On composite formulas for mathematical expectation of functionals of solution of the Ito equation in Hilbert space

A D Egorov

doi:10.29235/1561-2430-2019-55-2-158-168

Abstract

This article is devoted to constructing composite approximate formulas for calculation of mathematical expectation of nonlinear functionals of solution of the linear Ito equation in Hilbert space with additive noise. As the leading process, the Wiener process taking values in Hilbert space is examined. The formulas are a sum of the approximations of the nonlinear functionals obtained by expanding the leading random process into a series of independent Gaussian random variables and correcting approximating functional quadrature formulas that ensure an approximate accuracy of compound formulas for third-order polynomials. As a test example, the application of the obtained formulas to the case of a one-dimensional wave equation with a leading Wiener process indexed by spatial and temporal variables is considered. This article continues the research begun in [1].The problem is motivated by the necessity to calculate the nonlinear functionals of solution of stochastic partial differential equations. Approximate evaluation of mathematical expectation of stochastic equations with a leading random process indexed only by the time variable is considered in [2–11]. Stochastic partial equations in various interpretations are considered [12–16]. The present article uses the approach given in [12].

Highlights

This article is devoted to constructing composite approximate formulas for calculation of mathematical expectation of nonlinear functionals of solution of the linear Ito equation in Hilbert space with additive noise
The formulas are a sum of the approximations of the nonlinear functionals obtained by expanding the leading random process into a series of independent Gaussian random variables and correcting approximating functional quadrature formulas that ensure an approximate accuracy of compound formulas for third-order polynomials
The problem is motivated by the necessity to calculate the nonlinear functionals of solution of stochastic partial differential equations

Summary

Сходимость следует из непрерывности функционала и соотношений

X (t ) − X N,n (t ) =( X (t ) − X N (t )) + ( X N (t ) − X N,n (t )) → 0, где X (t ) → X N (t ) при N → ∞, X N,n (t ) → X N (t ) в L2 (Ω ×[0,T ], L2 (U0, H )) и JN,n ( F ) → JN ( F ). Для простоты изложения будем рассматривать частный случай, когда U – подпространство функций из L2 ( R), обращающихся в нуль вне отрезка (0,l), T = 1 и ek = ek ( x) – собственные функции оператора Λ. Что оператор Q имеет те же собственные функции, что и оператор Λ. Решение уравнения (17) определяется формулой (7), в которой S(t) имеет следующий вид (см. [12]):

Λt cos

Приближенные значения

Список использованных источников