BACKWARDS SDE WITH RANDOM TERMINAL TIME AND APPLICATIONS TO SEMILINEAR ELLIPTIC PDE

Suppose {F t } is the filtration induced by a Wiener process W in R d , τ is a finite {F t } stopping time (terminal time), ξ is an F τ -measurable random variable in R k (terminal value) and f(., y,z) is a coefficient process, depending on y ∈ R k and z ∈ L(R d ; R k ), satisfying (y - y)[f(s, y, z) - f(s, y, z)] ≤ - a| - y - y| 2 (f need not be Lipschitz in y), and |f(s, y, z) - f(s, y, z)| ≤ b∥z - z∥, for some real a and b, plus other mild conditions. We identify a Hilbert space, depending on τ and on the number y = b 2 - 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.