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Multi-dimensional reflected BSDEs driven by $G$-Brownian motion with diagonal generators

Hanwu Li, Guomin Liu

Abstract

We consider the well-posedness problem of multi-dimensional reflected backward stochastic differential equations driven by $G$-Brownian motion ($G$-BSDEs) with diagonal generators. Two methods, i.e., the penalization method and the Picard iteration argument, are provided to prove the existence and uniqueness of solutions. We also study its connection with the obstacle problem of a system of fully nonlinear PDEs.

Multi-dimensional reflected BSDEs driven by $G$-Brownian motion with diagonal generators

Abstract

We consider the well-posedness problem of multi-dimensional reflected backward stochastic differential equations driven by -Brownian motion (-BSDEs) with diagonal generators. Two methods, i.e., the penalization method and the Picard iteration argument, are provided to prove the existence and uniqueness of solutions. We also study its connection with the obstacle problem of a system of fully nonlinear PDEs.

Paper Structure

This paper contains 10 sections, 22 theorems, 117 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. $G$-expectation
  4. Multi-dimensional $G$-BSDEs
  5. One-dimensional reflected $G$-BSDEs
  6. Multi-dimensional reflected $G$-BSDEs
  7. A priori estimates
  8. Construction via penalization method
  9. Picard iteration method
  10. Multi-dimensional obstacle problems for fully nonlinear PDEs

Key Result

Theorem 2.1

There exists a weakly compact set $\mathcal{P}$ of probability measures on $(\Omega,\mathcal{B}(\Omega))$ such that $\mathcal{P}$ is called a set that represents $\hat{\mathbb{E}}$.

Theorems & Definitions (39)

  • Theorem 2.1: DHP11
  • Theorem 2.2: S11
  • Theorem 2.3: Liu-stochastics
  • Theorem 2.4: Liu-stochastics
  • Theorem 2.5: LPSHLS
  • Proposition 2.6: LPSH
  • Proposition 2.7: LPSH
  • Proposition 2.8: LPSHLS
  • Theorem 2.9: LPSHLS
  • Theorem 3.1
  • ...and 29 more