Stability of backward stochastic differential equations: the general case

Abstract

In this paper, we obtain stability results for backward stochastic differential equations with jumps (BSDEs) in a very general framework. More specifically, we consider a convergent sequence of standard data, each associated to their own filtration, and we prove that the associated sequence of (unique) solutions is also convergent. The current result extends earlier contributions in the literature of stability of BSDEs and unifies several frameworks for numerical approximations of BSDEs and their implementations. CONTENTS 1. INTRODUCTION 1 General notation 4 2. FRAMEWORK 5 3. STABILITY OF BACKWARD STOCHASTIC DIFFERENTIAL EQUATIONS 8 3.1. Outline of the proof 8 3.2. Uniform a priori BSDE estimates 9 3.3. Reducing the complexity of the induction steps and proving Theorem 3.1 10 3.4. The first step of the induction is valid 15 3.5. The p-th step of the induction is valid 18 3.6. On the nature of the conditions 25 4. EXAMPLES AND APPLICATIONS 26 4.1. Continuous martingales 27 4.2. Probabilistic numerical schemes for deterministic equations 27 4.3. PII martingales 27 4.4. BSDEs as dual problems 28 APPENDIX A. AUXILIARY RESULTS 28 A.1. Definitions and complete notation 28 A.2. Moore-Osgood's theorem 30 A.3. Weak convergence of measures on the positive real line 30 A.4. Some helpful lemmata for the proof of Theorem 3.1. 38 REFERENCES 41