Backwards SDE with random terminal time and applications to semilinear elliptic PDE

Suppose ₜ is the filtration induced by a Wiener process W in Rᵈ, τ is a finite ₜ stopping time (terminal time), ξ is an τ-measurable random variable in Rᵏ (terminal value) and f(·, y, z) is a coefficient process, depending on y ∈ Rᵏ and z ∈ L(Rᵈ, Rᵏ), satisfying (y - ỹ)[f(s, y, z) - f(s, ỹ, z)] ≤ - a|y - ỹ|² (f need not be Lipschitz in y), and |f(s, y, z) - f(s, y, z̃)| ≤ b||z - z̃||, for some real a and b, plus other mild conditions. We identify a Hilbert space, depending on τ and on the number γ ≡ b² - 2a, in which there exists a unique pair of adapted processes $(Y, Z)$ satisfying the stochastic differential equation dY(s) = 1_s ≤ τ [Z(s) dW(s) - f(s, Y(s), Z(s)) ds] with the given terminal condition Y(τ) - ξ, provided a certain integrability condition holds. This result is applied to construct a continuous viscosity solution to the Dirichlet problem for a class of semilinear elliptic PDE’s.