We derive the form of the Belavkin-Kushner-Stratonovich equation describing the filtering of a continuous observed quantum system via non-demolition measurements when the statistics of the input field used for the indirect measurement are in a general coherent state.

We define a discretized Langevin equation in Stratonovich-type calculus. We show that a generating functional with a field-dependent kernel can be written in mid-point prescription only when we calculate in the calculus. Moreover we investigate whether supersymmetry of the stochastic action with field-dependent kernel exists or not. – 1 –

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