Scheme of multiplicative linearization of a random process

Abstract

In this paper, we consider a stochastic differential equation in a Hilbert space. This is a natural situation when modeling a physical system having disturbances such as white noise. The equation describing such a process is a nonlinear stochastic equation in the form of Ito. In this paper, we consider the method of multiplicative linearization of such an equation. To this end, the time interval for the existence of a solution is divided into elementary ones in such a way that the diameter of the maximum partition tends to zero. At each of these intervals, the nonlinear coefficients of the equation are expanded in a Taylor series. This becomes possible when additional conditions for the coefficients of the equation are satisfied. It is required that the coefficients have Lipschitz uniformly bounded first and second order derivatives, and also that the fourth moment of the initial value be bounded. The conditions of the equivalence lemma of two multiplicative representations are verified. One of these multiplicative families is a stochastic strongly continuous evolutionary family of resolving operators of the original equation. Another family is the product of the resolving operators of linear equations on the corresponding elementary time intervals. Under the additional conditions on the coefficients of the equation, there exists a unique, up to stochastic equivalence, solution of a linear inhomogeneous equation. Using the Lagrange form for the remainder terms when expanding in a Taylor series and the Gronwall lemma, the conditions of the lemma are verified and the solution is proved to be a product of linear processes on each elementary interval. This multiplicative product can be interpreted as an approximate solution.