Reflected backward stochastic differential equations under stopping with an arbitrary random time

2021/07/25 by Safa Alsheyab, Alsheyab, Safa, Tahir Choulli +1 · 1 citation
Economics, Econometrics and Finance · #FOS: Economics and business #FOS: Mathematics #Mathematical Finance (q-fin.MF) #Probability (math.PR) #Stochastic processes and financial applications

paper · pdf · doi:10.48550/arxiv.2107.11896

openalex publication_date 2021/07/25 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

This paper addresses reflected backward stochastic differential equations (RBSDE hereafter) that take the form of \begincases dY_t=f(t,Y_t, Z_t)d(t\wedgeτ)+Z_tdW_t^τ+dM_t-dK_t, Y_τ=ξ, Y≥ S\quadon \Lbrack0,τ\Lbrack, ∫₀^τ(Y_s--S_s-)dK_s=0 P-a.s..\endcases Here τ is an arbitrary random time that might not be a stopping time for the filtration \mathbb F generated by the Brownian motion W. We consider the filtration \mathbb G resulting from the progressive enlargement of \mathbb F with τ where this becomes a stopping time, and study the RBSDE under \mathbb G. Precisely, we focus on answering the following problems: a) What are the sufficient minimal conditions on the data (f, ξ, S, τ) that guarantee the existence of the solution of the \mathbb G-RBSDE in L^p (p>1)? b) How can we estimate the solution in norm using the triplet-data (f, ξ, S)? c) Is there an RBSDE under \mathbb F that is intimately related to the current one and how their solutions are related to each other? We prove that for any random time, having a positive Azéma supermartingale, there exists a positive discount factor \widetilde\cal E that is vital in answering our questions without assuming any further assumption on τ, and determining the space for the triplet-data (f,ξ, S) and the space for the solution of the RBSDE as well.

Cited by

• Doubly Reflected BSDEs with default time under stochastic Lipschitz coefficients and Applications

Related

• Generalized BSDEs with random time horizon in a progressively enlarged\n filtration
• Generalized Reflected BSDEs with RCLL Random Obstacles in a General Filtration
• Linear reflected backward stochastic differential equations arising from vulnerable claims in markets with random horizon
• Backward Stochastic Differential Equations with Non-Markovian Singular Terminal Conditions with General Driver and Filtration
• Second order backward SDE with random terminal time