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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.

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Various paths properties of a stochastic process are obtained under mild conditions which allow for the integrability of the characteristic function of its increments and for the dependence among them. The main assumption is closely related to the notion of local asymptotic self-similarity. New results are obtained for the class of multifractional random processes .