Stability and Markov Property of Forward Backward Minimal Supersolutions

TL;DR

This paper investigates the stability and Markov properties of minimal supersolutions in forward-backward stochastic differential equations, establishing their relation to PDEs and demonstrating their stability under weak assumptions.

Contribution

It introduces new stability results for minimal supersolutions and proves their Markov property in the Markovian case, linking stochastic solutions to PDEs.

Findings

Minimal supersolutions are stable under weak assumptions.

Markovian supersolutions possess the Markov property.

Identified as unique viscosity supersolutions of related PDEs.

Abstract

We show stability and locality of the minimal supersolution of a forward backward stochastic differential equation with respect to the underlying forward process under weak assumptions on the generator. The forward process appears both in the generator and the terminal condition. Painlev\'e-Kuratowski and Convex Epi-convergence are used to establish the stability. For Markovian forward processes the minimal supersolution is shown to have the Markov property. Furthermore, it is related to a time-shifted problem and identified as the unique minimal viscosity supersolution of a corresponding PDE.