BSDES on finite and infinite horizon with time-delayed generators

We consider a backward stochastic differential equation with a generator that can be subjected to delay, in the sense that its current value depends on the weighted past values of the solutions, for instance a distorted recent average. Existence and uniqueness results are provided in the case of possibly infinite time horizon for equations with, and without refiection. Furthermore, we show that when the delay vanishes, the solutions of the delayed equations converge to the solution of the equation without delay. We argue that these equations are naturally linked to forward backward systems, and we exemplify a situation where this observation allows to derive results for quadratic delayed equations with non-bounded terminal conditions in multi- dimension.