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

This paper considers the setting governed by $(F,\tau )$ , where $F$ is the "public" flow of information, and τ is a random time which might not be $F$ -observable. This framework covers credit risk and life insurance. In this setting, $F$ is assumed to be generated by a d-dimensional Brownian motion W and ξ is a vulnerable claim, whose payment's policy depends essentially on the occurrence of τ. The hedging problems, in many directions, for this claim led to the question of studying the linear reflected-backward-stochastic differential equations (RBSDE hereafter), $\begin{array}{c}d{Y}_t=-f\left(t\right)d(t\wedge \tau )+Z_{t}d{W}_{t\wedge \tau }-d{M}_t-d{K}_t,Y_{\tau }=\xi ,\\ Y\ge S\text{on}〚0,\tau 〚,{\int }_0^{\tau }(Y_{s-}-S_{s-})d{K}_s=0P\text{-a.s.}.\end{array}$ This is the objective of this paper. For this RBSDE and without any further assumption on τ 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 $F$ -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.