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

• 成果类型: Article
• 署名作者: Darling, RWR; Pardoux, E
• 署名单位: State University System of Florida; University of South Florida; Aix-Marseille Universite
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• 发表日期: 1997
• 页码: 1135-1159
• 关键词: equations

摘要

Suppose {T-t} is the filtration induced by a Wiener process W in R-d, tau is a finite {T-t} stopping time (terminal time), xi is an T-tau-measurable random variable in R-k (terminal value) and f(., y, z) is a coefficient process, depending on y is an element of R-k and z is an element of L(R-d; R-k), satisfying (y - (y) over bar) [f(s, y, z) - f(s, (y) over bar, z)] less than or equal to -a/y - (y) over bar/(2) (f need not be Lipschitz in Y), and /f(s, y, z) -f(s, y (z) over bar/ less than or equal to b//z - (z) over bar//, for some real a and b, plus other mild conditions. We identify a Hilbert space, depending on tau 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 less than or equal to tau)) dW(s) - f(s, Y(s), Z(s)) ds] with the given terminal condition Y(tau) = xi, 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.