In this paper, we describe stochastic Runge–Kutta (SRK) methods with strong order 1.0 for strong solutions of Stratonovich stochastic differential equations (SDEs) which was first introduced by Burrage and Burrage in 1996. In particular, three new SRK methods with strong order 1.0 are constructed. They are an explicit two–stage method, an explicit three–stage method with minimum principal error...

We revisit the problem of well-defining rotation numbers for discrete random dynamical systems on S. We show that, contrasting with deterministic systems, the topological (i.e. based on Poincaré lifts) approach does depend on the choice of lifts (e.g. continuously for nonatomic randomness). Furthermore, the winding orbit rotation number does not agree with the topological rotation number. Exist...

We study existence and uniqueness of a variational solution in terms of stochastic variational inequalities (SVI) to stochastic nonlinear diffusion equations with a highly singular diffusivity term and multiplicative Stratonovich gradient-type noise. We derive a commutator relation for the unbounded noise coefficients in terms of a geometric Killing vector condition. The drift term is given by ...

• We use automatic differentiation and sparse-grid integration to automate the construction of projection filter. present methods for constructing filters multidimensional filtering problems using a non-Gaussian parametric density. show that practical performance filter is comparable particle finite difference based solutions Kushner–Stratonovich equation. An open-source implementation method a...

We present a mass-preserving scheme for the stochastic nonlinear Schrödinger equation with multiplicative noise of Stratonovich type. It is a splitting scheme and we present an explicit formula for solving the sub-step related to the nonlinear part. The scheme is unconditionally stable in the L2 norm. For the linear stochastic Schrödinger equation, we prove that the scheme has a strong converge...

A nonperturbative theory of the electron propagator is developed and used to calculate the one-particle Green’s function and tunneling density of states in strongly correlated electron systems. The method, which is based on a Hubbard–Stratonovich decoupling of the electron–electron interaction combined with a cumulant expansion of the resulting noninteracting propagator, provides a possible gen...

This paper compares the classical concept of assumed density lters (ADF) with a new class of approximate lters, the projection lters (PF). It is shown that the concept of ADF is inconsistent in the sense that the resulting lters vary according to the choice of a stochastic calculus, i.e. It^ o or McShane{Fisk{Stratonovich (MFS). The PF are based on the diierential geometric approach to statisti...

We study a class of singularly perturbed dynamical systems that have fast and slow components, 1 being the fast to slow timescale ratio. The fast components are governed by a strongly mixing discrete map, which is iterated at time intervals . The slow components are governed by a /rst-order /nite-di0erence equation that uses a time step . As tends to zero, the fast components may be eliminated,...

In this article we establish a large deviation principle for the family {ν ε : ε ∈ (0, 1)} of distributions of the scaled stochastic processes {P − log √ ε Z t } t≤1 , where (Z t) t∈[0,1] is a square-integrable martingale over Brownian filtration and (P t) t≥0 is the Ornstein-Uhlenbeck semigroup. The rate function is identified as well in terms of the Wiener-Itô chaos decomposition of the termi...

Abstract. In this article we establish a substitution theorem for semilinear stochastic evolution equations (see’s) depending on the initial condition as an infinite-dimensional parameter. Due to the infinitedimensionality of the initial conditions and of the stochastic dynamics, existing finite-dimensional results do not apply. The substitution theorem is proved using Malliavin calculus techni...