$L^{p}$-Variational Solutions of Multivalued Backward Stochastic Differential Equations

Abstract

The aim of the paper is to 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: \begin{equation*} \left\{ \begin{array}[c]{l} -dY_{t}+\partial_{y}Ψ(t,Y_{t})dQ_{t}\ni H(t,Y_{t},Z_{t})dQ_{t}-Z_{t}dB_{t},\;0\leq t<τ,\\[0.1cm] Y_τ=η, \end{array} \right. \end{equation*} where $τ$ is a stopping time, $Q$ is a progresivelly measurable increasing continuous stochastic process and $\partial_{y}Ψ$ is the subdifferential of the convex lower semicontinuous function $y\mapstoΨ(t,y).$