Multivalued Backward Stochastic Differential Equations with Time Delayed Generators

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.