Backward stochastic variational inequalities on random interval

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.