Infinite Anticipation Backward Stochastic Differential Equations

In this paper, we introduce a new type of backward stochastic differential equations (BSDEs) with infinite anticipation, where the generator depends on the entire future values of the solution in infinite horizon. We show that the new BSDEs has a unique solution and admits a comparison result. In the end, we solve a stochastic control problem via a duality between BSDEs with infinite anticipation and stochastic differential equations (SDEs) with infinite delay.