Constructing Gamma-Martingales with Prescribed Limit, Using Backwards SDE

Let VW and VY be Euclidean vector spaces and let VZ ≡ L(VW → VY). Given a Wiener process W on VW, with natural filtration ₜ\, and a IT-measurable random variable U in VY, we seek adapted processes $(Y, Z)$ in VY × VZ satisfying the SDE U = Y(t) + ∫(t,TZdW - ∫(t, T Γ(Y, ZZ⁾ ds/2, 0 ≤ t ≤ T, under local Lipschitz and convexity conditions on the map (y, A) → Γ(y, A). These conditions apply in particular in the case Γ(y, A) = ∑Γⁱⱼₖ(y)Aʲᵏ, where Γ is a linear connection on VY whose Christoffel symbols Γⁱⱼₖ are bounded and Lipschitz, and Γ has certain convexity properties. In that case the solution Y above is known as a Γ-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.