In this paper, the strong solutions (X,L) of multidimensional stochastic differential equations with reflecting boundary and possible anticipating initial random variables is established. The key is to obtain some substitution formula for Stratonovich integrals via a uniform convergence of the corresponding Riemann sums and to prove continuity of functionals of (X,L). MSC(2000): Primary 60H07, ...

The projection filter is an approximate nonlinear filter based on orthogonal projection in the tangent space of a manifold of densities. The Riemannian metric used is the Fisher information metric. In this paper it is shown that if one uses Gaussian densities the projection filter equals a McShane-Fisk-Stratonovich version of the socalled Assumed Density Filter.

In this paper, we establish the existence of the solutions (X,L) of reflected stochastic differential equations with possible anticipating initial random variables. The key is to obtain some substitution formula for Stratonovich integrals via a uniform convergence of the corresponding Riemann sums.

In the present paper, we prove that with probability one, the Stratonovich sigatures of a multidimensional diffusion process (possibly degenerate) over [0, 1], which is the collection of all iterated Stratonovich’s integrals of the diffusion process over [0, 1], determine the diffusion sample paths. MSC: 60J60; 60G17; 60J45

We consider the perturbation of parabolic operators of the form ∂t + P (x,D) by large-amplitude highly oscillatory spatially dependent potentials modeled as Gaussian random fields. The amplitude of the potential is chosen so that the solution to the random equation is affected by the randomness at the leading order. We show that, when the dimension is smaller than the order of the elliptic pseu...

A new class of energy-preserving numerical schemes for stochastic Hamiltonian systems with non-canonical structure matrix (in the Stratonovich sense) is proposed. These numerical integrators are of mean-square order one and also preserve quadratic Casimir functions. In the deterministic setting, our schemes reduce to methods proposed in [9] and [6].