On stochastic partial differential equations. Results on approximations

Abstract

We consider second order stochastic linear partial differential equations of parabolic type driven by continuous semimartingales. Approximating the driving process by continuous semimartingales in the topology of uniform convergences we get, under some conditions, the convergence of the resulting solutions to the solution of the original equation understood in Stratonovich's sense. Hence we establish a support theorem for SPDEs which is a generalization of the well known support theorem of Stroock and Varadhan. We present our results for SPDEs where the differential operators are given in terms of derivatives along certain vector fields in place of derivatives with respect to fixed coordinate vectors. By virtue of this generality we can treat, in particular, the case of SPDEs with unbounded coefficients which is important from the point of view of applications. We apply the results presented here to the problems of approximating both the nonlinear filter of partially observed diffusion processes and the equation of the hydromagnetic dynamo. This work is based on the papers [10]–[13]. The paper is organised as follows: 1. Second order SPDEs with unbounded coefficients 2. Approximations of SPDEs 3. Theorems on supports 4. Applications to nonlinear filtering 5. An application to a stochastic model of the hydromagnetic dynamo