288
K. A. Rybakov
multiplicity is considered. The special case of Walsh series has been discussed in
[11], and here it is suggested to apply orthogonal expansions with respect to arbitrary
complete orthonormal systems.
Obtained results may be used in the constructing high-order numerical methods
based on the Taylor–Stratonovich expansion for random processes [2–4, 12–15],
and also in numerical methods based on Taylor–Itô expansion due to the known
relationship between the iterated Itô and Stratonovich stochastic integrals [13, 15].
High-order numerical methods may be applied for the modeling stochastic dynamical
systems [3, 4, 12, 16], the solving optimal and suboptimal filtering problems [17–19],
and the optimizing dynamical systems of the joint estimation and control [20–23].
Iterated stochastic integrals can be used in the constructing high-order numerical
methods for non-commutative semilinear stochastic partial differential equations
[24].
The goal of this research is to obtain the representation of the iterated Stratonovich
stochastic integrals using the spectral form of mathematical description of signals
and control systems [25–28] and to construct the spectral method and corresponding
algorithm for modeling the iterated Stratonovich stochastic integrals.
The rest of this chapter is structured as follows. Section 20.2 provides definitions
of the iterated Itô and Stratonovich stochastic integrals. Elements of the spectral
form of mathematical description are described in Sect. 20.3. The main result of
the chapter, i.e., the representation of the iterated Stratonovich stochastic integrals
using the spectral form of mathematical description, is given in Sect. 20.4. Further,
some invariant relations based on results of Sect. 20.4 for expansion coefficients and
the iterated Stratonovich stochastic integrals are obtained in Sect. 20.5. Section 20.6
gives the tensor representation for the expansion coefficients. Finally, Sect. 20.7
presents the conclusions for this chapter.
20.2 Iterated Itô and Stratonovich Stochastic Integrals
Let (Ω, F, P) be a probability space, where Ω is the sample space, F is a σ-algebra
of subsets of Ω, and P is a probability measure, and let F t be a non-decreasing family
of σ-subalgebras of F, t 0.
The iterated Stratonovich stochastic integrals of multiplicity k 2 are defined as
follows:
I
∗( j 1 j 2 ... j k )
h
=
h
0
. . .
τ 3
0
τ 2
0
dw j 1 (τ 1 ) ◦ dw j 2 (τ 2 ) ◦ . . . ◦ dw j k (τ k ), j 1 , j 2 , . . . , j k = 1, 2, . . . , s,
(20.1)
K. A. Rybakov
multiplicity is considered. The special case of Walsh series has been discussed in
[11], and here it is suggested to apply orthogonal expansions with respect to arbitrary
complete orthonormal systems.
Obtained results may be used in the constructing high-order numerical methods
based on the Taylor–Stratonovich expansion for random processes [2–4, 12–15],
and also in numerical methods based on Taylor–Itô expansion due to the known
relationship between the iterated Itô and Stratonovich stochastic integrals [13, 15].
High-order numerical methods may be applied for the modeling stochastic dynamical
systems [3, 4, 12, 16], the solving optimal and suboptimal filtering problems [17–19],
and the optimizing dynamical systems of the joint estimation and control [20–23].
Iterated stochastic integrals can be used in the constructing high-order numerical
methods for non-commutative semilinear stochastic partial differential equations
[24].
The goal of this research is to obtain the representation of the iterated Stratonovich
stochastic integrals using the spectral form of mathematical description of signals
and control systems [25–28] and to construct the spectral method and corresponding
algorithm for modeling the iterated Stratonovich stochastic integrals.
The rest of this chapter is structured as follows. Section 20.2 provides definitions
of the iterated Itô and Stratonovich stochastic integrals. Elements of the spectral
form of mathematical description are described in Sect. 20.3. The main result of
the chapter, i.e., the representation of the iterated Stratonovich stochastic integrals
using the spectral form of mathematical description, is given in Sect. 20.4. Further,
some invariant relations based on results of Sect. 20.4 for expansion coefficients and
the iterated Stratonovich stochastic integrals are obtained in Sect. 20.5. Section 20.6
gives the tensor representation for the expansion coefficients. Finally, Sect. 20.7
presents the conclusions for this chapter.
20.2 Iterated Itô and Stratonovich Stochastic Integrals
Let (Ω, F, P) be a probability space, where Ω is the sample space, F is a σ-algebra
of subsets of Ω, and P is a probability measure, and let F t be a non-decreasing family
of σ-subalgebras of F, t 0.
The iterated Stratonovich stochastic integrals of multiplicity k 2 are defined as
follows:
I
∗( j 1 j 2 ... j k )
h
=
h
0
. . .
τ 3
0
τ 2
0
dw j 1 (τ 1 ) ◦ dw j 2 (τ 2 ) ◦ . . . ◦ dw j k (τ k ), j 1 , j 2 , . . . , j k = 1, 2, . . . , s,
(20.1)