Continuous dependence property of BSDE with constraints

TL;DR

This paper investigates the continuous dependence of solutions for backward stochastic differential equations with constraints, extending existing results to more general non-negative constraints and utilizing convex analysis in special cases.

Contribution

It establishes new continuous and semi-continuous properties of solutions for CBSDEs with general non-negative constraints, including convex cases.

Findings

Proved a continuous property from below for minimal super-solutions.

Established lower semi-continuity of solutions in the general case.

Derived continuous properties using convex analysis in the convex case.

Abstract

In this paper, we study continuous properties of adapted solutions for backward stochastic differential equations with constraints (CBSDEs in short). Comparing with many existing literatures about this topic, our case is very general in the sense that constraints are formulated by general non-negative real functions. In general case, we proved a continuous property from below and a lower semi-continuous property of the minimal super-solution of CBSDE in its effective domain. Furthermore, in the special convex case, we obtained a continuous property with the help of convex analysis.