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. Finally we show an application to a control problem.
Fulvia Confortola. Philippe Briand. "Quadratic BSDEs with Random Terminal Time and Elliptic PDEs in Infinite Dimension." Electron. J. Probab. 13 1529 - 1561, 2008. https://doi.org/10.1214/EJP.v13-514