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

TL;DR

This paper establishes the existence and uniqueness of $L^{p}$-variational solutions for a class of multivalued backward stochastic differential equations with $p$-integrable data, involving subdifferentials of convex functions.

Contribution

It introduces a novel framework for solving multivalued backward stochastic differential equations with $L^{p}$ data, extending previous results to a broader class of equations.

Findings

Proved existence of solutions under $L^{p}$ conditions.

Established uniqueness of solutions.

Extended the theory to include multivalued BSDEs with convex subdifferentials.

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}\Psi(t,Y_{t})dQ_{t}\ni H(t,Y_{t},Z_{t})dQ_{t}-Z_{t}dB_{t},\;0\leq t<\tau,\\[0.1cm] Y_{\tau}=\eta, \end{array} \right. \end{equation*} where $\tau$ is a stopping time, $Q$ is a progresivelly measurable increasing continuous stochastic process and $\partial_{y}\Psi$ is the subdifferential of the convex lower semicontinuous function $y\mapsto\Psi(t,y).$