2024/08/08 by Tahir Choulli, Choulli, T., Safa’ Alsheyab +1
Decision Sciences · Economics, Econometrics and Finance · Social Sciences · #FOS: Economics and business #FOS: Mathematics #Insurance, Mortality, Demography, Risk Management #Mathematical Finance (q-fin.MF) #Probability (math.PR) #Probability and Risk Models #Stochastic processes and financial applications
paper · pdf · doi:10.48550/arxiv.2408.04758
openalex publication_date 2024/08/08 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/01
This paper considers the setting governed by (\mathbbF,τ), where \mathbbF is the "public" flow of information, and τ is a random time which might not be \mathbbF-observable. This framework covers credit risk theory and life insurance. In this setting, we assume \mathbbF being generated by a Brownian motion W and consider a vulnerable claim ξ, whose payment's policy depends \itessentially 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), \beginsplit amp;dYt=f(t)d(t\wedgeτ)+ZtdWt\wedgeτ+dMt-dKt, Yτ=ξ,
amp; Y≥ S\quadon \Lbrack0,τ\Lbrack, ∫0τ(Ys--Ss-)dKs=0 P-a.s..\endsplit 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.