Linear reflected backward stochastic differential equations arising from vulnerable claims in markets with random horizon
Year: 2026
Type: article
Abstract: This paper considers the setting governed by $(\mathbb{F},\tau )$, where $\mathbb{F}$ is the “public” flow of information, and τ is a random time which might not be $\mathbb{F}$-observable. This framework covers credit risk and life insurance. In this setting, $\mathbb{F}$ 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), $$ \begin{aligned} &dY_{t}=-f(t)d(t\wedge \tau )+Z_{t}dW_{t\wedge{\tau}}-dM_{t}-dK_{t},\quad Y_{\tau}=\xi ,\\ & Y\geq S\quad \text{on}\quad [\!\![0,\tau [\!\![,\quad \displaystyle \int _{0}^{\tau}(Y_{s-}-S_{s-})dK_{s}=0\quad P\text{-a.s.}.\end{aligned} $$ 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 $\mathbb{F}$-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.
Authors: Tahir Choulli, Safa’ Alsheyab
Language: English
Subfield: Finance
Domain: Social Sciences
Open Access status: hybrid
Estimated APC paid: $1,890
OpenAlex ID: W7126258428