Our aim is to study the following new type of multivalued backward stochastic differential equation: $$\left\{\begin{array}{c}-dY\left(t\right)+\partial \phi \left(Y\left(t\right)\right)dt\ni F\left(t,Y\left(t\right),Z\left(t\right),Y_t,Z_t\right)dt+Z\left(t\right)dW\left(t\right),0⩽t⩽T,\\ Y\left(T\right)=\xi ,\end{array}\right.$$ where ∂φ is the subdifferential of a convex function and (Y t, Z t):= (Y(t + θ), Z(t + θ))θ∈[−T,0] represent the past values of the solution over the interval [0, t]. Our results are based on the existence theorem from Delong Imkeller, Ann. Appl. Probab., 2010, concerning backward stochastic differential equations with time delayed generators.

We establish the Euler-Lagrange inclusion of a nonsmooth integral functional defined on Orlicz-Sobolev spaces. This result is achieved through variational techniques in nonsmooth analysis and an integral representation formula for the Clarke generalized gradient of locally Lipschitz integral functionals defined on Orlicz spaces.