$L^{p}$ - Variational Solution of Backward Stochastic Differential Equation driven by subdifferential operators on a deterministic interval time

Abstract

Our aim is to study 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: \[ \left\{ \begin{align*} &-dY_{t}+\partial_{y}\Psi\left( t,Y_{t}\right) dQ_{t} \ni H\left( t,Y_{t},Z_{t}\right) dQ_{t}-Z_{t}dB_{t},\;t\in\left[ 0,T\right] ,\\ &Y_{T} =\eta, \end{align*} \right. \] where $Q$ is a progresivelly measurable increasing continuous stochastic process and $\partial_{y}\Psi$ is the subdifferential of the convex lower semicontinuous function $y\mapsto\Psi(t,y)$. In the framework $p\geq2$ of Maticiuc, R\u{a}\c{s}canu from [Bernoulli, 2015], the strong solution found it there is the unique variational solution, via the uniqueness property proved in the present article.