Multivalued backward stochastic differential equations with time delayed generators

摘要

Our aim is to study the following new type of multivalued backward stochastic differential equation: {?dY (t) + ?φ(Y (t))dt ? F(t, Y (t), Z(t), Y_t, Z_t)dt + Z(t)dW(t), 0 ≤ t ≤ T, Y (T) = ξ, 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.