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

Abstract

Suppose {ℑt} is the filtration induced by a Wiener process W in Rd, τ is a finite {ℑt} stopping time (terminal time), ξ is an ℑτ-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, z̄)| ≤ b||z - z̄||, 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.