Jensen's Inequality for Backward SDEs Driven by $G$-Brownian motion

TL;DR

This paper investigates Jensen's inequality within the context of nonlinear expectations generated by $G$-Brownian motion-driven backward SDEs, establishing conditions for when these inequalities hold in one and multiple dimensions.

Contribution

It provides necessary and sufficient conditions for Jensen's inequality to hold for $G$-BSDEs, extending understanding of nonlinear expectations in stochastic analysis.

Findings

One-dimensional Jensen inequality holds under specific conditions.

Multi-dimensional Jensen inequality holds only if the expectation is linear.

Characterization of when $G$-BSDEs satisfy Jensen's inequality.

Abstract

In this note, we consider Jensen's inequality for the nonlinear expectation associated with backward SDEs driven by $G$-Brownian motion ($G$-BSDEs for short). At first, we give a necessary and sufficient condition for $G$-BSDEs under which one-dimensional Jensen inequality holds. Second, we prove that for $n>1$, the $n$-dimensional Jensen inequality holds for any nonlinear expectation if and only if the nonlinear expectation is linear, which is essentially due to Jia (Arch. Math. 94 (2010), 489-499). As a consequence, we give a necessary and sufficient condition for $G$-BSDEs under which the $n$-dimensional Jensen inequality holds.