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.
Brahim Boufoussi. Jan Van Casteren. N. Mrhardy. "Generalized backward doubly stochastic differential equations and SPDEs with nonlinear Neumann boundary conditions." Bernoulli 13 (2) 423 - 446, May 2007. https://doi.org/10.3150/07-BEJ5092