  In this paper we study one dimensional backward stochastic differential
equations (BSDEs) with random terminal time not necessarily bounded or finite
when the generator F(t,Y,Z) has a quadratic growth in Z. We provide existence
and uniqueness of a bounded solution of such BSDEs and, in the case of infinite
horizon, regular dependence on parameters. The obtained results are then
applied to prove existence and uniqueness of a mild solution to elliptic
partial differential equations in Hilbert spaces.
