In this paper we discuss new types of differential equations which we call anticipated backward stochastic differential equations (anticipated BSDEs). In these equations the generator includes not only the values of solutions of the present but also the future. We show that these anticipated BSDEs have unique solutions, a comparison theorem for their solutions, and a duality between them and stochastic differential delay equations.
"Anticipated backward stochastic differential equations." Ann. Probab. 37 (3) 877 - 902, May 2009. https://doi.org/10.1214/08-AOP423