Chapter 20
Using Spectral Form of Mathematical
Description to Represent Iterated
Stratonovich Stochastic Integrals
Konstantin A. Rybakov
Abstract In this chapter, it is suggested to apply the spectral form of mathematical
description for the representation of the iterated Stratonovich stochastic integrals of
an arbitrary multiplicity. Some invariant relations for expansion coefficients and the
iterated Stratonovich stochastic integrals are obtained. An algorithm for modeling
the iterated Stratonovich stochastic integrals is discussed.
20.1 Introduction
Iterated stochastic integrals play a fundamental role in the constructing high-order
numerical methods for stochastic differential equations. These methods are based
on the Taylor–Itô expansion and the Taylor–Stratonovich expansion for random
processes. The first numerical method using iterated stochastic integrals of multiplicity 2 and orthogonal expansions into the trigonometric series was Milstein method
[1–3]. Iterated stochastic integrals of multiplicity 3 and orthogonal expansions into
the trigonometric series have also been used in [4] by Kloeden and Platen. In
Kuznetsov method, various complete orthonormal systems may be applied for the
representation of iterated stochastic integrals of an arbitrary multiplicity [5]. Orthogonal expansions into the trigonometric series and the Haar series for Milstein method
have been investigated in [6]. The numerical simulation of iterated stochastic integrals
is discussed in [2, 3, 7–9].
The paper [10] deals with orthogonal expansions for random processes with
respect to Milstein method by using the spectral form of mathematical description.
In this chapter, the application of the spectral form of mathematical description for
the representation of the iterated Stratonovich stochastic integrals of an arbitrary
K. A. Rybakov (B)
Moscow Aviation Institute (National Research University), 4, Volokolamskoe shosse, Moscow
125993, Russian Federation
e-mail: rkoffice@mail.ru
© The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2021
L. C. Jain et al. (eds.), Applied Mathematics and Computational Mechanics for Smart
Applications, Smart Innovation, Systems and Technologies 217,
https://doi.org/10.1007/978-981-33-4826-4_20
287
Using Spectral Form of Mathematical
Description to Represent Iterated
Stratonovich Stochastic Integrals
Konstantin A. Rybakov
Abstract In this chapter, it is suggested to apply the spectral form of mathematical
description for the representation of the iterated Stratonovich stochastic integrals of
an arbitrary multiplicity. Some invariant relations for expansion coefficients and the
iterated Stratonovich stochastic integrals are obtained. An algorithm for modeling
the iterated Stratonovich stochastic integrals is discussed.
20.1 Introduction
Iterated stochastic integrals play a fundamental role in the constructing high-order
numerical methods for stochastic differential equations. These methods are based
on the Taylor–Itô expansion and the Taylor–Stratonovich expansion for random
processes. The first numerical method using iterated stochastic integrals of multiplicity 2 and orthogonal expansions into the trigonometric series was Milstein method
[1–3]. Iterated stochastic integrals of multiplicity 3 and orthogonal expansions into
the trigonometric series have also been used in [4] by Kloeden and Platen. In
Kuznetsov method, various complete orthonormal systems may be applied for the
representation of iterated stochastic integrals of an arbitrary multiplicity [5]. Orthogonal expansions into the trigonometric series and the Haar series for Milstein method
have been investigated in [6]. The numerical simulation of iterated stochastic integrals
is discussed in [2, 3, 7–9].
The paper [10] deals with orthogonal expansions for random processes with
respect to Milstein method by using the spectral form of mathematical description.
In this chapter, the application of the spectral form of mathematical description for
the representation of the iterated Stratonovich stochastic integrals of an arbitrary
K. A. Rybakov (B)
Moscow Aviation Institute (National Research University), 4, Volokolamskoe shosse, Moscow
125993, Russian Federation
e-mail: rkoffice@mail.ru
© The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2021
L. C. Jain et al. (eds.), Applied Mathematics and Computational Mechanics for Smart
Applications, Smart Innovation, Systems and Technologies 217,
https://doi.org/10.1007/978-981-33-4826-4_20
287