Reflected backward stochastic differential equations under stopping with an arbitrary random time

Abstract

This paper addresses reflected backward stochastic differential equations (RBSDE hereafter) that take the form of \begin{eqnarray*} \begin{cases} dY_t=f(t,Y_t, Z_t)d(t\wedge\tau)+Z_tdW_t^{\tau}+dM_t-dK_t,\quad Y_{\tau}=\xi, Y\geq S\quad\mbox{on}\quad \Lbrack0,\tau\Lbrack,\quad \displaystyle\int_0^{\tau}(Y_{s-}-S_{s-})dK_s=0\quad P\mbox{-a.s..}\end{cases} \end{eqnarray*} Here $\tau$ is an arbitrary random time that might not be a stopping time for the filtration $\mathbb F$ generated by the Brownian motion $W$. We consider the filtration $\mathbb G$ resulting from the progressive enlargement of $\mathbb F$ with $\tau$ where this becomes a stopping time, and study the RBSDE under $\mathbb G$. Precisely, we focus on answering the following problems: a) What are the sufficient minimal conditions on the data $(f, \xi, S, \tau)$ that guarantee the existence of the solution of the $\mathbb G$-RBSDE in $L^p$ ($p>1$)? b) How can we estimate the solution in norm using the triplet-data $(f, \xi, S)$? c) Is there an RBSDE under $\mathbb F$ that is intimately related to the current one and how their solutions are related to each other? We prove that for any random time, having a positive Az\'ema supermartingale, there exists a positive discount factor ${\widetilde{\cal E}}$ that is vital in answering our questions without assuming any further assumption on $\tau$, and determining the space for the triplet-data $(f,\xi, S)$ and the space for the solution of the RBSDE as well.