Stochastic partial differential equations with singular terminal condition

TL;DR

This paper establishes existence and uniqueness results for stochastic partial differential equations with singular terminal conditions, including cases where the terminal data can be infinite on a set of positive measure.

Contribution

It extends the theory of BDSDEs and SPDEs to include singular terminal conditions, providing minimal solutions under monotonicity assumptions.

Findings

Existence and uniqueness of solutions for BDSDEs and SPDEs with singular terminal data.

Construction of minimal solutions in the presence of infinite terminal conditions.

Solutions are weak in the Sobolev sense.

Abstract

In this paper, we first prove existence and uniqueness of the solution of a backward doubly stochastic differential equation (BDSDE) and of the related stochastic partial differential equation (SPDE) under monotonicity assumption on the generator. Then we study the case where the terminal data is singular, in the sense that it can be equal to +$\infty$ on a set of positive measure. In this setting we show that there exists a minimal solution, both for the BDSDE and for the SPDE. Note that solution of the SPDE means weak solution in the Sobolev sense.