Lp-Variational solutions of multivalued backward stochastic differential equations

We prove the existence and uniqueness of the L p -variational solution, with p > 1, of the following multivalued backward stochastic differential equation with p -integrable data: {−dYt + ∂y Ψ(t,Yt )dQt ∋H (t,Yt ,Zt )dQt −Zt dBt ,0≤t = η , $\[ \left\{ \begin{array} [c]{l}% -dY_{t}+\partial_{y}\Psi(t,Y_{t}){\rm d}Q_{t}\ni H(t,Y_{t},Z_{t}){\rm d}Q_{t}-Z_{t}% {\rm d}B_{t},\;0\leq t is a stopping time, Q is a progressively measurable increasing continuous stochastic process and ∂ y Ψ 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.