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

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: $$\{\begin{array}{rl}{\displaystyle }& {\displaystyle -d{Y}_t+{\mathrm{\partial }}_y\mathrm{\Psi }(t,Y_t)d{Q}_t\ni H(t,Y_t,Z_t)d{Q}_t-Z_{t}d{B}_t,t\in [0,T],}\\ {\displaystyle }& {\displaystyle Y_T=\eta ,}\end{array}$$ where $Q$ is a progresivelly measurable increasing continuous stochastic process and ${\mathrm{\partial }}_y\mathrm{\Psi }$ is the subdifferential of the convex lower semicontinuous function $y\mapsto \mathrm{\Psi }(t,y)$. In the framework $p\ge 2$ 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.