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Doubly Reflected Backward SDEs Driven by G-Brownian Motions and Fully Nonlinear PDEs with Double Obstacles

Hanwu Li, Ning Ning

Abstract

In this paper, we introduce a new method to study the doubly reflected backward stochastic differential equation driven by G-Brownian motion (G-BSDE). Our approach involves approximating the solution through a family of penalized reflected G-BSDEs with a lower obstacle that are monotone decreasing. By employing this approach, we establish the well-posedness of the solution of the doubly reflected G-BSDE with the weakest known conditions, and uncover its relationship with the fully nonlinear partial differential equation with double obstacles for the first time.

Doubly Reflected Backward SDEs Driven by G-Brownian Motions and Fully Nonlinear PDEs with Double Obstacles

Abstract

In this paper, we introduce a new method to study the doubly reflected backward stochastic differential equation driven by G-Brownian motion (G-BSDE). Our approach involves approximating the solution through a family of penalized reflected G-BSDEs with a lower obstacle that are monotone decreasing. By employing this approach, we establish the well-posedness of the solution of the doubly reflected G-BSDE with the weakest known conditions, and uncover its relationship with the fully nonlinear partial differential equation with double obstacles for the first time.

Paper Structure

This paper contains 13 sections, 20 theorems, 150 equations.

Table of Contents

  1. Introduction
  2. Background
  3. Our contributions
  4. Organization of the paper
  5. Preliminaries
  6. $G$-expectation and $G$-Itô's calculus
  7. $G$-BSDEs
  8. Reflected $G$-BSDEs with a single obstacle
  9. Well-posedness of doubly reflected $G$-BSDEs
  10. Doubly reflected $G$-BSDEs
  11. Preliminary analysis
  12. Proof of Theorem \ref{['main']}
  13. Probabilistic representation of fully nonlinear PDEs with double obstacles

Key Result

Theorem 2.1

There exists a weakly compact set $\mathcal{P}$ of probability measures on $(\Omega_T,\mathcal{B}(\Omega_T))$, such that We call $\mathcal{P}$ a set that represents $\widehat{\mathbb{E}}$.

Theorems & Definitions (24)

  • Theorem 2.1: denis2011function
  • Definition 2.2
  • Proposition 2.3: hu2014comparison
  • Proposition 2.4: li2018supermartingale
  • Theorem 2.5: song2011some
  • Theorem 2.6: hu2014backward
  • Theorem 2.7: li2021backward
  • Theorem 2.8: hu2014comparison
  • Theorem 2.9: song2019properties
  • Theorem 2.10: li2018reflected
  • ...and 14 more