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

Abstract

This paper considers the setting governed by $(\mathbb{F},\ddot{I}„)$, where $\mathbb{F}$ is the "public" flow of information, and $\ddot{I}„$ 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 $\hat{I}¾$, whose payment's policy depends {\it{essentially}} on the occurrence of $\ddot{I}„$. 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\wedgeτ)+Z_tdW_{t\wedgeτ}+dM_t-dK_t,\quad Y_τ=ξ,\\ & Y\geq S\quad\mbox{on}\quad \Lbrack0,τ\Lbrack,\quad \displaystyle\int_0^τ(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 $\ddot{I}„$ that might neglect any risk intrinsic to its stochasticity, we answer the following: a) What are the sufficient minimal conditions on the data $(f,\hat{I}¾,S,\ddot{I}„)$ that guarantee the existence of the solution to this RBSDE? b) How can we estimate the solution in norm using $(f,\hat{I}¾,S)$? 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.

arXiv admin note: substantial text overlap with arXiv:2301.09836