Wong-Zakai Approximation Research Articles

Wiener Chaos and Nonlinear Filtering

We investigate the rate of convergence of the Wong-Zakai approximations for second-order stochastic PDEs of parabolic type driven by a multi-dimensional Wiener process W.

Wong-Zakai approximations of stochastic evolution equations

Theorems on weak convergence of the laws of the Wong-Zakai approximations for evolution equation $$ \begin{aligned} dX(t) & = (AX(t) + F(X(t)))dt + G(X(t))dW(t) X(0) & = x \in H \end{aligned} $$ are proved. The operator A in the equation generates an analytic semigroup of linear operators on a Hilbert space H. The tightness of the approximating sequence is established using the stochastic factorisation formula. Applications to strongly damped wave and plate equations as well as to stochastic invariance are discussed.

On Wong–Zakai approximations with δ–martingales

We present a class of families of absolutely continuous stochastic processes approximating the Wiener process. We estimate the rate of convergence of the solutions of the stochastic differential equations driven by these families.

A Wong–Zakai Type Approximation for Multiple Wiener–Stratonovich Integrals

We present an extension of the Wong-Zakai type approximation theorem for a multiple stochastic integral. Using a piecewise linear approximation w (n) of a Wiener process w, we prove that the multiple integral process where f is a given symmetric function in the space 𝒞([0, T] m ), converge to the multiple Stratonovich integral of f in the uniform L 2-sense.

A Wong–Zakai type approximation for two-parameter processes1

We present an extension of the Wong–Zakai approximation theorem for a stochastic differential equation on the plane driven by a two-parameter Wiener process. For an approximation of the two-parameter Wiener process, we use a two-parameter version of the one-parameter piecewise linear approximation. By our approximation to the two-parameter Wiener process we show that the solution of an ordinary differential equation converges, in the uniform L 2-sense, to that of a stochastic differential equation obtained by using Stratonovich integral. *This research was supported (in part) by KOSEF through Statistical Research Center for Complex Systems at Seoul National University.

Wong-zaksi approximations for reflecting stochastic differential equations

We show, under convenient conditions, existence of a solution to a stochastic differential equation reflecting on the boundary of a convex set by using Wong-Zakai approximations. This means that solutions of RSDEs can be seen as the limit of solutions of differential inclusions, where the Brownian motion is replaced by its polygonal approximation

Long-time behavior of solutions to a class of stochastic parabolic equations with homogeneous white noise: Stratonovitch's case

In this article we prove new results concerning the long-time behavior of random fields that are solutions in some sense to a class of semilinear parabolic equations subjected to a homogeneous and multiplicative white noise. Our main results state that these random fields eventually homogeneize with respect to the spatial variable and finally converge to a non-random global attractor which consists of two spatially and temporally homogeneous asymptotic states. More precisely, we prove that the random fields either stabilize exponentially rapidly with probability one around one of the asymptotic states, or that they set out to oscillate between them. In the first case we can also determine exactly the corresponding Lyapunov exponents. In the second case we prove that the random fields are in fact recurrent in that they can reach every point between the two asymptotic states in a finite time with probability one. In both cases we also interpret our results in terms of stability properties of the global attractor and we provide estimates for the average time that the random fields spend in small neighborhoods of the asymptotic states. Our methods of proof rest upon the use of a suitable regularization of the Brownian motion along with a related Wong-Zakai approximation procedure.

Wong-Zakai approximations for stochastic differential equations

The aim of this paper is to give a wide introduction to approximation concepts in the theory of stochastic differential equations. The paper is principally concerned with Zong-Zakai approximations. Our aim is to fill a gap in the literature caused by the complete lack of monographs on such approximation methods for stochastic differential equations; this will be the objective of the author's forthcoming book. First, we briefly review the currently-known approximation results for finite- and infinite-dimensional equations. Then the author's results are preceded by the introduction of two new forms of correction terms in infinite dimensions appearing in the Wong-Zakai approximations. Finally, these results are divided into four parts: for stochastic delay equations, for semilinear and nonlinear stochastic equations in abstract spaces, and for the Navier-Stokes equations. We emphasize in this paper results rather than proofs. Some applications are indicated.

An approximation theorem of wong-zakai type for nonlinear stochastic partial differential equations

We present an extension of the Wong-Zakai approximation theorem for nonlinear 984 given by the Wiener process and a martingale. By approximating these disturbances we obtain in the limit equation the Ito correction term for the infinite dimensional case. Such form of the correction term connected with the Wiener process was proved in the author's papers [21] and [22], where the approximation theorem for semilinear stochastic evolution equations in Hilbert spaces was studied. Our model here is similar as the one considered by Pardoux [17]

On Wong-Zakai approximation of stochastic differential equations

An approximation theorem of stochastic differential equations driven by semimartingales is proved, based on approximation of semimartingales by a sequence of processes with piecewise monotonic sample functions.