Rate of convergence of Wong-Zakai approximations for SDEs and SPDEs
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.