BSDE on an infinite horizon and elliptic PDEs in infinite dimension

Abstract.

In this paper, we study the existence and uniqueness of mild solutions to a possibly degenerate elliptic partial differential equation ${\mathcal{L}}u(x) + \psi(x, u(x), \nabla u(x)G(x)) - \lambda u(x) = 0$ in Hilbert spaces. Our aim is, in the case in which ψ(·, 0, 0) is bounded, to drop the assumptions on the size of λ needed in [11]. The main tool will be existence, uniqueness and regular dependence on parameters of a bounded solution to a suitable backward stochastic differential equation with infinite horizon. Finally we apply the result to study an optimal control problem.