Sensitivity Analysis of Distributionally Robust BSDEs and RBSDEs

TL;DR

This work addresses sensitivity analysis for distributionally robust optimization in non-Markovian BSDE/RBSDE settings with drift uncertainty. It develops first-order sensitivity formulas for both $\mathbb{L}^\infty$ and $\mathbb{L}^2$ drift perturbations, showing that derivatives can be expressed through the baseline $Z$-process and Doléans-Dade-type exponentials, and extends these results to mixed control/stopping problems via RBSDEs. The authors establish saddle-point representations and duality results, derive explicit derivative formulas such as $V'_\,\infty(0)=\|Z\|_{L^1(\mathbb{P}^{\lambda^*,k^*},k^*)}$ and $V'_2(0)=\|Z\|_{L^2(\mathbb{P}^{\lambda^*,k^*},k^*)}$, and validate the theory with an optimal liquidation example using a Deep BSDE solver. These results provide robust hedging and model-risk quantification tools in non-Markovian stochastic control and stopping contexts.

Abstract

We examine the sensitivity properties of backward stochastic differential equations and reflected backward stochastic differential equations, which naturally arise in the context of optimal control and optimal stopping problems. Motivated by issues of sensitivity analysis in distributionally robust optimization (DRO) control and optimal stopping problems, we establish explicit formulas for the corresponding sensitivities under drift reference measure uncertainty. Our work is closely related to \citeauthor{bartl2023sensitivity} \cite{bartl2023sensitivity}. In contrast to the existing literature, our analysis is carried out within a general non-Markovian framework.

Figures (1)

• Figure 1: Sensitivities for $\rho = 0.5$.

Theorems & Definitions (22)

• Theorem 3.2
• Remark 3.3
• Remark 3.5
• Theorem 3.7
• Corollary 3.8
• Theorem 3.10
• Corollary 3.11
• Remark 5.1
• Lemma 5.2
• Remark 5.3