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.
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Notes
Note that this barrier is equal to \(E(A'_T-A'_t + C'_{T^-} - C'_{t^-} \vert {\mathscr {F}}_{t^-}) - {\tilde{\xi }}^{g,p}_t\) if \(t<T\), and 0 if \(t=T\).
Note that this barrier is equal to \(E(A_T-A_t + C_{T^-} - C_{t^-} \vert {\mathscr {F}}_{t^-}) - {\tilde{\zeta }}^{g,p}_t\) if \(t<T\), and 0 if \(t=T\).
We omit the exponent g in the notation for \({\mathbf {J}}^{p,n}\) and \(\bar{{\mathbf {J}}}^{p,n}\) for sake of simplicity.
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The authors thank the anonymous Referee for his valuable comments and suggestions from which the manuscript greatly benefited.
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Appendix
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
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
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 \(S \in {\mathscr {T}}^p_0\)
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
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:
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
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:
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
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
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
holds. By letting n go to \(+\infty \), we get
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,
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. \(\square \)
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Arharas, I., Bouhadou, S. & Ouknine, Y. Doubly Reflected Backward Stochastic Differential Equations in the Predictable Setting. J Theor Probab 35, 115–141 (2022). https://doi.org/10.1007/s10959-020-01070-5
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DOI: https://doi.org/10.1007/s10959-020-01070-5
Keywords
- Doubly reflected backward stochastic differential equations
- Predictable DRBSDEs
- Non-quasi-left continuous
- Picard iteration method
- Fixed point theorem