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

Paper: arXiv 2408.04758

Authors: T. Choulli, S. Alsheyab

Abstract

This paper considers the setting governed by $(\mathbb{F},τ)$, 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 theory and life insurance. In this setting, we assume $\mathbb{F}$ being generated by a Brownian motion $W$ and consider a vulnerable claim $ξ$, whose payment’s policy depends {\it{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{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 $τ$ that might neglect any risk intrinsic to its stochasticity, we answer the following: a) What are the sufficient minimal conditions on the data $(f, ξ, S, τ)$ that guarantee the existence of the solution to this RBSDE? b) How can we estimate the solution in norm using $(f, ξ, S)$? 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.

Complexity vs Empirical Score

• Math Complexity: 9.0/10
• Empirical Rigor: 1.0/10
• Quadrant: Lab Rats — theoretically deep, empirically untested

Why this score: The paper is a rigorous mathematical investigation of reflected backward stochastic differential equations (RBSDEs) under a random horizon without relying on standard immersion (H-hypothesis) assumptions. It is purely theoretical with advanced measure theory and stochastic calculus, but completely lacks empirical data or backtesting.

Research Flowchart

  flowchart TD
    A[Research Goal: Solve Linear RBSDE for Vulnerable Claims under Random Horizon τ] --> B[Input Data: Parameters f, ξ, S, W, and Random Time τ]
    B --> C[Methodology: Progressive Enlargement of Filtration & Stochastic Calculus]
    C --> D[Derive Minimal Conditions for Existence & Uniqueness]
    C --> E[Formulate Associated F-RBSDE Reduction & Norm Estimates]
    D --> F[Outcomes: Rigorous RBSDE Framework for Credit Risk & Life Insurance Hedging]
    E --> F