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Stochastic Differential Equations Driven by G-Brownian Motion with Mean Reflections

Hanwu Li, Ning Ning

TL;DR

This paper analyzes mean-reflected stochastic differential equations driven by $G$-Brownian motion, where the reflection constraint is imposed via $\widehat{\mathbb{E}}[l(t,X_t)] \ge 0$ rather than pathwise conditions. It develops a Skorokhod problem with mean reflection under $G$-expectation and proves existence and uniqueness through two constructive approaches, enabling a contraction-mapping framework. These Skorokhod results are then used to establish well-posedness and regularity for the forward mean-reflected $G$-SDE, including moment bounds for the solution and, under smooth loss $l$, Lipschitz continuity of the compensator $A$. The work extends mean-reflected SDE theory to the $G$-framework, addressing volatility/Knightian uncertainty, and provides tools applicable to risk management and nonlinear PDE connections.

Abstract

In this paper, we study the mean reflected stochastic differential equations driven by G-Brownian motion, where the constraint depends on the expectation of the solution rather than on its paths. Well-posedness is achieved by first investigating the Skorokhod problem with mean reflection under G-expectation. Two approaches to constructing the solution are introduced, both offering insights into desired properties and aiding in the application of the contraction mapping method.

Stochastic Differential Equations Driven by G-Brownian Motion with Mean Reflections

TL;DR

This paper analyzes mean-reflected stochastic differential equations driven by -Brownian motion, where the reflection constraint is imposed via rather than pathwise conditions. It develops a Skorokhod problem with mean reflection under -expectation and proves existence and uniqueness through two constructive approaches, enabling a contraction-mapping framework. These Skorokhod results are then used to establish well-posedness and regularity for the forward mean-reflected -SDE, including moment bounds for the solution and, under smooth loss , Lipschitz continuity of the compensator . The work extends mean-reflected SDE theory to the -framework, addressing volatility/Knightian uncertainty, and provides tools applicable to risk management and nonlinear PDE connections.

Abstract

In this paper, we study the mean reflected stochastic differential equations driven by G-Brownian motion, where the constraint depends on the expectation of the solution rather than on its paths. Well-posedness is achieved by first investigating the Skorokhod problem with mean reflection under G-expectation. Two approaches to constructing the solution are introduced, both offering insights into desired properties and aiding in the application of the contraction mapping method.
Paper Structure (10 sections, 17 theorems, 94 equations)

This paper contains 10 sections, 17 theorems, 94 equations.

Table of Contents

  1. Introduction
  2. Background
  3. Our contributions
  4. Organization of the paper
  5. Preliminaries
  6. The Skorokhod problem with mean reflection
  7. Definition and an illustration
  8. First construction of solutions
  9. Second construction of solutions
  10. Mean reflected $G$-SDEs

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 (21)

  • Theorem 2.1: DHP11
  • Proposition 2.2: LL
  • Lemma 2.3: LiuW
  • Lemma 2.4: DHP11
  • Definition 2.5
  • Proposition 2.6: P19
  • Proposition 2.7: LPS
  • Definition 3.1
  • Theorem 3.2
  • Example 3.3
  • ...and 11 more