The Stochastic Differential Equation in the Infinite Dimensional Space and Its Application to the Filtering Problems

Abstract

For the analytical approach to optimization problems of the distributed parameter systems, it is advantageous to use the methods of the functional analysis. This means that we at first transform the partial differential equations, which describe the state of the system with distributed parameters, into the ordinary differential equations in the function space and solve the optimization problems, and then we transform again the results obtained above into the partial differential equations and we obtain the optimum solutions for the distributed parameter systems described by the partial differential equations.
In this paper, we define in the infinite dimensional space the stochastic integral of Stratonovich type, which is useful for the modeling of the physical processes in the finite dimensional space, and we shall discuss the convergence and other properties of the integral. Furthermore, using these results, we can derive the nonlinear filter in the infinite dimensional space, under such a criterion as the unbiased and least square estimation. As a special application of the above, we shall derive the optimal filter for the linear distributed parameter systems with the state dependent noise, moreover we shall discuss the moment equations required for this case.