[Paper Review] Reflected Solutions of BSDEs Driven by G-Brownian Motion

Hanwu Li, Shigē Péng|arXiv (Cornell University)|May 31, 2017

Stochastic processes and financial applications14 references3 citations

TL;DR

This paper introduces a novel formulation of reflected backward stochastic differential equations (RBSDEs) driven by G-Brownian motion, replacing the classical Skorohod condition with a martingale condition to ensure uniqueness. Using a penalization approximation method under the G-framework, the authors prove existence and uniqueness of solutions, establishing a robust framework for modeling model uncertainty in nonlinear expectations and fully nonlinear PDEs.

ABSTRACT

In this paper, we study the reflected solutions of one-dimensional backward stochastic differential equations driven by G-Brownian motion (RGBSDE for short). The reflection keeps the solution above a given stochastic process. In order to derive the uniqueness of reflected GBSDEs, we apply a "martingale condition" instead of the Skorohod condition. Similar to the classical case, we prove the existence by approximation via penalization.

Motivation & Objective

• To address the lack of uniqueness in reflected G-BSDEs under the classical Skorohod condition by reformulating the problem using a martingale condition.
• To establish existence and uniqueness of reflected solutions in the G-framework, which extends classical BSDE theory to handle model uncertainty.
• To develop a constructive method based on penalization to approximate the solution of reflected G-BSDEs despite the failure of standard convergence theorems under G-expectation.
• To ensure the solution is universally defined across a non-dominated class of probability measures, aligning with the robust nature of G-expectation.
• To provide a probabilistic representation for fully nonlinear PDEs with obstacles, extending classical results to the nonlinear and uncertain volatility setting.

Proposed method

• Reformulate the reflected G-BSDE using a martingale condition: the process $\{-\int_0^t (Y_s - S_s) dA_s\}_{t \in [0,T]}$ is required to be a non-increasing G-martingale.
• Define the solution as a triplet $(Y, Z, A)$ in the space $\mathcal{S}_G^\alpha(0,T)$, with $Y_t \geq S_t$ and $A$ continuous and non-decreasing.
• Construct the solution via a sequence of penalized G-BSDEs, where the penalization term drives the solution toward the obstacle $S_t$.
• Establish convergence of the penalized solutions using the uniform continuity of processes in $S_G^p(0,T)$, overcoming the lack of weak compactness in $M_G^p(0,T)$.
• Utilize the extended conditional G-expectation and optional stopping theorem for $*$-stopping times to analyze the solution dynamics.
• Apply the non-dominated probability framework to ensure the solution holds $P$-a.s. for all $P \in \mathcal{P}$, preserving robustness.

Experimental results

Research questions

• RQ1 Can a unique solution be established for reflected G-BSDEs when the classical Skorohod condition fails to ensure uniqueness?
• RQ2 How can the penalization method be adapted to the G-framework, given the absence of dominated convergence and weak compactness?
• RQ3 What alternative condition can replace the Skorohod condition to guarantee uniqueness in the context of G-Brownian motion?
• RQ4 How does the martingale condition ensure the minimal intervention of the increasing process $A_t$ in the reflected solution?
• RQ5 To what extent can the solution of a reflected G-BSDE be interpreted as a value function in a robust optimal stopping problem under model uncertainty?

Key findings

• The solution of the reflected G-BSDE is uniquely characterized by the martingale condition: $\{-\int_0^t (Y_s - S_s) dA_s\}$ is a non-increasing G-martingale.
• Existence of the solution is proven via approximation through penalized G-BSDEs, with convergence established using uniform continuity in $S_G^p(0,T)$.
• The solution satisfies the dynamic programming principle: $Y_0 = \hat{\mathbb{E}}[\int_0^D f(s,Y_s,Z_s)ds + S_D I_{\{D<T\}} + \xi I_{\{D=T\}}]$, where $D = \inf\{t: Y_t = S_t\} \wedge T$.
• The process $A_t$ is continuous and non-decreasing, and the integral $\int_0^T (Y_t - S_t) dA_t = 0$ holds almost surely under all $P \in \mathcal{P}$, ensuring minimal reflection.
• The framework allows for a robust probabilistic interpretation of fully nonlinear PDEs with obstacles, extending classical results to model uncertainty.
• The solution is universally defined across a non-dominated class of measures, ensuring consistency under Knightian uncertainty.