Multivalued Backward Stochastic Differential Equations with Time Delayed Generators

TL;DR

This paper introduces a new class of multivalued backward stochastic differential equations with time-delayed generators, extending existing theories to include past values and subdifferential operators, with proven existence results.

Contribution

It develops an existence theorem for multivalued backward stochastic differential equations with time delays and subdifferential operators, expanding the scope of stochastic differential equation theory.

Findings

Established existence of solutions for the new class of equations

Extended previous results to include time delays and multivalued operators

Built on the framework of Delong & Imkeller (2010)

Abstract

Our aim is to study the following new type of multivalued backward stochastic differential equation: \[ \left\{\begin{array} [c]{r}-dY\left(t\right) +\partial\varphi\left(Y\left(t\right)\right) dt\ni F\left(t,Y\left(t\right),Z\left(t\right),Y_{t},Z_{t}\right) dt+Z\left(t\right) dW\left(t\right),\;0\leq t\leq T,\medskip\\ \multicolumn{1}{l}{Y\left(T\right) =\xi,}\end{array} \right. \] where $\partial\varphi$ is the subdifferential of a convex function and $\left(Y_{t},Z_{t}\right):=(Y(t+\theta),Z(t+\theta))_{\theta\in\lbrack-T,0]}$ represent the past values of the solution over the interval $\left[ 0,t\right] $. Our results are based on the existence theorem from Delong & Imkeller, Ann. Appl. Probab., 2010, concerning backward stochastic differential equations with time delayed generators.