20 Using Spectral Form of Mathematical Description …
299
− I
∗( j 5 j 4 j 3 )
h
I
∗( j 2 j 1 )
h
− I
∗( j 5 j 4 j 3 j 2 )
h
I
∗( j 1 )
h
+ I
∗( j 5 )
h
I
∗( j 4 )
h
I
∗( j 3 j 2 j 1 )
h
+ I
∗( j 5 )
h
I
∗( j 4 j 3 j 2 )
h
I
∗( j 1 )
h
+ I
∗( j 5 j 4 j 3 )
h
I
∗( j 2 )
h
I
∗( j 1 )
h
+ I
∗( j 5 )
h
I
∗( j 4 j 3 )
h
I
∗( j 2 j 1 )
h
+ I
∗( j 5 j 4 )
h
I
∗( j 3 )
h
I
∗( j 2 j 1 )
h
+ I
∗( j 5 j 4 )
h
I
∗( j 3 j 2 )
h
I
∗( j 1 )
h
− I
∗( j 5 )
h
I
∗( j 4 )
h
I
∗( j 3 )
h
I
∗( j 2 j 1 )
h
− I
∗( j 5 )
h
I
∗( j 4 )
h
I
∗( j 3 j 2 )
h
I
∗( j 1 )
h
− I
∗( j 5 )
h
I
∗( j 4 j 3 )
h
I
∗( j 2 )
h
I
∗( j 1 )
h
− I
∗( j 5 j 4 )
h
I
∗( j 3 )
h
I
∗( j 2 )
h
I
∗( j 1 )
h
+ I
∗( j 5 )
h
I
∗( j 4 )
h
I
∗( j 3 )
h
I
∗( j 2 )
h
I
∗( j 1 )
h
.
Finally, it can be shown that for an arbitrary k:
C i k ...i 2 i 1 =
k
l=1
(−1)
k−l
C
l−1
k−1
m=1
M lm ,
where C
l−1
k−1 is the binomial coefficient [32], and the set of elements M lm for a fixed
l is formed by the products of expansion coefficients defined by Eq. 20.4 that are
required to represent l iterated stochastic integrals of the total multiplicity k:
M 11 = C i 1 i 2 ...i k ,
M 21 = C i 1 i 2 ...i k−1 C i k , M 22 = C i 1 i 2 ...i k−2 C i k−1 i k , ..., M 2,k−1 = C i 1 C i 2 i 3 ...i k ,
M lm = C i 1 ...i p C i p+1 ...i q . . . C i r +1 ...i k , l = 3, . . . , k − 2,
M k−1,1 = C i 1 i 2 C i 3 . . . C i k , M k−1,2 = C i 1 C i 2 i 3 C i 4 . . . C i k , ...,
M k−1,k−1 = C i 1 . . . C i k−2 C i k−1 i k ,
M k1 = C i 1 C i 2 . . . C i k ,
where the ordered set of indices i 1 . . . i p i p+1 . . . i q . . . i r +1 . . . i k (1 p < q r <
k) coincides with i 1 i 2 . . . i k , i.e., the summation over m is the summation over all
possible partitions of the set of indices i 1 i 2 . . . i k into l subsets with saving their order.
Therefore, for the iterated Stratonovich stochastic integrals we have:
I
∗( j 1 j 2 ... j k )
h
=
k
l=1
(−1)
k−l
C
l−1
k−1
m=1
M
∗
lm ,
where
M
∗
11 = I
∗( j k ... j 2 j 1 )
h
,
M
∗
21 = I
∗( j k )
h
I
∗( j k−1 ... j 2 j 1 )
h
, M
∗
22 = I
∗( j k j k−1 )
h
I
∗( j k−2 ... j 2 j 1 )
h
, ...,
M
∗
2,k−1 = I
∗( j k ... j 3 j 2 )
h
I
∗( j 1 )
h
,
M
∗
lm = I
∗( j k ... j r +1 )
h
. . . I
∗( j q ... j p+1 )
h
I
∗( j p ... j 1 )
h
, l = 3, . . . , k − 2,
M
∗
k−1,1 = I
∗( j k )
h
. . . I
∗( j 3 )
h
I
∗( j 2 j 1 )
h
, M
∗
k−1,2 = I
∗( j k )
h
. . . I
∗( j 4 )
h
I
∗( j 3 j 2 )
h
I
∗( j 1 )
h
, ...,
299
− I
∗( j 5 j 4 j 3 )
h
I
∗( j 2 j 1 )
h
− I
∗( j 5 j 4 j 3 j 2 )
h
I
∗( j 1 )
h
+ I
∗( j 5 )
h
I
∗( j 4 )
h
I
∗( j 3 j 2 j 1 )
h
+ I
∗( j 5 )
h
I
∗( j 4 j 3 j 2 )
h
I
∗( j 1 )
h
+ I
∗( j 5 j 4 j 3 )
h
I
∗( j 2 )
h
I
∗( j 1 )
h
+ I
∗( j 5 )
h
I
∗( j 4 j 3 )
h
I
∗( j 2 j 1 )
h
+ I
∗( j 5 j 4 )
h
I
∗( j 3 )
h
I
∗( j 2 j 1 )
h
+ I
∗( j 5 j 4 )
h
I
∗( j 3 j 2 )
h
I
∗( j 1 )
h
− I
∗( j 5 )
h
I
∗( j 4 )
h
I
∗( j 3 )
h
I
∗( j 2 j 1 )
h
− I
∗( j 5 )
h
I
∗( j 4 )
h
I
∗( j 3 j 2 )
h
I
∗( j 1 )
h
− I
∗( j 5 )
h
I
∗( j 4 j 3 )
h
I
∗( j 2 )
h
I
∗( j 1 )
h
− I
∗( j 5 j 4 )
h
I
∗( j 3 )
h
I
∗( j 2 )
h
I
∗( j 1 )
h
+ I
∗( j 5 )
h
I
∗( j 4 )
h
I
∗( j 3 )
h
I
∗( j 2 )
h
I
∗( j 1 )
h
.
Finally, it can be shown that for an arbitrary k:
C i k ...i 2 i 1 =
k
l=1
(−1)
k−l
C
l−1
k−1
m=1
M lm ,
where C
l−1
k−1 is the binomial coefficient [32], and the set of elements M lm for a fixed
l is formed by the products of expansion coefficients defined by Eq. 20.4 that are
required to represent l iterated stochastic integrals of the total multiplicity k:
M 11 = C i 1 i 2 ...i k ,
M 21 = C i 1 i 2 ...i k−1 C i k , M 22 = C i 1 i 2 ...i k−2 C i k−1 i k , ..., M 2,k−1 = C i 1 C i 2 i 3 ...i k ,
M lm = C i 1 ...i p C i p+1 ...i q . . . C i r +1 ...i k , l = 3, . . . , k − 2,
M k−1,1 = C i 1 i 2 C i 3 . . . C i k , M k−1,2 = C i 1 C i 2 i 3 C i 4 . . . C i k , ...,
M k−1,k−1 = C i 1 . . . C i k−2 C i k−1 i k ,
M k1 = C i 1 C i 2 . . . C i k ,
where the ordered set of indices i 1 . . . i p i p+1 . . . i q . . . i r +1 . . . i k (1 p < q r <
k) coincides with i 1 i 2 . . . i k , i.e., the summation over m is the summation over all
possible partitions of the set of indices i 1 i 2 . . . i k into l subsets with saving their order.
Therefore, for the iterated Stratonovich stochastic integrals we have:
I
∗( j 1 j 2 ... j k )
h
=
k
l=1
(−1)
k−l
C
l−1
k−1
m=1
M
∗
lm ,
where
M
∗
11 = I
∗( j k ... j 2 j 1 )
h
,
M
∗
21 = I
∗( j k )
h
I
∗( j k−1 ... j 2 j 1 )
h
, M
∗
22 = I
∗( j k j k−1 )
h
I
∗( j k−2 ... j 2 j 1 )
h
, ...,
M
∗
2,k−1 = I
∗( j k ... j 3 j 2 )
h
I
∗( j 1 )
h
,
M
∗
lm = I
∗( j k ... j r +1 )
h
. . . I
∗( j q ... j p+1 )
h
I
∗( j p ... j 1 )
h
, l = 3, . . . , k − 2,
M
∗
k−1,1 = I
∗( j k )
h
. . . I
∗( j 3 )
h
I
∗( j 2 j 1 )
h
, M
∗
k−1,2 = I
∗( j k )
h
. . . I
∗( j 4 )
h
I
∗( j 3 j 2 )
h
I
∗( j 1 )
h
, ...,