CONSTRUCTING GAMMA-MARTINGALES WITH PRESCRIBED LIMIT, USING BACKWARDS SDE

• 成果类型: Article
• 署名作者: DARLING, RWR
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• DOI: 10.1214/aop/1176988182
• 发表日期: 1995
• 页码: 1234-1261
• 关键词: Stochastic differential utility; HARMONIC MAPS; probability; EXISTENCE

摘要

Let V-W and V-Y be Euclidean Vector spaces and let V-Z = L(V-W --> V-Y). Given a Wiener process W on V-W, with natural filtration {T-t}, and a T-T-measurable random variable U in V-Y, we seek adapted processes (Y, Z) in V-Y X V-Z satisfying the SDE U = Y(t) + integral((t,T]) ZdW - integral((t,T])Gamma(Y,ZZ*)ds/2, 0 less than or equal to t less than or equal to T, under local Lipschitz and convexity conditions on the map (y, A) --> Gamma(y, A). These conditions apply in particular in the case Gamma(y, A) = Sigma Gamma(jk)(i)(y)A(jk) where Gamma is a linear connection on V-Y whose Christoffel symbols Gamma(jk)(i) are bounded and Lipschitz, and Gamma has certain convexity properties. In that case the solution Y above is known as a Gamma-martingale with terminal value U. The solution (Y, Z) is constructed explicitly using the Pardoux-Peng theory of backwards SDE's. Applications include the Dirichlet problem and the heat equation for harmonic mappings, and other PDE's.