Abstract
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.
Highlights
Let τ be a stopping time which is not necessarily bounded or finite
We look for a pair of processes (Yt, Zt)t≥0 progressively measurable which satisfy ∀t ≥ 0, ∀T ≥ t
backward stochastic differential equations (BSDEs) with random terminal time have been treated by several authors
Summary
Let τ be a stopping time which is not necessarily bounded or finite. We look for a pair of processes (Yt, Zt)t≥0 progressively measurable which satisfy ∀t ≥ 0, ∀T ≥ t. The result on BSDE is exploited to study existence and uniqueness of a mild solution (see Section 5 for the definition) to the following elliptic partial differential equation in Hilbert space H. where F is a function from H × R × Ξ∗ to R strictly monotone with respect the second variable and with quadratic growth in the gradient of the solution and L is the second order operator: Lφ(x). The paper is organized as follows: the Section is devoted to notations; in Section 3 we deal with quadratic BSDEs with random terminal time; in Section 4 we study the forward backward system on infinite horizon; in Section 5 we show the result about the solution to PDE. The last Section is devoted to the application to the control problem
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