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

TL;DR

This paper studies linear reflected backward stochastic differential equations (RBSDEs) in a setting with a random horizon, relevant for credit risk and life insurance, establishing conditions for solutions and their properties.

Contribution

It provides minimal sufficient conditions for the existence of solutions to RBSDEs with random horizons without additional assumptions on the random time.

Findings

Established existence conditions for RBSDE solutions.

Derived norm estimates for solutions based on data.

Explored relationships between RBSDEs under different filtrations.

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.