Spectral representation of stochastic integration operators

Abstract

The spectral representation for stochastic integration operators with respect to the Wiener process is proposed in the form of a composition of spectral characteristics used in the spectral form of mathematical description for control systems. This spectral representation can be defined relative to the various orthonormal bases. For given deterministic square-integrable kernels, the spectral characteristic of a stochastic integration operator is determined as an infinite random matrix. The main applications of such a representation suppose solving linear stochastic differential equations and modeling multiple or iterated Stratonovich stochastic integrals. Specific formulas are provided that allow to represent the spectral characteristic for the stochastic integration operator, the kernel of which is the Heaviside function, relative to Walsh functions and trigonometric functions.