Uncovering Market Disorder and Liquidity Trends Detection

Assets liquidity is an important factor in ensuring the efficient functioning of a market. Glosten and Harris [1] define liquidity as the ability of an asset to be traded rapidly, in significant volumes and with minimal price impact. Measuring liquidity, therefore, involves three aspects of the trading process: time, volume and price. This is reflected in Kyle’s description of liquidity as a measure of the tightness of the bid-ask spread, the depth of the limit order book and its resilience [2]. Hence, as highlighted in Lybek and Sarr [3] and in Binkowski and Lehalle [4], it is essential for liquidity-driven variables to not only capture the transaction cost associated with a relatively small quantity of shares but also to assess the depth accessible to large market participants. Additionally, measures related to the pace or speed of the market are considered since trading slowly can help mitigate implicit transaction costs. Typical indicators in this category include value traded and volatility over a reference time interval.

Several studies have been devoted to identifying different market regimes rapidly and in an automated fashion. Hamilton [5] proposes the initial notion of regime switches, wherein he establishes a relationship between cycles of economic activity and business cycle regimes. The idea remains relevant, with ongoing exploration of novel approaches. Hovath et al. [6] suggest an unsupervised learning algorithm for partitioning financial time series into different market regimes based on the Wassertein k-means algorithm for example. Bucci et al. [7] employ covariance matrices to identify market regimes and identify shifts in volatile states.

This paper introduces a novel liquidity regime change detection methodology aimed at assessing the resilience of an order book using tick-by-tick market data. More precisely, our objective is to study methods derived from the theory of disorder detection and to develop a liquidity proxy that effectively captures its inherent characteristics. This will enable us to thoroughly examine how the distribution of the liquidity proxy evolves over time. By employing these methods, we seek to gain a detailed understanding of the dynamics involved in liquidity changes and their impact within the distribution patterns of our chosen proxy. To accomplish this, our methodology centers around the introduction of a liquidity proxy known as ”trades-through” (see Definition 2.1). Trades-through possess a high informational content, which is of interest in this study. Specifically, trades-through serve as an indicator of the order book’s resilience, allowing the examination of activity at various levels of depth. They can also provide insightful information on the volatility of the financial instrument under examination. Additionally, trades-through can indicate whether an order book has been depleted or not in comparison to its previous states and to which extent. Pomponio and Abergel [8] investigate several stylized facts related to trades-through, which are similar to other microstructural features observed in the literature. Their research highlights the statistical robustness of trades-through concerning the size of orders placed. This implies that trades-though are not solely a result of low quantities available on the best limits and that the information they provide is of significant importance. Another noteworthy result they observe is the presence of self-excitement and clustering patterns in trades-through. In other words, a trade-through is more likely to occur soon after another trade-through than after any other trade. This result supports the idea of modelling trades-through using Hawkes processes as in Ioane Muni Toke and Fabrizio Pomponio [9]. This approach is particularly interesting given that Hawkes processes fall into the class of branched processes. Their dynamics can therefore be illustrated by a representation of immigration and birth that allows interpretation. Indeed, the intensity, at which events take place, is formed by an exogenous term representing the arrival of ”immigrants” and a self-reflexive endogenous term representing the fertility of ”immigrants” and their ”descendants”. We may refer to Brémaud et al. [10] for more comprehensive details.

Our focus here will be to use the Marked-Hawkes process to model trades-through rather than standard Hawkes processes in order to capture the impact of the orders’ volumes on the intensity of the counting processes. Our research centers on employing the Marked-Hawkes process as a modelling technique for trades-through, as opposed to conventional Hawkes processes. The rationale behind this choice lies in our intention to account for the influence of order volumes on the intensity of the counting processes.

Upon the conclusion of the modelling phase, we will utilize this proxy to identify intraday liquidity regimes⁴⁴4Moments of transition between liquidity regimes will be defined as instances where the distribution of trade-throughs experiences a change.. In order to achieve this, our model involves detecting the times at which the distribution of liquidity (represented by a counting process) undergoes changes as fast as possible. This entails comparing the intensities of two doubly stochastic Poisson processes. This type of problem is commonly known in literature as ”Quickest change-point detection” or ”Disorder detection”. Disorder detection has been studied through two research paths, namely the Bayesian approach and the minimax approach. The Bayesian perspective, originally proposed by Girschick and Rubin [11], typically assumes that the change point is a random variable and provides prior knowledge about its distribution. As an example, Dayanik et al. [12] conduct their study on compound Poisson processes assuming an exponential prior distribution for the disorder time. In their work, Bayraktar et al. [13] address a Poisson disorder problem with an exponential penalty for delays, assuming an exponential distribution for the delay. However, the minimax approach does not include any prior on the distribution of the disorder times, see for instance, Page [14] and Lorden [15]. Due to the scarcity of relevant literature and data pertaining to disorder detection problems, accurately estimating the prior distribution in the context of market microstructure poses a challenging task. Hence, our preference lies in adopting a non-Bayesian min-max approach. El Karoui et al. [16] prove the optimality of the CUSUM⁵⁵5The abbreviation CUSUM stands for CUmulative SUM. procedure 3.1 for $\rho<1$ and $\rho>1$ for doubly stochastic poisson processes which extends the proof proposed by Moustakides [17] and Poor and Hadjiliadis [18] for $\rho<1$. However, the proposed solution is limited to cases where point processes do not exhibit simultaneous jumps. This presents a problem in our case since definition 2.1 suggests that each time a $n$-limit trade-through event is observed, $n-1$ other trade-through events occur simultaneously. Hence, we need to demonstrate the optimality of these results by considering scenarios in which point processes exhibit a finite number of simultaneous arrival times which is precisely what we obtained. We are also able to produce a tractable formula of the average run delay. More importantly, our findings encompass the intricacies inherent to modelling limit order books and investigating market microstructure. We use our results to identify changes in liquidity regimes within an order book. To do this, we apply our detection procedure to real market data, enabling us to achieve convincing detection performance. To the best of our knowledge, we believe that this is the first attempt to quantify market liquidity using a methodology combining the notion of trades-through, disorder detection theory, and marked-hawkes processes.

The article is structured in the following manner. In Section 2, the primary aim is to establish a model for trades-through using Marked Hawkes Processes (MHP) and subsequently provide the relevant mathematical framework. In Section 3, various results related to the optimality of the CUSUM procedure within the framework of sequential test analysis for a simultaneous jump Cox process will be presented. Finally, in Section 4, we provide an overview of the goodness-of-fit results obtained from applying a Marked Hawkes Process to analyze trades-through data in the Limit Order Book of BNP Paribas stock. Additionally, we present the outcomes of utilizing our disorder detection methodology to identify various liquidity regimes.

The aim of this section is to build a model for trades-through by means of Marked Hawkes Processes (MHP) and to present the corresponding mathematical framework. We begin by defining the concept of trades-through.

Let $(\Omega,\mathbb{F},\mathcal{F}=\{\mathcal{F}_{t}\}_{t\geq 0},\mathbb{P})$ be a complete filtered probability space endowed with right continuous filtration $\{\mathcal{F}_{t}\}_{t\geq 0}$. Going forward, whenever we mention equalities between random variables, we mean that they hold true almost surely under the probability measure $\mathbb{P}$. We may not explicitly indicate this notion ($\mathbb{P}$-a.s) in most cases. Let $0<\tau^{A,i}_{0}<\tau^{A,i}_{1}<\tau^{A,i}_{2}<\dots$ (resp. $0<\tau^{B,i}_{0}<\tau^{B,i}_{1}<\tau^{B,i}_{2}<\dots$) form a sequence of increasing $\mathcal{F}$-measurable stopping times and $(v^{A,i}_{k})_{k\geq 0}$ (resp. $(v^{B,i}_{k})_{k\geq 0}$) be a sequence of i.i.d $\mathbb{R}_{+}$-valued random variables for each $i\in\{1,\dots,M\}$. Here, $(\tau^{A,i}_{k})_{k\geq 0}$ (resp. $(\tau^{B,i}_{k})_{k\geq 0}$) represent the arrival times of trade-throughs at limit $i$ on the ask (resp. bid) side of the limit order book while $(v^{A,i}_{k})_{k\geq 0}$ (resp. $(v^{B,i}_{k})_{k\geq 0}$) represent the sizes of volume jumps of trades-though for these arrival times.

Our goal is to design a model that takes into account the volumes of the trades-through for all pertinent limits and that replicates the clustering phenomenon observed in the corresponding stylized facts (See [8]). To achieve this, we suggest using a modified version of Cox processes⁶⁶6Also referred to as doubly stochastic Poisson process., specifically Marked Hawkes Processes. We define a 2-dimensional marked point process $N^{A\times B}=\left(N^{A},N^{B}\right)$ where $N^{A}$ (resp. $N^{B}$) is the counting measure on the measurable space $\left(\mathbb{R}_{+}\times\mathbb{R}_{+},\mathcal{B}\otimes\mathcal{L}\right)$ associated to the set of points $\left\{(\tau_{k}^{A},v^{A}_{k});k\geq 0\right\}$ (resp. $\left\{(\tau_{k}^{B},v^{B}_{k});k\geq 0\right\}$), $\mathcal{B}$ being the Borel $\sigma$-field on $\mathbb{R}_{+}$ while $\mathcal{L}$ is the Borel $\sigma$-field on $\mathbb{R}_{+}$. This means that $\forall C\in\mathcal{B}\otimes\mathcal{L}$  :

	$\displaystyle N^{A}(C)$	$\displaystyle=\sum_{k\geq 0}\mathbf{1}_{C}(\tau_{k}^{A},v^{A}_{k})\quad\quad\quad\quad\quad,$	$\displaystyle N^{B}(C)$	$\displaystyle=\sum_{k\geq 0}\mathbf{1}_{C}(\tau_{k}^{B},v^{B}_{k})$
		$\displaystyle=\sum_{1\leq i\leq M}\sum_{k\geq 0}\mathbf{1}_{C}(\tau_{k}^{A,i},v^{A,i}_{k})$		$\displaystyle=\sum_{1\leq i\leq M}\sum_{k\geq 0}\mathbf{1}_{C}(\tau_{k}^{B,i},v^{B,i}_{k})$

Let $\mathcal{F}^{A\times B}=\left(\mathcal{F}_{t}^{A\times B}\right)_{t\geq 0}$ be a sub-$\sigma$-field of the complete $\sigma$-field $\mathcal{F}$ where
$\mathcal{F}_{t}^{A\times B}=\sigma\left(N^{A\times B}(s,v);\\ s\in[0,t],v\in\mathbb{R}_{+}\right)$. Simply put, $\mathcal{F}^{A\times B}_{t}$ contains all the information about the process $N^{A\times B}$ up to time $t$. The collection of all such sub-$\sigma$-fields, denoted by $\left(\mathcal{F}_{t}^{A\times B},t\geq 0\right)$, is called the internal history of the process $N^{A\times B}$. One of the core concepts underlying the dynamics of $N^{A\times B}$ is the notion of conditional intensity. Specifically, the dynamics of $N^{A\times B}$ are characterized by the non-negative, $\mathcal{F}^{A\times B}$-progressively measurable processes $\lambda^{A}$ and $\lambda^{B}$, which model the conditional intensities. These processes satisfy the following conditions:

		$\displaystyle\mathbb{E}\left[\int_{K}N^{A}(t,v)-N^{A}(s,v)\mathrm{d}v\big{\lvert}\mathcal{F}^{A\times B}_{s}\right]=\mathbb{E}\left[\int_{[s,t[\times K}\lambda^{A}(u,v)\mathrm{d}u\mathrm{d}v\big{\lvert}\mathcal{F}^{A\times B}_{s}\right],\quad 0\leq s\leq t$		(1)
		$\displaystyle\mathbb{E}\left[\int_{K}N^{B}(t,v)-N^{B}(s,v)\mathrm{d}v\big{\lvert}\mathcal{F}^{A\times B}_{s}\right]=\mathbb{E}\left[\int_{[s,t[\times K}\lambda^{B}(u,v)\mathrm{d}u\mathrm{d}v\big{\lvert}\mathcal{F}^{A\times B}_{s}\right],\quad 0\leq s\leq t$

Moving forward, the point process $N^{A\times B}$ will be represented as a bivariate Hawkes process, with its intensity wherein its intensity is dependent on the marks $(v^{A}_{k})_{k\geq 0}$ and $(v^{B}_{k})_{k\geq 0}$. By viewing the marked point processes $N^{A}$ and $N^{B}$ as a processes on the product space $\mathbb{R}_{+}\times\mathbb{R}_{+}$, we can derive the marginal processes $N^{A}_{g}(.)=N^{A}(.,\mathbb{R}_{+})$ and $N^{B}_{g}(.)=N^{B}(.,\mathbb{R}_{+})$, which refer to as ground process. In other worlds, for $t\in\mathbb{R}_{+}$,

$$N^{A}_{g}(t)=\int_{[0,t[\times\mathbb{R}_{+}}N^{A}(\mathrm{~d}u\times\mathrm{d}v^{A}),\quad N^{B}_{g}(t)=\int_{[0,t[\times\mathbb{R}_{+}}N^{B}(\mathrm{~d}u\times\mathrm{d}v^{B})$$

(2)

These ground processes therefore describe solely the arrival times of the trades-through and can be understood as being Hawkes processes as well. In instances where $N^{A\times B}$ exhibit sufficient regularity (See The definition 6.4.III and section 7.3 in Daley and Vere-Jones [19]), it is possible to factorize its conditional intensities $\lambda^{A}$ and $\lambda^{B}$ without presupposing stationarity and under the assumption that the mark’s conditional distributions at time $t$ given $\mathcal{F}^{A\times B}_{t^{-}}$ have densities of $\mathbb{R}_{+}\rightarrow\mathbb{R}_{+}:~v\mapsto f_{A}(v\big{\lvert}\mathcal{F}^{A\times B}_{t^{-}})$ and $\mathbb{R}_{+}\rightarrow\mathbb{R}_{+}:~v\mapsto f_{B}(v\big{\lvert}\mathcal{F}^{A\times B}_{t^{-}})$. The conditional intensity factors out as :

$$\lambda^{A}(t,v)=\lambda^{A}_{g}(t)f_{A}(v\big{\lvert}\mathcal{F}^{A\times B}_{t^{-}}),\quad\lambda^{B}(t,v)=\lambda^{B}_{g}(t)f_{B}(v\big{\lvert}\mathcal{F}^{A\times B}_{t^{-}}),\quad(t,v)\in\mathbb{R}_{+}\times\mathbb{R}_{+}$$

(3)

where $\lambda_{g}(t)$ is referred to as the $\mathcal{F}^{A\times B}_{t^{-}}$-intensity of the ground intensity.
In a heuristic manner, one can summarize equation 3 for $(t,v)\in\mathbb{R}_{+}\times\mathbb{R}_{+}$ as being in the form of :

		$\displaystyle\lambda^{A}(t,v)\mathrm{d}t\mathrm{d}v\approx\mathbb{E}\left[N^{A}(\mathrm{d}t\times\mathrm{d}v)\big{\lvert}\mathcal{F}^{A\times B}_{t^{-}}\right]\approx\lambda^{A}_{g}(t)f_{A}(v\big{\lvert}\mathcal{F}^{A\times B}_{t^{-}})\mathrm{d}t\mathrm{d}v$
		$\displaystyle\lambda^{B}(t,v)\mathrm{d}t\mathrm{d}v\approx\mathbb{E}\left[N^{B}(\mathrm{d}t\times\mathrm{d}v)\big{\lvert}\mathcal{F}^{A\times B}_{t^{-}}\right]\approx\lambda^{B}_{g}(t)f_{B}(v\big{\lvert}\mathcal{F}^{A\times B}_{t^{-}})\mathrm{d}t\mathrm{d}v$

We assume that the conditional intensities $\lambda_{g}^{A}$ and $\lambda_{g}^{B}$ have the following form :

	$\displaystyle\lambda_{g}^{i}(t)$	$\displaystyle=\mu^{i}(t)+\sum_{j=A,B}\sum_{k\geq 0}\gamma_{ij}(t-\tau_{k}^{j},v_{k}^{j})$		(4)
		$\displaystyle=\mu^{i}(t)+\sum_{j=A,B}\int_{[0,t[\times\mathbb{R}_{+}}\gamma_{ij}(t-s,v)N^{j}(\mathrm{~d}s\times\mathrm{d}v),\quad i=A,B$

where $\mu^{i}:\mathbb{R}_{+}\rightarrow\mathbb{R}_{+}$ and $\gamma_{ij}:\mathbb{R}_{+}\times\mathbb{R}_{+}\rightarrow\mathbb{R}_{+}$ are non-negative measurable functions.
We choose a standard factorized form for the ground intensity kernel $\gamma_{ij}(t-u,v)=\alpha_{ij}e^{\beta_{ij}(t-u)}\times g_{j}(v)$ to describe the weights of the marks, $g_{j}$ being a measurable function defined on $\mathbb{R}_{+}$ that characterizes the mark’s impact on the intensity. The exponential shape of the kernel is derived from the findings of Toke et al. [9]. The properties of functions $g_{A}$ and $g_{B}$ will be examined in Section 4.

Thus, the conditional ground intensities of this model have the following integral representation :

	$\displaystyle\lambda^{A}_{g}(t)=\mu^{A}(t)+$	$\displaystyle\int_{[0,t[\times\mathbb{R}_{+}}\alpha_{AA}e^{-\beta_{AA}(t-u)}g_{A}\left(v\right)N^{A}(\mathrm{d}u\times\mathrm{d}v)$		(5)
		$\displaystyle+\int_{[0,t[\times\mathbb{R}_{+}}\alpha_{AB}e^{-\beta_{AB}(t-u)}g_{B}\left(v\right)N^{B}(\mathrm{d}u\times\mathrm{d}v),\quad t\geq 0$

	$\displaystyle\lambda^{B}_{g}(t)=\mu^{B}(t)+$	$\displaystyle\int_{[0,t[\times\mathbb{R}_{+}}\alpha_{BA}e^{-\beta_{BA}(t-u)}g_{A}\left(v\right)N^{A}(\mathrm{d}u\times\mathrm{d}v)$		(6)
		$\displaystyle+\int_{[0,t[\times\mathbb{R}_{+}}\alpha_{BB}e^{-\beta_{BB}(t-u)}g_{B}\left(v\right)N^{B}(\mathrm{d}u\times\mathrm{d}v),\quad t\geq 0$

where $\left(\alpha_{ij},\beta_{ij}\right)_{(i,j)\in\{A,B\}^{2}}$ are non-negative reals.
Conventionally, the impact functions $g_{i}$ have to satisfy the following normalizing condition which enhances the overall stability of the numerical results (see section 1.3 of [20] for further details) :

$$\int_{0}^{+\infty}g_{i}(v)f_{i}(v)\mathrm{d}v=1,\quad i=A,B.$$

(7)

This means that the impact functions of the Marked-Hawkes based model would be equal to $g_{i}(v)=\frac{v^{\eta_{i}}}{\mathbb{E}\left(v^{\eta_{i}}\right)}$. The following stability conditions of our model are based on Theroem 8 of Brémaud et al. [21].

We now shift our focus towards developing a methodology that can distinguish between distinct liquidity regimes. This can be done by detecting changes or disruptions in the distribution of a given liquidity proxy, which, in our case, is represented by the number of trades-through. Here, we will focus on identifying disruptions that affect the intensity of the marked multivariate Hawkes process discussed in the previous section. This will enable us to compare the resilience of our order book to that of previous days over a finite horizon $T$. The sequential detection methodology involves comparing the distribution of the observations to a predefined target distribution. The objective is to detect changes in the state of the observations as rapidly as possible while minimizing the number of false alarms which correspond to sudden changes in the arrival rate in the case of Poisson processes with stochastic intensity. To be specific, let us consider a general point process $N$ with a known rate $\lambda$, where the arrivals of certain events are associated to this process. At a certain point in time $\theta$, the rate of occurrence of events of the process $N$ undergoes an abrupt shift from $\lambda$ to $\rho\lambda$. However, the value of the disorder time $\theta$ is unobservable. The goal is to identify a stopping time $\tau$ that relies exclusively on past and current observations of the point process $N$ and can detect the occurrence of the disorder time $\theta$ swiftly. Thus, we will focus in the following on solving the sequential hypothesis testing problems in the aforementioned general form :

$$\begin{array}[]{lll}H_{0}:&\lambda(t),\\ H_{1}:&\check{\lambda}^{\theta_{1}}(t)=\lambda(t)\mathbbm{1}_{\{t<\theta_{1}\}}+\rho_{1}\lambda(t)\mathbbm{1}_{\{t\geq\theta_{1}\}},\quad\rho_{1}>1\\ \end{array}$$

(8)

and,

$$\begin{array}[]{lll}H_{0}:&\lambda(t),\\ H_{2}:&\check{\lambda}^{\theta_{2}}(t)=\lambda(t)\mathbbm{1}_{\{t<\theta_{2}\}}+\rho_{2}\lambda(t)\mathbf{1}_{\{t\geq\theta_{2}\}},\quad\rho_{2}<1\\ \end{array}$$

(9)

We propose the introduce a change-point detection procedure that is applicable to our model. The goal is to study the distribution of the process $N^{A\times B}$ process. The problem at hand can be tackled through various approaches. Here, we opt for a non-Bayesian approach, specifically the min-max approach introduced by Page [14], which assumes that the timing of the change is unknown and non-random. Previous research has shown the optimality of the CUSUM procedure in solving this problem (see [22][23][16]). However, these demonstrations are confined to scenarios where the process involves Cox jumps of size 1. This constraint poses significant limitations in our case, as the definition 2.1 of trade-throughs implies for of simultaneous jumps. Therefore, the objective of this chapter is to demonstrate the optimality of the CUSUM procedure in a broader context that considers multiple jumps and to formulate the relevant detection delay.

The identification of different liquidity regimes in an order book can be done in various ways. One can apply our disorder detection methodology separately on processes $N^{A}$ and $N^{B}$ to study liquidity either on the bid or on the ask, or consider process $N^{A}+N^{B}$ and take both into account at the same time. To enhance the clarity and maintain the general applicability of our results, we consider the point process $N$ as a general Cox process that decomposes into the form $\sum_{i=1}^{D}N^{i}$ where $0<D<+\infty$ and $N^{i}$ is a finite point process defined on the same probability space introduced in the previous section and whose arrivals are non-overlapping. We denote $\lambda^{i}$ the $\mathcal{F}$-intensity of $N^{i}$ and by $t\mapsto\Lambda^{i}(t)=\int_{0}^{t}\lambda^{i}(s)\mathrm{d}s$ it’s compensator for $i=1,\dots,D$.

We denote the probability measure for the case with no disorder by $\mathbb{P}_{\infty}$, where $\theta=+\infty$. The probability measure for the case with a disorder that starts from the beginning, where $\theta=0$, is denoted by $\mathbb{P}_{0}$. If there is a disorder at the instant $\theta$, represented by the value of the intensity of the counting process equal to $\check{\lambda}^{\theta}(t)$, then the probability measure is denoted by $\mathbb{P}_{\theta}$. The constructed measure satisfies the following conditions :

$$\mathbb{P}_{\theta}=\begin{cases}\mathbb{P}_{0},&\text{ if }\theta=0\\ \mathbb{P}_{\infty},&\text{ if }\theta=+\infty\end{cases}$$

We initiate the discourse by introducing a few notations that will be instrumental in presenting our results. Let $U^{i}=N^{i}-\beta(\rho)\Lambda^{i}$ and $i\in\{1,\dots,D\}$. The process $\sum_{i=1}^{D}U^{i}$ is referred to as the Log Sequential Probability Ratio (LSPR) between the two probability measure $\mathbb{P}_{\infty}$ and $\mathbb{P}_{0}$ where $\beta(\rho)=(\rho-1)/\log\rho$ and $\Lambda^{i}(t)=\int_{0}^{t}\lambda^{i}(s)\mathrm{d}s$ for all $0\leq t\leq T$. The probability measure $\mathbb{P}_{0}$ is defined as the measure equivalent to $\mathbb{P}_{\infty}$ with density :

$$\frac{\mathrm{d}\mathbb{P}_{0}}{\mathrm{d}\mathbb{P}_{\infty}}\big{\lvert}_{\mathcal{F}_{t}}=\rho^{\sum_{i=1}^{D}U^{i}(t)},~~~~0\leq t\leq T$$

(10)

Or, more generally :

$$\begin{split}\frac{\mathrm{d}\mathbb{P}_{\theta}}{\mathrm{d}\mathbb{P}_{\infty}}\big{\lvert}_{\mathcal{F}_{t}}&=\prod_{i=1}^{D}\frac{\rho^{U^{i}(t)}}{\rho^{U^{i}(\theta)}}\quad~~,~~~~0\leq\theta\leq t\\ &=\rho^{\sum_{i=1}^{D}U^{i}(t)-U^{i}(\theta)}\end{split}$$

(11)

According to Girsanov’s theorem, these densities are defined if $\mathbb{E}\left(\rho^{\sum_{i=1}^{D}U^{i}(t)-U^{i}(\theta)}\right)=1$ which is the case since $\rho^{\sum_{i=1}^{D}U_{i}(t)}=e^{\log(\rho)\sum_{i=1}^{D}N^{i}(t)-(\rho-1)\sum_{i=1}^{D}\Lambda^{i}(t)}$ is a local martingale if $\sum_{i=1}^{D}\Lambda^{i}(t)$ is càdlàg (see Øksendal et al. [24]). Hence, $\sum_{i=1}^{D}N^{i}(t)-\rho\Lambda^{i}(t)$ is a $\mathbb{P}_{\theta}$-martingale iff $\rho^{\sum_{i=1}^{D}U^{i}(t)-U^{i}(\theta)}$ is an $\mathbb{P}_{\infty}$-martingale. We can therefore now define the log-likelihood ratio :

$$\begin{split}\ell_{t,\theta}:&=\sum_{i=1}^{D}\int_{\theta}^{t}\log\left(\frac{\check{\lambda}^{i}_{\theta}(s)}{\lambda^{i}(s)}\right)\mathrm{d}N^{i}(s)-\sum_{i=1}^{D}\int_{\theta}^{t}\left(\check{\lambda}^{i}_{\theta}(s)-\lambda^{i}(s)\right)\mathrm{d}s\\ &=\log\left(\rho\right)\left(\sum_{i=1}^{D}\left(N^{i}(t)-N^{i}(\theta)\right)-\frac{\rho-1}{\log\left(\rho\right)}\left(\Lambda^{i}(t)-\Lambda^{i}(\theta)\right)\right)\\ &=\log\left(\rho\right)\left(\sum_{i=1}^{D}U^{i}(t)-U^{i}(\theta)\right)\end{split}$$

(12)

with

$$\check{\lambda}^{i}_{\theta}(t)=\begin{cases}\lambda^{i}(t),&0\leq t\leq\theta\\ \rho\lambda^{i}(t),&t>\theta\end{cases}$$

which is the intensity of the node $i$ if the disorder takes place at the time $\theta$.

In statistical problems, it is common to observe two distinct components of loss. The first component is typically associated with the costs of conducting the experiment, or any delay in reaching a final decision that may incur additional losses. The second component is related to the accuracy of the analysis. The min-max problem (see [14]) can be formulated as finding a stopping rule that minimizes the worst-case expected cost under the Lorden criterion. The objective is therefore to find the stopping time that minimizes the following cost :

$$\begin{split}\inf_{\tau\in\mathcal{T}_{\pi}}C(\tau)\end{split}$$

(13)

Where $C(\tau)=\sup_{\theta\in[0,+\infty]}{\text{ ess sup }}\mathbb{E}^{\theta}\left[(\tau-\theta)^{+}\big{\lvert}\mathcal{F}_{\theta}\right]$, $\mathcal{T}_{\pi}$ is the class of $\mathcal{F}$-stopping times that statisfies $\mathbb{E}(\tau)\geq\pi$ and $\pi>0$ a constant.
As explained by Lorden [15], the formulation of the problem under criterion $C(\tau)$ is robust in the sense that it seeks to estimate the worst possible delay before the disorder time. The false alarm rate can be controlled by controlling $\mathbb{E}(\tau)$ through $\pi$. It has been observed that Lorden’s optimality criterion is characterized by a high level of stringency as it necessitates the optimization of the maximum average detection delay over all feasible observation paths leading up to and following the changepoint. This rigorous approach often produces conservative outcomes. However, it is worth noting that the motivation behind the use of min-max optimization can be linked to the Kullback-Leibler divergence. This has notably been the case in the works of Moustakides [23] for detecting disorder occurring in the drift of Ito processes. In our case, for $\tau\in\mathcal{T}_{\pi}$ and $0\leq\theta\leq\tau$, equation 12 yields :

$$\begin{split}\mathbb{E}^{\theta}\left[\log\left(\frac{\mathrm{d}\mathbb{P}_{\theta}}{\mathrm{d}\mathbb{P}_{\infty}}\right)\big{\lvert}\mathcal{F}_{\theta}\right]&=\log\left(\rho\right)\mathbb{E}^{\theta}\left[\sum_{i=1}^{D}\left(N^{i}(\tau)-N^{i}(\theta)\right)-\frac{\rho-1}{\log\left(\rho\right)}\left(\Lambda^{i}(\tau)-\Lambda^{i}(\theta)\right)\big{\lvert}\mathcal{F}_{\theta}\right]\\ &=\left(\log\left(\rho\right)-\rho+1\right)\sum_{i=1}^{D}\mathbb{E}^{\theta}\left[\left(N^{i}(\tau)-N^{i}(\theta)\right)^{+}\big{\lvert}\mathcal{F}_{\theta}\right]\end{split}$$

(14)

Hence, the following minimax formulation of the detection problem is proposed :

$$\begin{split}\inf_{\tau}\widetilde{C}(\tau)&=\inf_{\tau}\sup_{\theta\in[0,+\infty]}{\text{ ess sup }}\sum_{i=1}^{D}\mathbb{E}^{\theta}\left[\left(N^{i}(\tau)-N^{i}(\theta)\right)^{+}\big{\lvert}\mathcal{F}_{\theta}\right]\\ &\text{with the constraint}~~\sum_{i=1}^{D}\mathbb{E}\left(N^{i}(\tau)\right)\geq\pi\end{split}$$

(15)

El Karoui et al. [16] suggests another justification of this relaxation based on the Time-rescaling theorem (see Daley and Vere-Jones [19]) which allows us to transform a Cox process into a Poisson process verifying :

$$\mathbb{E}^{\theta}\left[\sum_{i=1}^{D}\left(N^{i}(\tau)-N^{i}(\theta)\right)^{+}\big{\lvert}\mathcal{F}_{\theta}\right]=\rho\sum_{i=1}^{D}\mathbb{E}^{\theta}\left[\int^{\tau}_{\tau\wedge\theta}\lambda^{i}(s)\mathrm{d}s\big{\lvert}\mathcal{F}_{\theta}\right]$$

(16)

Heuristically, this is equivalent to the formulation referenced in 15. Indeed, $\sum_{i=1}^{D}\mathbb{E}^{\theta}\left[\int^{\tau}_{\tau\wedge\theta}\lambda^{i}(s)\mathrm{d}s\big{\lvert}\mathcal{F}_{\theta}\right]=\left(\sum_{i=1}^{D}\lambda^{i,cst}\right)\mathbb{E}^{\theta}\left[\left(\tau-\theta\right)^{+}\big{\lvert}\mathcal{F}_{\theta}\right]$ in the case of a Poisson process with constant intensity.

Given that the likelihood ratio test is considered the most powerful test for comparing simple hypotheses according to the Neyman-Pearson lemma, we will utilize $\rho^{\sum_{i=1}^{D}U^{i}(t)-U^{i}(\theta)}$ to assess the shift in this distribution. The log-likelihood ratio $\sum_{i=1}^{D}U^{i}(t)-U^{i}(\theta)$ usually exhibits a negative trend before a change and a positive trend after a change (see section 8.2.1 of Tartakovski et al. [28] for more details). Consequently, the key factor in detecting a change is the difference between the current value of $\sum_{i=1}^{D}U^{i}(t)-U^{i}(\theta)$ and its minimum value. As a result, we are prompted to introduce the CUSUM process.

One noteworthy characteristic of $\sum_{i=1}^{D}N^{i}(\widetilde{T}_{\mathrm{C}})$ and $\sum_{i=1}^{D}N^{i}(\hat{T}_{\mathrm{C}})$ is that their conditional expectations can be explicitly calculated based on their initial values and the parameter $m$. As a result, the theorems 3.1 and 3.2 offer a valuable tool for controlling the ARL constraint through the parameter $m$, i.e, expressing $\sum_{i=1}^{D}\mathbb{E}\left(N^{i}(\widetilde{T}_{\mathrm{C}})\right)$ and $\sum_{i=1}^{D}\mathbb{E}\left(N^{i}(\hat{T}_{\mathrm{C}})\right)$ as tractable formulas for $\rho>1$ and $\rho<1$.