Linear reflected backward stochastic differential equations arising from vulnerable claims in markets with random horizon

Abstract: This paper considers the setting governed by $(\mathbb{F},\tau)$, where $\mathbb{F}$ is the "public" flow of information, and $\tau$ is a random time which might not be $\mathbb{F}$-observable. This framework covers credit risk theory and life insurance. In this setting, we assume $\mathbb{F}$ being generated by a Brownian motion $W$ and consider a vulnerable claim $\xi$, whose payment's policy depends {\it{essentially}} on the occurrence of $\tau$. The hedging problems, in many directions, for this claim led to the question of studying the linear reflected-backward-stochastic differential equations (RBSDE hereafter), \begin{equation*} \begin{split} &dY_t=f(t)d(t\wedge\tau)+Z_tdW_{t\wedge{\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{split} \end{equation*} This is the objective of this paper. For this RBSDE and without any further assumption on $\tau$ that might neglect any risk intrinsic to its stochasticity, we answer the following: a) What are the sufficient minimal conditions on the data $(f, \xi, S, \tau)$ that guarantee the existence of the solution to this RBSDE? b) How can we estimate the solution in norm using $(f, \xi, S)$? c) Is there an $\mathbb F$-RBSDE that is intimately related to the current one and how their solutions are related to each other? This latter question has practical and theoretical leitmotivs.