In this paper a new class of generalized backward doubly stochastic differential equations is investigated. This class involves an integral with respect to an adapted continuous increasing process. A probabilistic representation for viscosity solutions of semi-linear stochastic partial differential equations with a Neumann boundary condition is given.Keywords: Backward doubly stocastic equations; stochastic partial differential equations This is an electronic reprint of the original article published by the ISI/BS in Bernoulli, 2007, Vol. 13, No. 2, 423-446. This reprint differs from the original in pagination and typographic detail.
We prove a weak convergence result for a sequence of backward stochastic di erential equations related to a semilinear parabolic partial di erential equation; under the assumption that the di usion corresponding to the PDEs is obtained by penalization method converging to a normal re ected di usion on a smooth and bounded domain D. As a consequence we give an approximation result to the solution of semilinear parabolic partial di erential equations with nonlinear Neumann boundary conditions. A similar result in the linear case was obtained by Lions et al. in 1981.
In this paper we lift fundamental topological structures on probability measures and random variables, in particular the weak topology, convergence in law and finite-dimensional convergence to an isometric level. This allows for an isometric quantitative study of important concepts such as relative compactness, tightness, stochastic equicontinuity, Prohorov's theorem and σ -smoothness. In doing so we obtain numerical results which allow for the development of an intrinsic approximation theory and from which moreover all classical topological results follow as easy corollaries.
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