294
K. A. Rybakov
20.4 Using Spectral Form of Mathematical Description
to Represent Iterated Stratonovich Stochastic
Integrals
Since functions {q(i, t)}
∞
i=0 form the orthonormal basis, their spectral characteristics
are columns of the infinite identity matrix E, i.e.,
S[q(i, t)] = E i , i = 0, 1, 2, . . .
Consider functions q(i 1 , t), q(i 2 , t), …, q(i k , t) as input signals of the dynamical
system defined by SODE (Eq. 20.8), i.e., g l (t) = q(i l , t), l = 1, 2, . . . , k. Then,
we obtain relations for expansion coefficients defined by Eq. 20.4: C i k ...i 2 i 1 = x k (h).
Thus,
C i k ...i 2 i 1 = E
T
i k
P
−1
(V E i k−1 ) . . . P
−1
(V E i 2 )P
−1 E i 1
(20.13)
that follows from Eq. 20.12 with G l = E i l , l = 1, 2, . . . , k.
The product V E i l is a section of the three-dimensional infinite matrix V when
any index of three indices is fixed at i l (since V is the symmetric three-dimensional
infinite matrix with respect to any pair of indices), and the product P
−1 E i l is the
section of the infinite matrix P
−1 at the second fixed index, i.e., its i l th column. We
denote these sections by V ∗∗i l and P
−1
∗i l
, respectively. Similarly, the product E
T
i k
P
−1
is the section of the infinite matrix P
−1 at the first fixed index, i.e., its i l th row, which
we denote P
−1
i l ∗ . Consequently,
C i k ...i 2 i 1 = P
−1
i k ∗ V ∗∗i k−1 . . . P
−1 V ∗∗i 2 P
−1
∗i 1
.
Thus, all expansion coefficients C i k ...i 2 i 1 needed for modeling the iterated
Stratonovich stochastic integrals are expressed in terms of the infinite matrix P
−1
and the three-dimensional infinite matrix V .
Iterated stochastic integrals can be expressed by the same matrices P
−1 and V . It
is easy to see that the iterated stochastic integral I
∗( j 1 j 2 ... j k )
h
determined by Eq. 20.1
can be found as a solution to the following system of the Stratonovich stochastic
differential equations
dx 1 (t) = dw j 1 (t), x 1 (0) = 0,
dx 2 (t) = x 1 (t) ◦ dw j 2 (t), x 2 (0) = 0,
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
dx k (t) = x k−1 (t) ◦ dw j k (t), x k (0) = 0,
or the Langevin equations
˙
x 1 (t) = v j 1 (t), x 1 (0) = 0,
K. A. Rybakov
20.4 Using Spectral Form of Mathematical Description
to Represent Iterated Stratonovich Stochastic
Integrals
Since functions {q(i, t)}
∞
i=0 form the orthonormal basis, their spectral characteristics
are columns of the infinite identity matrix E, i.e.,
S[q(i, t)] = E i , i = 0, 1, 2, . . .
Consider functions q(i 1 , t), q(i 2 , t), …, q(i k , t) as input signals of the dynamical
system defined by SODE (Eq. 20.8), i.e., g l (t) = q(i l , t), l = 1, 2, . . . , k. Then,
we obtain relations for expansion coefficients defined by Eq. 20.4: C i k ...i 2 i 1 = x k (h).
Thus,
C i k ...i 2 i 1 = E
T
i k
P
−1
(V E i k−1 ) . . . P
−1
(V E i 2 )P
−1 E i 1
(20.13)
that follows from Eq. 20.12 with G l = E i l , l = 1, 2, . . . , k.
The product V E i l is a section of the three-dimensional infinite matrix V when
any index of three indices is fixed at i l (since V is the symmetric three-dimensional
infinite matrix with respect to any pair of indices), and the product P
−1 E i l is the
section of the infinite matrix P
−1 at the second fixed index, i.e., its i l th column. We
denote these sections by V ∗∗i l and P
−1
∗i l
, respectively. Similarly, the product E
T
i k
P
−1
is the section of the infinite matrix P
−1 at the first fixed index, i.e., its i l th row, which
we denote P
−1
i l ∗ . Consequently,
C i k ...i 2 i 1 = P
−1
i k ∗ V ∗∗i k−1 . . . P
−1 V ∗∗i 2 P
−1
∗i 1
.
Thus, all expansion coefficients C i k ...i 2 i 1 needed for modeling the iterated
Stratonovich stochastic integrals are expressed in terms of the infinite matrix P
−1
and the three-dimensional infinite matrix V .
Iterated stochastic integrals can be expressed by the same matrices P
−1 and V . It
is easy to see that the iterated stochastic integral I
∗( j 1 j 2 ... j k )
h
determined by Eq. 20.1
can be found as a solution to the following system of the Stratonovich stochastic
differential equations
dx 1 (t) = dw j 1 (t), x 1 (0) = 0,
dx 2 (t) = x 1 (t) ◦ dw j 2 (t), x 2 (0) = 0,
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
dx k (t) = x k−1 (t) ◦ dw j k (t), x k (0) = 0,
or the Langevin equations
˙
x 1 (t) = v j 1 (t), x 1 (0) = 0,