Penalization schemes for BSDEs and reflected BSDEs with generalized driver

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). The goal of this work is 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 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 constrained optimal stopping problem or a constrained Dynkin game in which the set of minimizer’s admissible exercise times is constrained to the right support of the measure generated by $A$.

Keywords

Generalized BSDEs / Reflected generalized BSDEs / Penalization scheme / Constrained optimal stopping / Constrained Dynkin game

Acknowledgements

The authors are very grateful to the anonymous reviewer for careful reading of their submission and criticism leading to an improvement of their work. The research of M. Rutkowski was supported by the Australian Research Council Discovery Project (Grant No. DP220103106).

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