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

TL;DR

This paper establishes well-posedness and comparison results for generalized BSDEs and reflected generalized BSDEs, motivated by pricing vulnerable options, and explores penalization schemes leading to optimal stopping and Dynkin game formulations.

Contribution

It provides new well-posedness results and comparison theorems for generalized BSDEs with nondecreasing drivers, and introduces penalization schemes for constrained optimal stopping and Dynkin games.

Findings

Proved well-posedness and comparison theorems for generalized BSDEs.

Developed penalization schemes converging to optimal stopping and Dynkin games.

Applied results to pricing vulnerable European and American options.

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 $A$.