Well-posedness and penalization schemes for generalized BSDEs and reflected generalized BSDEs

2022/12/25 by Li, Libo, Liu, Ruyi, Rutkowski, Marek
#FOS: Economics and business #FOS: Mathematics #Mathematical Finance (q-fin.MF) #Probability (math.PR)

paper · doi:10.48550/arxiv.2212.12854

Abstract

The paper is directly motivated by the pricing of vulnerable European and American options in a general hazard process setup and a related study of the corresponding pre-default backward stochastic differential equations (BSDE) and pre-default reflected backward stochastic differential equations (RBSDE). We work with a generic filtration \FF for which the martingale representation property is assumed to hold with respect to a square-integrable martingale M and the goal of this work is of twofold. First, we aim to establish the well-posedness results and comparison theorems for a generalized BSDE and a reflected generalized BSDE with a continuous and nondecreasing driver A. Second, we study extended penalization schemes for a generalized BSDE and a reflected generalized BSDE in which we penalize against the driver in order to obtain in the limit either a particular optimal stopping problem or a Dynkin game in which the set of admissible exercise time is constrained to the right support of the measure generated by A.

Related

• Stochastic Control Representations for Penalized Backward Stochastic Differential Equations
• Doubly reflected BSDEs driven by Inhomogeneous simple Levy processes: Applications to generalized Dynkin games
• Generalized BSDE With 2-Reflecting Barriers and Stochastic Quadratic Growth
• Generalized Reflected BSDEs with RCLL Random Obstacles in a General Filtration
• Mean-field reflected backward stochastic differential equations