Backward doubly stochastic differential equations with infinite time horizon. (Q1941787)

In the paper, a class of infinite horizon backward doubly stochastic differential equations (BDSDEs) is studied. The existence and uniqueness of the solution is proved using a standard fixed point argument under (global in space, local in time) Lipschitz assumptions. The assumption of exponential decay of the solution (see, e.g., [\textit{Q. Zhang} and \textit{H. Zhao}, J. Funct. Anal. 252, No. 1, 171--219 (2007; Zbl 1127.60059)]) is replaced by assuming a square integrable terminal condition. The next result is a proof of continuous dependence on the terminal condition of the solution couple and a convergence result for the solutions to the BDSDEs for different terminal conditions converging in mean square. Finally, the last result is used to demonstrate the relation between finite and infinite time horizon BDSDEs.