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

Libo LiRuyi LiuMarek Rutkowski
Dec 2022
摘要
The paper is directly motivated by the pricing of vulnerable European andAmerican options in a general hazard process setup and a related study of thecorresponding pre-default backward stochastic differential equations (BSDE) andpre-default reflected backward stochastic differential equations (RBSDE). Wework with a generic filtration $\FF$ for which the martingale representationproperty 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 thewell-posedness results and comparison theorems for a generalized BSDE and areflected generalized BSDE with a continuous and nondecreasing driver $A$.Second, we study extended penalization schemes for a generalized BSDE and areflected generalized BSDE in which we penalize against the driver in order toobtain in the limit either a particular optimal stopping problem or a Dynkingame in which the set of admissible exercise time is constrained to the rightsupport of the measure generated by $A$.
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