Doubly Reflected Backward Stochastic Differential Equations in the Predictable Setting

Abstract

In this paper, we introduce a specific kind of doubly reflected backward stochastic differential equations (in short DRBSDEs), defined on probability spaces equipped with general filtration that is essentially non quasi-left continuous, where the barriers are assumed to be predictable processes. We call these equations predictable DRBSDEs. Under a general type of Mokobodzki’s condition, we show the existence of the solution (in consideration of the driver’s nature) through a Picard iteration method and a Banach fixed point theorem. By using an appropriate generalization of Itô’s formula due to Gal’chouk and Lenglart we provide a suitable a priori estimates which immediately implies the uniqueness of the solution.

Appendix

Let T be a fixed positive real number. Let $\xi =(\xi _t)_{t\in [0,T]}$ be a predictable process in ${\mathbf {S}}^{2,p}$, called obstacle or barrier in ${\mathbf {S}}^{2,p}$.

Definition 9

(One barrier predictable reflected BSDE with driver 0) A process $(Y,Z,M,A,B) \in {\mathbf {S}}^{2,p} \times {\mathbf {H}}^2 \times {\mathbf {M}}^{2,\perp } \times ({\mathbf {S}}^{2,p})^2$ is said to be solution to the predictable reflected BSDE with (lower) barrier $\xi$ and driver 0, if

$$\begin{aligned} Y_\tau =\xi _T - \int _\tau ^T Z_s dWs - (M_{T^-}-M_{\tau ^-}) + A_T-A_\tau + B_{T^-}-B_{\tau ^-}, \end{aligned}$$

(5.1)

with

• (i) $\xi _\tau \le Y_\tau$ a.s. for all $\tau \in {\mathscr {T}}_0^p$,

• (ii) A is a non-decreasing right-continuous process with $A_0=0$ and such that

  $$\begin{aligned} \int _0^T \mathbf{1 }_{\lbrace Y_t > \xi _t \rbrace } \mathrm{d}A^c_t=0 \quad \text {a.s. and}\quad (Y_{\tau ^-} - \xi _{\tau ^-})(A^d_\tau - A^d_{\tau ^-})=0 \quad \text {a.s. for all}\quad \tau \in {\mathscr {T}}_0^p,\nonumber \\ \end{aligned}$$

  (5.2)

• (iii) B is a non-decreasing right-continuous adapted purely discontinuous process with $B_{0^-}=0$, and such that

  $$\begin{aligned} (Y_\tau -\xi _\tau )(B_\tau -B_{\tau ^-})=0 \quad \text {a.s. for all}\quad \tau \in {\mathscr {T}}_0^p. \end{aligned}$$

  (5.3)

The following result established by S. Bouhadou and Y. Ouknine in [2] (see Theorem 2, p. 10):

Proposition 4

Let $\xi$ be a process in ${\mathbf {S}}^{2,p}$. There exists a unique solution $(Y,Z,M,A,B) \in {\mathbf {S}}^{2,p} \times {\mathbf {H}}^2 \times {\mathbf {M}}^{2,\perp } \times ({\mathbf {S}}^{2,p})^2$ of the predictable reflected BSDE from Definition 9, and for each stopping time $\tau \in {\mathscr {T}}^p_0$, we have

$$\begin{aligned} Y_\tau = \mathop {\mathrm{ess\, sup}}\limits _{S \in {\mathscr {T}}^p_\tau } E(\xi _S \vert {\mathscr {F}}_{\tau ^-}) \quad \text {a.s.} \end{aligned}$$

Here are some elementary properties of this operator.

Lemma 6

The operator ${\mathscr {P}}{} \textit{re}$ is non-decreasing, i.e., for $\xi , \xi ' \in {\mathbf {S}}^{2,p}$ such that $\xi \le \xi '$ we have ${\mathscr {P}}{} \textit{re}[\xi ] \le {\mathscr {P}}{} \textit{re}[\xi ']$. Further, for each $\xi \in {\mathbf {S}}^{2,p}$, ${\mathscr {P}}{} \textit{re}[\xi ]$ is a predictable strong supermartingale and satisfies ${\mathscr {P}}{} \textit{re}[\xi ] \ge \xi$.

Proof

By definition, ${\mathscr {P}}{} \textit{re}[\xi ]$ is the first component of the solution of the predictable reflected BSDE (5.1). Hence, Theorem 2 in [2] shows that ${\mathscr {P}}{} \textit{re}[\xi ]$ is the predictable value function associated with the reward $\xi$, that is for each stopping time $S \in {\mathscr {T}}^p_0$

$$\begin{aligned} {\mathscr {P}}{} \textit{re}[\xi ]_S = \mathop {\mathrm{ess\, sup}}\limits _{\tau \in {\mathscr {T}}^p_S} E(\xi _\tau \vert {\mathscr {F}}_{S^-}). \end{aligned}$$

Thus, the operator ${\mathscr {P}}{} \textit{re}$ is non-decreasing and the process $({\mathscr {P}}{} \textit{re}[\xi ])_{t\in [0,T]}$ is characterized as the predictable Snell envelope associated with the process $(\xi )_{t\in [0,T]}$, that is the smallest strong predictable supermartingale greater than or equal to $\xi$ (cf. [2, Lemma 15, p. 34]) and the lemma follows. $\square$

Remark 13

If $\xi \in {\mathscr {T}}^p_0$ is a predictable strong supermartingale, then ${\mathscr {P}}{} \textit{re}[\xi ]=\xi$. Indeed, it remains to show that ${\mathscr {P}}{} \textit{re}[\xi ] \le \xi$. Let $S \in {\mathscr {T}}^p_0$, since $\xi$ is a predictable strong supermartingale, for each stopping time $\tau \in {\mathscr {T}}^p_S$, we have

$$\begin{aligned} E(\xi _\tau \vert {\mathscr {F}}_{S^-}) \le \xi _S. \end{aligned}$$

By definition of the essential supremum, we get ${\mathscr {P}}{} \textit{re}[\xi ]_S \le \xi _S$. Consequently, ${\mathscr {P}}{} \textit{re}[\xi ] = \xi$.

Remark 14

The limit of a non-decreasing sequence of predictable strong supermartingales is also a predictable strong supermartingale (It can be shown using Lebesgue’s dominated convergence theorem and the fact that every trajectory of a predictable strong supermartingale is bounded on all compact interval of ${\mathbb {R}}^+$).

Proof of Lemma 3.16

Let $n\in {\mathbb {N}}$, we begin by proving that the processes ${\mathbf {J}}^{p,n}$ and $\bar{{\mathbf {J}}}^{p,n}$ are valued in $[0,+\infty ]$. By definition, we have:

$$\begin{aligned} {\mathbf {J}}^{p,n}_T=\bar{{\mathbf {J}}}^{p,n}_T=0 \quad \text {a.s. for each} \quad n. \end{aligned}$$

(5.4)

Hence, ${\mathbf {J}}^{p,n}$ and $\bar{{\mathbf {J}}}^{p,n}$ are nonnegative since they are predictable strong supermartingales. From ${\tilde{\xi }}^{p,g}_T={\tilde{\zeta }}^{p,g}_T=0$, it follows that $(\bar{{\mathbf {J}}}^{p,n} + {\tilde{\xi }}^{g,p} ) \mathbf{1 }_{[0,T)}=(\bar{{\mathbf {J}}}^{p,n} + {\tilde{\xi }}^{g,p} )$ and $({\mathbf {J}}^{p,n} -{\tilde{\zeta }}^{g,p} ) \mathbf{1 }_{[0,T)}=({\mathbf {J}}^{p,n} -{\tilde{\zeta }}^{g,p} )$. We prove that $({\mathbf {J}}^{p,n})_{n\in {\mathbb {N}}}$ and $(\bar{{\mathbf {J}}}^{p,n})_{n\in {\mathbb {N}}}$ are non-decreasing sequences of processes.

We have ${\mathbf {J}}^{p,0}=0 \le {\mathbf {J}}^{p,1}$ and $\bar{{\mathbf {J}}}^{p,0}=0 \le \bar{{\mathbf {J}}}^{p,1}$. Suppose that ${\mathbf {J}}^{p,n-1} \le {\mathbf {J}}^{p,n}$ and $\bar{{\mathbf {J}}}^{p,n-1} \le \bar{{\mathbf {J}}}^{p,n}$. The non-decreasingness of the operator ${\mathscr {P}}{} \textit{re}$ gives

$$\begin{aligned}&{\mathscr {P}}{} \textit{re}[{\mathbf {J}}^{p,n-1} -{\tilde{\zeta }}^{g,p} ] \le {\mathscr {P}}{} \textit{re}[{\mathbf {J}}^{p,n} -{\tilde{\zeta }}^{g,p} ] \\&{\mathscr {P}}{} \textit{re}[\bar{{\mathbf {J}}}^{p,n-1} + {\tilde{\xi }}^{g,p}] \le {\mathscr {P}}{} \textit{re}[\bar{{\mathbf {J}}}^{p,n} + {\tilde{\xi }}^{g,p}]. \end{aligned}$$

Thus, ${\mathbf {J}}^{p,n-1} \le {\mathbf {J}}^{p,n}$ and $\bar{{\mathbf {J}}}^{p,n-1} \le \bar{{\mathbf {J}}}^{p,n}$, which is the desired conclusion.

The processes ${\mathbf {J}}^p := \lim \uparrow {\mathbf {J}}^{p,n}$ and $\bar{{\mathbf {J}}}^p := \lim \uparrow \bar{{\mathbf {J}}}^{p,n}$ are predictable (valued in $[0,+\infty ]$) as the limit of sequences of predictable nonnegative processes. By (5.4), we get ${\mathbf {J}}^p _T = \bar{{\mathbf {J}}}^p_T=0$ a.s. Moreover, ${\mathbf {J}}^p$ and $\bar{{\mathbf {J}}}^p$ are strong supermartingales valued in $[0, +\infty ]$ (cf. Remark 14).

We next prove that ${\mathbf {J}}^p$ and $\bar{{\mathbf {J}}}^p$ belong to ${\mathbf {S}}^{2,p}$. For this purpose, consider $H^{p}$ and ${\bar{H}}^{p}$ the nonnegative predictable strong supermartingales that come from Mokobodzki’s condition for $(\xi ,\zeta )$. Then, we define two processes $H^{g,p}$ and ${\bar{H}}^{g,p}$ as follows:

$$\begin{aligned} H^{g,p}_t&:= H^p_t + E[\xi ^-_T \vert {\mathscr {F}}_{t^-}] + E\left[ \int _t^T g^-(s) \mathrm{d}s \vert {\mathscr {F}}_{t^-}\right] \\ {\bar{H}}^{g,p}_t&:= {\bar{H}}^p_t + E[\xi ^+_T \vert {\mathscr {F}}_{t^-}] + E\left[ \int _t^T g^+(s) \mathrm{d}s, \vert {\mathscr {F}}_{t^-}\right] . \end{aligned}$$

It is easy to check that $H^{g,p}$ and ${\bar{H}}^{g,p}$ are nonnegative predictable strong supermartingales in ${\mathbf {S}}^{2,p}$. From Mokobodzki’s condition, we get

$$\begin{aligned} {\tilde{\xi }}^{g,p} \le H^{g,p} - {\bar{H}}^{g,p} \le {\tilde{\zeta }}^{g,p}. \end{aligned}$$

(5.5)

Let us now show by induction that ${\mathbf {J}}^{p,n} \le H^{g,p}$ and $\bar{{\mathbf {J}}}^{p,n} \le {\bar{H}}^{g,p}$, for all $n \in {\mathbb {N}}$. First, we have ${\mathbf {J}}^{p,0}=0 \le H^{g,p}$ and $\bar{{\mathbf {J}}}^{p,0}=0 \le {\bar{H}}^{g,p}$. Suppose that, for a fixed $n \in {\mathbb {N}}$, we have ${\mathbf {J}}^{p,n} \le H^{g,p}$ and $\bar{{\mathbf {J}}}^{p,n} \le {\bar{H}}^{g,p}$. From Eq. (5.5), we get ${\mathbf {J}}^{p,n} \le {\bar{H}}^{g,p} + {\tilde{\zeta }}^{g,p}$ and $\bar{{\mathbf {J}}}^{p,n} \le H^{g,p} - {\tilde{\xi }}^{g,p}$. As the operator ${\mathscr {P}}{} \textit{re}$ is a non-decreasing operator (see Lemma 6), we get

$$\begin{aligned}&{\mathbf {J}}^{p,n+1} = {\mathbf {P}}{} \textit{re}[\bar{{\mathbf {J}}}^{p,n} + {\tilde{\xi }}^{g,p}] \le {\mathscr {P}}{} \textit{re}[H^{g,p}]\quad \text {and}\\&\bar{{\mathbf {J}}}^{p,n+1} = {\mathbf {P}}{} \textit{re}[{\mathbf {J}}^{p,n} - {\tilde{\zeta }}^{g,p}] \le {\mathscr {P}}{} \textit{re}[{\bar{H}}^{g,p}]. \end{aligned}$$

Since $H^{g,p}$ and ${\bar{H}}^{g,p}$ are predictable strong supermartingales, it follows by Remark 13 that ${\mathscr {P}}{} \textit{re}[H^{g,p}]= H^{g,p}$ and ${\mathscr {P}}{} \textit{re}[{\bar{H}}^{g,p}]= {\bar{H}}^{g,p}$. Hence, ${\mathbf {J}}^{p,n+1} \le H^{g,p}$ and $\bar{{\mathbf {J}}}^{p,n+1} \le {\bar{H}}^{g,p}$, which is the desired conclusion.

By letting n tend to $+ \infty$ in ${\mathbf {J}}^{p,n} \le H^{g,p}$ and $\bar{{\mathbf {J}}}^{p,n} \le {\bar{H}}^{g,p}$, we get ${\mathbf {J}}^{p} \le H^{g,p}$ and $\bar{{\mathbf {J}}}^{p} \le {\bar{H}}^{g,p}$. Hence, ${\mathbf {J}}^{p}$ and $\bar{{\mathbf {J}}}^{p}$ belong to ${\mathbf {S}}^{2,p}$.

The proof is completed by showing that the processes ${\mathbf {J}}^{p}$ and $\bar{{\mathbf {J}}}^{p}$ satisfy the system (3.11). Note that $(\bar{{\mathbf {J}}}^{p,n} + {\tilde{\xi }}^{g,p})_{n\in {\mathbb {N}}}$ is a non-decreasing sequence of processes belonging to ${\mathbf {S}}^{2,p}$. As the operator ${\mathscr {P}}{} \textit{re}$ is a non-decreasing, the sequence $({\mathscr {P}}{} \textit{re}[\bar{{\mathbf {J}}}^{p,n} + {\tilde{\xi }}^{g,p}])_{n\in {\mathbb {N}}}$ is also non-decreasing. Hence, for each $n \in {\mathbb {N}}$, the following property

$$\begin{aligned} {\mathbf {J}}^{p,n+1} ={\mathscr {P}}{} \textit{re}[\bar{{\mathbf {J}}}^{p,n} + {\tilde{\xi }}^{g,p}] \le {\mathscr {P}}{} \textit{re}[\bar{{\mathbf {J}}}^p + {\tilde{\xi }}^{g,p}], \end{aligned}$$

holds. By letting n go to $+\infty$, we get

$$\begin{aligned} {\mathbf {J}}^{p} \le {\mathscr {P}}{} \textit{re}[\bar{{\mathbf {J}}}^p + {\tilde{\xi }}^{g,p}]. \end{aligned}$$

(5.6)

Now, by definition of ${\mathscr {P}}{} \textit{re}[\bar{{\mathbf {J}}}^{p,n} + {\tilde{\xi }}^{g,p}]$ as the solution of the predictable reflected BSDE with obstacle $\bar{{\mathbf {J}}}^{p,n} + {\tilde{\xi }}^{g,p}$, we have ${\mathscr {P}}{} \textit{re}[\bar{{\mathbf {J}}}^{p,n} + {\tilde{\xi }}^{g,p}] \ge \bar{{\mathbf {J}}}^{p,n} + {\tilde{\xi }}^{g,p}$, for all $n\in {\mathbb {N}}$. Thus, by letting n go to $+\infty$, we get ${\mathbf {J}}^{p} \ge \bar{{\mathbf {J}}}^p + {\tilde{\xi }}^{g,p}$. Hence,

$$\begin{aligned} {\mathscr {P}}{} \textit{re}[{\mathbf {J}}^{p}] \ge {\mathscr {P}}{} \textit{re}[\bar{{\mathbf {J}}}^p + {\tilde{\xi }}^{g,p}] \end{aligned}$$

(5.7)

Since ${\mathbf {J}}^{p}$ is a predictable strong supermartingale, Remark 13 implies ${\mathscr {P}}{} \textit{re}[{\mathbf {J}}^{p}]= {\mathbf {J}}^{p}$. From inequality (5.6), It follows that ${\mathbf {J}}^{p}= {\mathscr {P}}{} \textit{re}[\bar{{\mathbf {J}}}^p + {\tilde{\xi }}^{g,p}]$. We show similarly that $\bar{{\mathbf {J}}}^{p}= {\mathscr {P}}{} \textit{re}[{\mathbf {J}}^p + {\tilde{\zeta }}^{g,p}]$. Since, ${\mathbf {J}}^{p}_T= \bar{{\mathbf {J}}}^{p}_T=0$, we conclude that ${\mathbf {J}}^{p}$ and $\bar{{\mathbf {J}}}^{p}$ are solutions of the system (3.11), and the lemma follows. $\square$