Lp-Variational solutions of multivalued backward stochastic differential equations

Abstract

We prove the existence and uniqueness of the Lp-variational solution, with p > 1, of the following multivalued backward stochastic differential equation with p-integrable data: {−dYt + ∂yΨ(t,Yt)dQt∋H(t,Yt,Zt)dQt−ZtdBt,0≤t<τ, Yτ = η, where τ is a stopping time, Q is a progressively measurable increasing continuous stochastic process and ∂yΨ is the subdifferential of the convex lower semicontinuous function y↦Ψ(t, y). In the framework of [14] (the case p ≥ 2), the strong solution found it there is the unique variational solution, via the uniqueness property proved in the present article.