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

Paper: arXiv 2212.12854

Authors: Libo Li, Ruyi Liu, Marek Rutkowski

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 $A$.

Complexity vs Empirical Score

• Math Complexity: 9.2/10
• Empirical Rigor: 1.5/10
• Quadrant: Lab Rats — theoretically deep, empirically untested

Why this score: The paper presents rigorous mathematical foundations for generalized BSDEs and penalization schemes relevant to vulnerable option pricing. It features high mathematical complexity with formal proofs, but lacks empirical backtesting or real-market data analysis.

Research Flowchart

  flowchart TD
    A[Goal: Study well-posedness & penalization for Generalized BSDEs/RBSDEs in Option Pricing] --> B[Inputs: Generic Filtration F, Martingale M, Continuous Driver A]
    B --> C[Methodology: Formulate Pre-default Generalized BSDEs & RBSDEs]
    C --> D[Establish Well-Posedness & Comparison Theorems]
    C --> E[Apply Extended Penalization Schemes against Driver A]
    D --> F[Outcomes: Pricing Vulnerable European & American Options]
    E --> F
    E --> G[Outcomes: Constrained Optimal Stopping & Dynkin Games]