The family of admissible controls are related to this filtration. That is, in this section, the family $\mathcal{A}(t,x)$ of admissible controls is the set of $\tilde{\mathbf{G}}$-progressively measurable processes $u\in L^{2}([0,T]\times\Omega)$, for which (20) has a unique solution with $X_{t}=x$, for all $x\in\mathbb{R}$. Note that, in particular, we have $u\sigma\in\text{Dom }\delta^{-}$. We also observe that if the filtration $\tilde{\mathbf{G}}$ agrees with the filtration $\mathbf{G}$ introduced in (1) and $B$ is the $\mathbf{G}$-semimartingale given in (2), then equation (20) is nothing else than equation (4) due to Remark 4.2.ii). This is the reason why the forward integral was used for the first time in [13] to solve problems related to financial markets.

Remember that we are interested in the optimal control problem defined on (11), where we consider the cost functional given by

In other words, we take the classic quadratic running cost function $L(t,x,u)=au^{2}$ and the final cost is $\psi(t,x)=-e^{\int_{0}^{t}-r_{s}ds}x$, which can be interpreted as the present value of quantity $x$.

The objective is to prove that if there exists an admissible optimal control $u^{*}\in\mathcal{A}(t,x)$ for the problem given in (11) via the cost functional (21), thus we can conclude that the $\mathbf{F}-$Brownian motion $B$ is a semimartingale in the bigger filtration $\tilde{\mathbf{G}}$. For achieving this result we will use the following hypothesis which is inspired in [3, Theorem 3.5].

Concerning point 1 of Hypothesis 4.4, we observe the following. Consider the $\mathbf{F}$-adapted random field

for $\tilde{t}\in[0,T]$ and $y\in\mathbb{R}$. The classical Itô formula implies that $X(y)$ is a solution to the $\mathbf{F}$-adapted stochastic differential equation

Therefore, Remarks 4.2.i) and 4.2.iii) yield that, for a $\tilde{\mathbf{G}}$-random variable $\theta$, the $\tilde{\mathbf{G}}$-adapted process $X(\theta)$ is a solution to equation (20) with $u=\theta\chi_{(t,t+h]}$. Moreover, proceeding as in León et. al. [11], we can show that, in this case, equation (20) has a unique solution of the form $\pi(\theta)$, where $\phi=\{\pi_{s}(y):(s,y)\in[0,T]\times\mathbb{R}\}$ is an $\mathbf{F}$-adapted random field satisfying suitable conditions. We observe that we suppose that, in Hypothesis 4.4.1, the process $(s,\omega)\mapsto\chi_{(t,t+h]}(s)\theta_{0}(\omega)$ is an admissible control because we do not know the form of all the solutions to equation (20). In other words, we are assuming the uniqueness of the solution to (20) for controls of the form $(s,\omega)\mapsto\chi_{(t,t+h]}(s)\theta_{0}(\omega)$.

Now, we can prove the main result of this section.