Accelerated finite difference schemes for stochastic partial differential equations in the whole space

We give sufficient conditions under which the convergence of finite difference approximations in the space variable of the solution to the Cauchy problem for linear stochastic PDEs of parabolic type can be accelerated to any given order of convergence by Richardson's method.Comment: 24 page

Localization errors in solving stochastic partial differential equations in the whole space

Cauchy problems with SPDEs on the whole space are localized to Cauchy problems on a ball of radius $R$. This localization reduces various kinds of spatial approximation schemes to finite dimensional problems. The error is shown to be exponentially small. As an application, a numerical scheme is presented which combines the localization and the space and time discretisation, and thus is fully implementable.Comment: Some details added; published versio

Accelerated finite elements schemes for parabolic stochastic partial differential equations

For a class of finite elements approximations for linear stochastic parabolic PDEs it is proved that one can accelerate the rate of convergence by Richardson extrapolation. More precisely, by taking appropriate mixtures of finite elements approximations one can accelerate the convergence to any given speed provided the coefficients, the initial and free data are sufficiently smooth.Comment: 1 figur

A Note on Euler Approximations for Stochastic Differential Equations with Delay

An existence and uniqueness theorem for a class of stochastic delay differential equations is presented, and the convergence of Euler approximations for these equations is proved under general conditions. Moreover, the rate of almost sure convergence is obtained under local Lipschitz and also under monotonicity conditions

On randomized stopping

A general result on the method of randomized stopping is proved. It is applied to optimal stopping of controlled diffusion processes with unbounded coefficients to reduce it to optimal control problem without stopping. This is motivated by recent results of Krylov on numerical solutions to the Bellman equation

First derivatives estimates for finite-difference schemes

Abstract. We give sufficient conditions under which solutions of discretized in space second-order parabolic and elliptic equations, perhaps degenerate, admit estimates of the first derivatives in the space variables independent of the mesh size. 1

On L_p-Solvability of Stochastic Integro-Differential Equations

A class of (possibly) degenerate stochastic integro-differential equations of parabolic type is considered, which includes the Zakai equation in nonlinear filtering for jump diffusions. Existence and uniqueness of the solutions are established in Bessel potential spaces