This paper studies the stochastic heat equation with multiplicative noises: ∂u ∂t = 1 2 ∆u+uẆ , where Ẇ is a mean zero Gaussian noise and uẆ is interpreted both in the sense of Skorohod and Stratonovich. The existence and uniqueness of the solution are studied for noises with general time and spatial covariance structure. Feynman-Kac formulas for the solutions and for the moments of the solutio...

The signature of a d-dimensional Brownian motion is a sequence of iterated Stratonovich integrals along the Brownian paths, an object taking values in the tensor algebra over R. In this article, we derive the exact rate of convergence for the expected signatures of piecewise linear approximations to Brownian motion. The computation is based on the identification of the set of words whose coeffi...

The stochastic differential equations of many diffusion processes which arise in studies of population growth in random environments can be transformed, if the Stratonovich stochastic calculus is employed, to the equation of the Wiener process. If the transformation function has certain properties then the transition probability density function and quantities relating to the time to first atta...

The aim of this paper is to study the d-dimensional stochastic heat equation with a multiplicative Gaussian noise which is white in space and it has the covariance of a fractional Brownian motion with Hurst parameter H ∈ (0, 1) in time. Two types of equations are considered. First we consider the equation in the Itô-Skorohod sense, and later in the Stratonovich sense. An explicit chaos developm...

Using the method of steps, we describe stochastic processes with delays in terms of Markov diffusion processes. Thus, multivariate Langevin equations and Fokker-Planck equations are derived for stochastic delay differential equations. Natural, periodic, and reflective boundary conditions are discussed. Both Ito and Stratonovich calculus are used. In particular, our Fokker-Planck approach recove...

A unified canonical operator formalism for quantum stochastic differential equations, including the quantum stochastic Liouville equation and the quantum Langevin equation both of the Itô and the Stratonovich types, is presented within the framework of Non-Equilibrium Thermo Field Dynamics (NETFD). It is performed by introducing an appropriate martingale operator in the Schrödinger and the Heis...

(1) ∂X ∂t − ν∆X + (f · ∇)X + (X · ∇)g = ∇p in (0,∞)×O, ∇ ·X = 0 in (0,∞)×O, X(0, ξ) = x(t), ξ ∈ O, X = 0 on (0,∞)× ∂O. Here, O is an open and bounded subset of R, d = 2, 3, with smooth boundary ∂O and f, g ∈ C2(O;Rd) are given functions. In the special case g ≡ 0, system (1) describes the dynamic of a fluid Stokes flow with partial inclusion of convection acceleration (f ·∇)X (X is the velocity...

The context is the general stochastic differential equation (SDE) model dN/dt=N(g(N)+sigmaepsilon(t)) for population growth in a randomly fluctuating environment. Here, N=N(t) is the population size at time t, g(N) is the 'average' per capita growth rate (we work with a general almost arbitrary function g), and sigmaepsilon(t) is the effect of environmental fluctuations (sigma>0, epsilon(t) sta...

Consider the class of Ensemble Square Root filtering algorithms for numerical approximation posterior distribution nonlinear Markovian signals partially observed with linear observations corrupted independent measurement noise. We analyze asymptotic behavior these in large ensemble limit both discrete and continuous time. identify limiting mean-field processes on level members, prove correspond...

We review the manifold projection method for stochastic nonlinear filtering in a more general setting than in our previous paper in Geometric Science of Information 2013. We still use a Hilbert space structure on a space of probability densities to project the infinite dimensional stochastic partial differential equation for the optimal filter onto a finite dimensional exponential or mixture fa...