Minimal supersolutions of convex BSDEs

TL;DR

This paper investigates the properties of minimal supersolutions in convex backward stochastic differential equations, establishing foundational results like existence, uniqueness, and comparison principles for these solutions.

Contribution

It introduces a comprehensive analysis of minimal supersolutions for convex BSDEs, including new existence, uniqueness, and comparison results under broad conditions.

Findings

Proved existence and uniqueness of minimal supersolutions.

Established a comparison principle for these solutions.

Demonstrated monotone convergence and lower semicontinuity properties.

Abstract

We study the nonlinear operator of mapping the terminal value $\xi$ to the corresponding minimal supersolution of a backward stochastic differential equation with the generator being monotone in $y$, convex in $z$, jointly lower semicontinuous and bounded below by an affine function of the control variable $z$. We show existence, uniqueness, monotone convergence, Fatou's lemma and lower semicontinuity of this operator. We provide a comparison principle for minimal supersolutions of BSDEs.