Optimal Stopping of Marked Point Processes and Reflected Backward Stochastic Differential Equations

Abstract

We define a class of reflected backward stochastic differential equation (RBSDE) driven by a marked point process (MPP) and a Brownian motion, where the solution is constrained to stay above a given càdlàg process. The MPP is only required to be non-explosive and to have totally inaccessible jumps. Under suitable assumptions on the coefficients we obtain existence and uniqueness of the solution, using the Snell envelope theory. We use the equation to represent the value function of an optimal stopping problem, and we characterize the optimal strategy.

Some Remarks on the Snell Envelope Theory

The Snell envelope theory has been treated in various works. [14] considers the case for a positive process without any restrictions on the filtration, obtaining general results. For a bit less general results, but still enough for our work, [27] develops the theory for non-negative càdlàg processes, while [35] treats the case where the process is càdlàg and left continuous over stopping times, and satisfies the condition

$$\begin{aligned} \mathbb {E}\left[ \sup _t|\eta _t|\right] <\infty . \end{aligned}$$

(A.1)

The recent work [28] treats the subject in the framework of family of random variables indexed by stopping times, using quite general assumptions. In the following, let $(\varOmega ,\mathscr {F},\mathbb {P})$ be a probability space and let $\mathbb {F}=\left( \mathscr {F}_t\right) _{t\ge 0}$ be a filtration satisfying the usual conditions. Let $\eta$ be a cadlag process. Several properties that hold for positive processes can be shown under the condition (A.1), as we will see in Proposition A.1.

We recall the following definition:

Definition A.1

An optional process R of class [D] is said to be regular if $R_{t^-}={^p}R_t$ for any $t<T$, where ${^p}X$ indicates the predictable projection.

Proposition A.1

Let $\eta$ be a càdlàg process satisfying (A.1). Define

$$\begin{aligned} R_t=\mathop {\mathrm{ess\,sup}}\limits _{\tau \in \mathscr {T}_t}\mathbb {E}\left[ \eta _\tau \bigg \vert \mathscr {F}_t\right] \end{aligned}$$

(A.2)

It holds that

• (i) $R_t$ is the Snell envelope of $\eta _t$. This means it is the smallest càdlàg supermartingale that dominates $\eta _t$, i.e. $R_t\ge \eta _t$ for all $t\ \mathbb {P}$-a.s.

• (ii) A stopping time $\tau ^*$ is optimal in (A.2) (i.e. $R_t=\mathbb {E}\left[ \eta _{\tau ^*}\bigg \vert \mathscr {F}_t\right]$) if and only if one of the following conditions hold

  • $R_{\tau ^*}=\eta _{\tau ^*} \text { and } R_{s\wedge \tau ^*} \text {is a }\mathbb {F}\text {-martingale}$

  • $\mathbb {E}[R_t]=\mathbb {E}[\eta _\tau ^*]$

• (iii) $R_t$ is of class [D], hence it admits decomposition

  $$\begin{aligned} R_t=M_t-K_t, \end{aligned}$$

  where M is a martingale, K a predictable increasing process with $K_0=0$. K can be decomposed as $K=K_t^c+K_t^d$, where $K^c$ indicates the continuous part and $K^d$ the discontinuous part. Moreover we have, a.s.

  $$\begin{aligned} \begin{array}{c} \left\{ t:\varDelta K_t>0\right\} \subset \left\{ t: R_{t^-}=\eta _{t^-}\right\} \\ \text {or equivalently, }\quad \varDelta K_t=\varDelta K_t \mathbb {1}_{\{R(\eta )_{t^-}=\eta _{t^-} \}}, \quad t\ge 0. \end{array} \end{aligned}$$

• (iv) If the process $R_t$ is regular in the sense that $R_{t^-}={^p}R_t$, where ${^p}R$ indicates the predictable projection, defining the stopping time

  $$\begin{aligned} D_t^*=\inf \lbrace s\ge t : R_s\ne M_s\rbrace , \end{aligned}$$

  then $D_t^*$ is an optimal stopping time and it is in fact the largest optimal stopping time.

Proof

Define

$$\begin{aligned} I=\inf \limits _{t\in [0,T]}\eta _t \qquad \text { and } \qquad N_t=\mathbb {E}\left[ I\bigg \vert \mathscr {F}_t\right] , \end{aligned}$$

and since $\eta _t-I\ge 0$ for all t, we have $\eta _t-N_t\ge 0$ for all t. $N_t$ is a uniformly integrable martingale thanks to (A.1). Consider $\tilde{\eta }_t=\eta _t-N_t\ge 0$ and $\tilde{R}_t=R_t-N_t$. Notice that then

$$\begin{aligned} \tilde{R}_t=R_t-N_t=\mathop {\mathrm{ess\,sup}}\limits _{\tau \in \mathscr {T}_t} \mathbb {E}\left[ \eta _\tau -N_\tau \bigg \vert \mathscr {F}_t\right] =\mathop {\mathrm{ess\,sup}}\limits _{\tau \in \mathscr {T}_t} \mathbb {E}\left[ \tilde{\eta }_\tau \bigg \vert \mathscr {F}_t\right] , \end{aligned}$$

i.e. $\tilde{R}$ is the Snell envelope of the positive process $\tilde{\eta }$. R inherits all the properties from $\tilde{R}$. Let us see why the fourth property holds, as the rest are obtained similarly.

If $R_t$ is regular, so is $\tilde{R}_t$ because we are adding a uniformly integrable martingale, which is regular (all uniformly quasi-left-continuous integrable càdlàg martingales are regular, see [25] Def 5.49). The result then holds by [14], p. 140. $\square$