In the work we estimate the rate of convergence of the Wong-Zakai type of approximations for SDEs and SPDEs. Two cases are studied: SDEs in finite dimensional settings and evolution stochastic systems (SDEs in the infinite dimensional case). The latter result is applied to the second order SPDEs of parabolic type and the filtering problem. Roughly, the result is the following. Let Wn be a sequence of continuous stochastic processes of finite variation on an interval [0, T]. Assume that for some a > 0 the processes Wn converge almost surely in the supremum norm in [0, T] to W with the rate n-k for each k < a. Then the solutions Un of the differential equations with Wn converge almost surely in the supremum norm in [0, T] to the solution u of the "Stratonovich" SDE with W with the same rate of convergence, n-k for each k < a, in the case of SDEs and with the rate of convergence n-k/2 for each k < a, in the case of evolution systems and SPDEs. In the final chapter we verify that the two most common approximations of the Wiener process, smoothing and polygonal approximation, satisfy the assumptions made in the previous chapters.

In this dissertation, we first prove that for a random differential equation with the multiplicative driving noise constructed from a Q-Wiener process and the Wiener shift, which is an approximation to a stochastic evolution equation, there exists a unique solution that generates a local dynamical system. There also exist a local center, unstable, stable, centerunstable, center-stable manifold, and a local stable foliation, an unstable foliation on the center-unstable manifold, and a stable foliation on the center-stable manifold, the smoothness of which depend on the vector fields of the equation. In the second half of the dissertation, we show that any two arbitrary local center manifolds constructed as above are conjugate. We also show the same conjugacy result holds for a stochastic evolution equation with the multiplicative Stratonovich noise term as u â—¦ dW