Integro-partial differential equations with singular terminal condition

TL;DR

This paper establishes a probabilistic representation for the minimal viscosity solution of integro-partial differential equations with singular terminal conditions using backward stochastic differential equations, exploring various regularity types.

Contribution

It introduces a novel link between backward stochastic differential equations and integro-PDEs with singular terminal conditions, expanding understanding of solution regularity.

Findings

Probabilistic representation of singular integro-PDE solutions

Analysis of Sobolev, Hölder, and strong regularity

Minimal solutions characterized via backward stochastic differential equations

Abstract

In this paper, we show that the minimal solution of a backward stochastic differential equation gives a probabilistic representation of the minimal viscosity solution of an integro-partial differential equation both with a singular terminal condition. Singularity means that at the final time, the value of the solution can be equal to infinity. Different types of regularity of this viscosity solution are investigated: Sobolev, H{\"o}lder or strong regularity.