Backward stochastic variational inequalities on random interval

TL;DR

This paper establishes existence and uniqueness results for backward stochastic variational inequalities in infinite-dimensional spaces, with applications to certain backward stochastic partial differential equations with boundary conditions.

Contribution

It introduces a framework for solving multivalued backward stochastic differential equations on random intervals in infinite dimensions, extending previous finite-dimensional results.

Findings

Proved existence and uniqueness of solutions in the infinite-dimensional setting.

Applied results to backward stochastic PDEs with boundary conditions.

Extended the theory to include random, possibly infinite, time intervals.

Abstract

The aim of this paper is to study, in the infinite dimensional framework, the existence and uniqueness for the solution of the following multivalued generalized backward stochastic differential equation, considered on a random, possibly infinite, time interval: \[\cases{\displaystyle -\mathrm{d}Y_t+\partial_y\Psi (t,Y_t)\,\mathrm{d}Q_t\ni\Phi (t,Y_t,Z_t)\,\mathrm{d}Q_t-Z_t\,\mathrm{d}W_t,\qquad 0\leq t<\tau,\cr \displaystyle{Y_{\tau}=\eta,}}\] where $\tau$ is a stopping time, $Q$ is a progressively measurable increasing continuous stochastic process and $\partial_y\Psi$ is the subdifferential of the convex lower semicontinuous function $y\longmapsto\Psi (t,y)$. As applications, we obtain from our main results applied for suitable convex functions, the existence for some backward stochastic partial differential equations with Dirichlet or Neumann boundary conditions.