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DIFFERENTIAL EQUATIONS AND
CONTROL PROCESSES N. 4, 2022 Electronic Journal, reg. N&C77-39410 at 15.04.2010 ISSN 1817-2172
r
http://diffjournal. spbu. ru /
e-mail: jodiff@mail.ru
Stochastic differential equations Numerical methods Computer modeling in dynamical and control systems
A new approach to the series expansion of iterated Stratonovich stochastic integrals of arbitrary multiplicity with respect to components of the multidimensional Wiener process. II
Abstract. The article is devoted to the development of a new approach to the series expansion of iterated Stratonovich stochastic integrals with respect to components of the multidimensional Wiener process. This approach is based on multiple Fourier series in complete orthonormal systems of Legendre polynomials and trigonometric functions in Hilbert space. The theorem on the mean-square convergent expansion for the iterated Stratonovich stochastic integrals of multiplicity 6 is formulated and proved. In the first part of the paper, expansions of iterated Stratonovich stochastic integrals of multiplicities 1 to 5 were obtained. These results allow us to construct efficient approximation procedures for iterated Stratonovich stochastic integrals that are necessary for the implementation of strong numerical methods with orders 1.0, 1.5, 2.0, 2.5, and 3.0 for Ito stochastic differential equations with non-commutative noise (in the framework of the approach based on the Taylor-Stratonovich expansion). Key words: iterated Stratonovich stochastic integral, iterated Ito stochastic integral, Ito stochastic differential equation, generalized multiple Fourier series,
Dmitriy F. Kuznetsov
Peter the Great Saint-Petersburg Polytechnic University e-mail: sde_kuznetsov@inbox.ru
multiple Fourier-Legendre series, multiple trigonometric Fourier series, mean-square convergence, expansion.
Contents
1 Introduction 136
2 Preliminary Results 138
2.1 Expansion of Iterated Ito Stochastic Integrals of Arbitrary Multiplicity k (k G N)
Based on Generalized Multiple Fourier Series Converging in the Mean . . . ... 138
2.2 Stratonovich Stochastic Integral...........................141
2.3 Expansion of Iterated Stratonovich Stochastic Integrals of Arbitrary Multiplicity
k (k E N) Under the Condition of Convergence of Trace Series..........14
2.4 Expansions of Iterated Stratonovich Stochastic Integrals of Multiplicities 1 to 5 . 149
3 Main Result 151
3.1 Theorem on Expansion of Iterated Stratonovich Stochastic Integrals of Multi-
plicity 6........................................151
3.2 Proof of Theorem on Expansion of Iterated Stratonovich Stochastic Integrals of
Multiplicity 6 ..................................... 152
4 Conclusion 186
References 186
1 Introduction
Let (Q, F, p) be a complete probability space, let {Ft,t G [0, T]} be a non-decreasing right-continous family of a-algebras of F, and let wt be a standard m-dimensional Wiener stochastic process, which is Ft-measurable for any t G [0,T]. We assume that the components wt(i) (i = 1,... , m) of this process are independent. Consider an Ito stochastic differential equation (SDE) in the integral form
t t
xt = x0 + y a(xT,t)dr + J B(xT,t)dwT, x0 = x(0,w), u G Q. (1) 0 0
Here xt is some n-dimensional stochastic process satisfying the equation (1). The nonrandom functions a : Rn x [0,T] ^ Rn, B : Rn x [0,T] ^ Rnxm guarantee the existence and uniqueness up to stochastic equivalence of a solution of the equation (1) [1]. The second integral on the right-hand side of (1) is interpreted as the Ito stochastic integral. Further, xo be an n-dimensional random
f 21
variable, which is F0-measurable and m{ |x01 j < to (m denotes a mathematical expectation). We assume that x0 and wt — w0 are independent when t > 0.
Consider the following iterated Ito and Stratonovich stochastic integrals
t t2
J [^]Trfc) = / h (tk) ..J ^i(ti)dwt(il)... dw^, (2)
t t * T * t2
j*[h(k)}Tit-ik) = | ik(tk)..J h(t1)dwi;i)...dwi:k), (3) tt
where ^1(r),... ,hk(t) are nonrandom functions on [t,T], wT;) (i = 1,... ,m) are independent standard Wiener processes and w[0) = t, i1,... ,ik = 0, 1, ..., m,
and
denote Ito and Stratonovich stochastic integrals, respectively.
It is well known that the above stochastic integrals, with a special choice of weight functions h (t),... , hk(t), play an important role when solving the problem of numerical integration of Ito SDEs using an approach based on the Taylor-Ito and Taylor-Stratonovich expansions [2]-[5]. The importance of the noted problem is explained by a wide range of applications of Ito SDEs [2]-[5].
More pecisely, 11(t),..., hk(t) = 1, i1,... ,ik = 0,1,...,m in the classical Taylor-Ito and Taylor-Stratonovich expansions [2]-[8] and h(t) = (t — t)qi, ql = 0,1,... (l = 1,... ,k), i1,...,ik = 1,... ,m in the unified Taylor-Ito and Taylor-Stratonovich expansions [9]-[13 .
The development of a new effective approach [14] to the series expansion and mean-square approximation of the iterated Stratonovich stochastic integrals (3) composes the subject of the article.
We also note other approaches to the mean-square approximation of the iterated Ito and Stratonovich stochastic integrals (2) and (3) [2]-[5], [15]-[32 .
2 Preliminary Results
2.1 Expansion of Iterated Ito Stochastic Integrals of Arbitrary Multiplicity k (k E N) Based on Generalized Multiple Fourier Series Converging in the Mean
The results of this section are auxiliary to the proof of the main result of this article (see Theorem 8 below).
Suppose that 1 (t),..., 1k(t) E L2([t, T]). Define the following function on the hypercube [t,T]k
K(ti,..., tk) = 1{t1<...<ifc}(ti)... 1k(tk), (4)
where ti,...,tk E [t,T] (k > 2), K(ti) = 1 (ti) for ti E [t,T], and 1A is the indicator of the set A.
Assume that (x)}°=0 is a complete orthonormal system of functions in the space L2([t, T]). It is well known that the generalized multiple Fourier series of K(ti,...,tk) E L2([t,T]k) is converging to K(ti,...,tk) in the hypercube [t,T]k in the mean-square sense, i.e.
lim
PlvvPfc ^^
K — K
K Kpl •••Pfc
= 0,
L2([t,T ]k)
where
/ .
1/2
f
li2([t,T ]k)
J f2(ti,...,tk )dti ...dtk
V[t,T ]k y
Pi Pk k
Kpi...pk ... ,tk) = X ■ ■ • ^jk-ji II ^(t/^
ji=0 jk=0 1=1
/k
K(ti, ■ ■ ■ ,tk^(ti)dti ...dtk (5)
k i=i
is the Fourier coefficient.
Consider the partition {Tj j^Lo of [t, T] such that
t = t0 < ... < tn = T, AN = max Arj ^ 0 if N ^ œ, ATj = Tj+1 — Tj.
0<j<N—1
(6)
Theorem 1 [11] (2006), [12], [13], [33]-[53]. Suppose that every ^ (t) (l = 1, ...,
k) is a continuous nonrandom function on [t,T] and {j(x)}°=0 is a complete
orthonormal system of continuous functions in the space L2([t,T]). Then
pi pk / k (n---ifc) _ i • ^ \ A \ A I I I Aii)
= J.^ EC
ji=0 jfc=0 \/=1
l:i:m: ^ j fa ^W^ . . . j , (7)
(ii.....ik№ 1 k J
(¿1-ifc) T,t
¿1,..., ik = 0,1,... ,m, l.i.m. is a limit in the mean-square sense, J[^(k)] is defined by (2),
Gk = Hk\Lk, Hk = {(/i,...,/k) : 1i,...,1k = 0, 1,..., N - 1},
Lk = {fa...,4 ): li ,...,1k = 0, 1,..., N -1; lg = lr (g = r); g,r = 1,...,k},
T
j = J j(t)dw«
t
are independent standard Gaussian random variables for various i or j (in the case when i = 0), Cjk...j is the Fourier coefficient (5), Awlj = wTj++i — wTj (i = 0, 1, ..., m), fa}=0 is a partition of the interval [t, T], which satisfies the condition (6).
A number of generalizations and modifications of Theorem 1 can be found in [13], Chapter 1 (see also bibliography therein).
Let us consider corollaries from Theorem 1 (see (7)) for k = 1,..., 5 [11]
pi
J [V/1^ = l:i.m £ Cji Cf , (8)
' pi^œ ' * J1 ji =0
Pi P2 / \
j[v(2)]-Ti,iti2) = E E cj:;')cj22) - i{!i=,2=o}i{ji=j2> , (9)
ji=0 j2=0 V /
Pi P2 P3 /
J[v(3,]if3) = i:i.m YT. Ej Ä,Zj(:s)-
ji=0 j2=0 73 =0 \
-1{n=i2=0}l{ji=j2}Cii3) - 1{i2=i3=0}l{j2=j3} d;i} - 1{ii = i3=0}l{ji=j3}Cj2i2) I , (10)
J
;i...;4)
Pi
P4
l.i.m.
E-Lc^i n c
(;i) ji
ji=0 j4=0
J=i
1 1 A (;3V (;4) 1 1 Z (;2V (;4)
-1{;i=;2=0}1{ji=j2}Sj3 zj'4 - 1{;i=;3=0}1{ji=j3}Sj2 j -
1 i a(;2v(;3) i i /-(;iv(;4 )
1{;i=;4=0}1{ji=j4}Zj2 zj3 - 1{;2=;3=0}1{j2=j3}zji j -
1 1 ^(;i V(;3) 1 1 A;i)A;2) I
1{i2=i4=0}1{j2 =j4}Zji j - 1{;3 = ;4=0}1{j3=j4}Zji j +
+ 1{;i=;2=0}1{ji=j2}1{;3=;4=0}1{j3=j4} + 1{;i=;3=0}1{ji=j3} 1{;2=;4=0}1{j2=j4} +
+ 1{;i=;4=0}1{ji=j4}1{;2=;3=0}1{j2=j3}
(11)
J
{;i=;2=0} {;i=;4=0} {;2=;3=0} {i2 = i5=0} {;3=;5=0}
ii...;5)
Pi
P5
= l.i.m.
ji=0
E^-m zj
j5=0
(;i) ji
z (;3)z (;4)z (;5
{ji=j2}Zj3 zj4 zj5
z (;2)z (;3)z (;5
{ji=j4}zj2 z j3 zj5
z (;i)z (;4)z (;5
{j2=j3} zji zj4 zj5
z (;i)z (;3)z (;4
{j2=j5} zji z j3 zj4
z (;i)z (;2)z (;4
{j3=j5 } zji zj2 z j4
+ 1{;i= =i2=0} {ji = =j2} {;3 = i4=0} { j 3 ■ iz(;5 =j4} z j5
+ 1{;i= =i2=0} {ji = =j2} {;4 = i5=0} {j4 z (;3 =j5}zj3
+ 1{;i= =i3=0} j =j3} {;2 = i5=0} { j 2 ■ iz(;4 =j5} z j4
+ 1{;i= =i4=0} {ji = =j4} {;2 = i3=0} { j 2 z (;5 =j3}zj5
+ 1{;i= =i4=0} {ji = =j4} {;3 = i5=0} { j 3 z (;2 =j5}zj2
+ 1{;i= =i5=0} j =j5} {;2 = i4=0} { j 2 . lz(;3 =j4}zj3
+ 1{i2 = =i3=0} { j 2 =j3} {;4= = i5=0} {j4 =j5} zji
1
{;i=
1{;i= 1{i2 =
1
1
{i3=
{;4=
+ 1{;i= + 1{;i= + 1{;i= + 1{;i= + 1{;i= + 1{;i= + 1{i2 =
;3=0} i5=0} ;4=0} ;4=0} ;5=0} ;2=0} ;3=0} ;3=0} ;4=0} ;5=0} ;5=0} ;4=0}
,1=1
z (;2
{ji=j3}zj2
z (;2
{ji=j5}Sj2
z (;i
{j2=j4} zji
z (;i
{j3=j4}zji
z(i4)z(i5 zj4 z
j4 j5
z (;3)z (;4
zj3 zj4
z (;3)z (;5
zj3 zj5
z (;2)z (;5
z 72 z
j2 j5
z (;i)z (;2)z (;3) +
{j4=j5}zji z j2 zj3 +
{ji=j2} {ji=j3} {ji=j3} {ji=j4} {ji=j5} {ji=j5} {j2=j4}
{i3 = = i5=0} { j 3 ■ iz(;4 =j5 } zj4 +
{;2 = i4=0} { j 2 z (;5 =j4}zj5 +
{i4 = = i5=0} {j4 ■ T,z (i2 =j5 } z j2 +
{;2 = i5=0} { j 2 z (;3 =j5}zj3 +
{;2 = i3=0} { j 2 ■ iz(;4 =j3} zj4 +
{i3 = = i4=0} { j 3 ■ X(i2 =j4} zj2 +
{i3= = i5=0} { j 3 =j5} zji +
4
5
+ 1{i2=i5=0}l{j2=j5}1{i3 = i4=0}l{j3=j4}C]1i1^ , (12)
where 1A is the indicator of the set A.
Consider a generalization of Theorem 1 for the case of an arbitrary complete orthonormal system of functions in L2([t,T]) and ^i(t),... (t) E L2([t,T]) 54], [13], [45]. At that we will base on our notations [12] (2009), [13],
Theorem 2 [13] (Sect. 1.11), [45] (Sect. 15). Suppose that ),... (t) E L2([t,T]) and (x)}°=0 is an arbitrary complete orthonormal system of functions in the space L2([t,T]). Then the following expansion
Pi Pk ( k [k/2]
J №(k)]^Tir,k) = p l.i.m. V ... £ Ck.J II cji'' + E(-1)'x
ji=0 jk=0 \Z=1 r=1
r fc—2r
X X n 1{lg2s-1 = 1g2s =0} 1{jg2s-1 = jg2s } II ^ M (13)
({{si,32},...,{s2r-1,32r }},{?1,...,9fc-2r}) s = l 1=1 /
{si,S2,...,S2r-1,S2r,91,...,9fc-2r } = {1,2,.",fc}
that converges in the mean-square sense is valid, where [x] is an integer part of a real number x, the sum in the second line of the formula (13) means the sum with respect to all possible permutations of the set
({{gl, g2}, . . . , {g2r—1, g2r}}, . . . , qk—2r}) (14)
braces mean an unordered set, and parentheses mean an ordered set, {g1, g2,..., g2r-1, g2r, q1, • • •, qk-2r} = {1, 2,...,k}; another notations are the same as in Theorem 1.
2.2 Stratonovich Stochastic Integral
def
Let M2([t,T]) (0 < t < T < to) be the class of random functions £(t, w) = : [t, T] x Q ^ R, which satisfy the following conditions: £(t, w) is measurable with respect to the pair of variables (t, w), is FT-measurable for all t E [t, T], is independent with increments ws+A — ws for s > t, A > 0, and
t
j m {(^t)2} dT< to, m )2} < to for all t e [t,T].
t
We introduce the class Q4([t,T]) of Ito processes nT^, t G [t,T ], i = 1,... ,m of the form
T T
nTi) = nt(i) + J asds + J bsdw(i) w. p. 1, (15)
t t
where (as)4, (bs)4 G M2([t,T]) and lim m{|bs - bT|4} = 0 for all t G [t,T]. The
s—YT
second integral on the right-hand side of (15) is the Ito stochastic integral. Here and further, w. p. 1 means with probability 1.
Consider a function F(x, t) : R x [t, T] — R for fixed t from the class C2(R) consisting of twice continuously differentiable in x functions on R such that the first two derivatives are bounded.
The mean-square limit
N-1 , .
l i m Y,F[ \ + C) '^ ) (<> " O = / ^r)dw? (16) j=0 \ j j
is called [55] the Stratonovich stochastic integral with respect to the component
j }j=0
wT^ (/ = 1,... ,m) of the multidimensional Wiener process wT, where {Tj}N is a partition of the interval [t, T], which satisfies the condition (6).
It is known [55] (also see [2]) that under proper conditions, the following relation between Stratonovich and Ito stochastic integrals holds
* T t T
I F{i^\r)d^ = J + l-l[l=l} J^-(1]T,r)bTdT (17)
t t t w. p. 1, where 1A is the indicator of the set A and i, I = 1,..., m.
A possible variant of conditions under which the formula (17) is correct, for example, consists of the conditions n-T^ G Q4([t,T]), F(n-r^T) G M2([t,T]), F (x, t ) G C2(R) (for fixed t ), where i = 1,..., m.
2.3 Expansion of Iterated Stratonovich Stochastic Integrals of Arbitrary Multiplicity k (k G N) Under the Condition of Convergence of Trace Series
In this section, we consider the expansion of iterated Stratonovich stochastic integrals (3) of arbitrary multiplicity k (k G N) under the condition of convergence of trace series [14] (we also note approaches to the expansion of multiple
Stratonovich stochastic integrals based on other definitions of these integrals [56], [57]).
Let us introduce some notations. Consider the Fourier coefficient
T t2
Cjfc ...j1 = J ^ (tk) j (tk)... J fa (ti)j (ti)dti.. .dtk (18)
t t
corresponding to the kernel (4), where (x)}°=0 is a complete orthonormal system of functions in the space Lo{[t,T}). At that we suppose (f)o(x) = 1 /\/T — I.
Denote
C......
CJk-Ji+Ui JiJi-2-Ji
def
T t+2 ti + 1
= J 1(tk(tk) . . J (t/+0 ^ 1(t/M—i(t/)x t t t t t2
x J 1г_2(£г_2)фЛ-2(t/—2)... У 1i(ti)j(ti)dti... dt/—2dt/t/+i... dt* =
tt
T t;+2 t;+i
= J ClAli/jOj, (I I..) ...J ф1+1(и+1)фл+1(и+1) J Ф1Ш1- 1Шо(и)х t t t tl t2
x J1/—2(t/_2)0ji-2(t/_2).. ^1i(ti)0ji(ti)dti.. .dt/—2dt/t/+i.. .dtk, (19)
tt
i.e. (19) is again the Fourier coefficient of type (18) but with a new shorter multi-index jk ... j/+i0j/—2 ... j and new weight functions 1i(t), ..., 1/—2(т),
[t yipi{r ), -0/+i(r), ..., (also we suppose that {/, I — 1} is one
of the pairs {gi,^},---, {g2r—i,g2r} (see (14))). Let
Cjfc •••jq•••j1
q=g1,g2,-,g2r-1,g2r œ œ œ œ
=f E Cjk-л . (20)
jS2r-1 =p+i jS2r-3 =p+i j'as =p+i jS1 =p+i
J31 JS2 '•"'JS2r-l JS2r
Introduce the following notation
Sil a
(p)
jk-jq •••j1
def 1
1
jg2i=g2i-i+i}
oo oo
E E
jfl2r-1 =P+1 jfl2r-3 =P+1
y^ y^ • • • S Cjfc .ji
jS2i + 1 =P+1 j's2i-3 =P+1 j'flS =P+1 j'fll =P+1
(j32l jB2l-1 W0jg1 jS2 '•••'jS2r-1 jS2r
Note that the operation S/ (l = 1, 2,..., r) acts on the value
/ (P)
jk •••jq •••j1
(21)
q=01,02,-,02r-1,g2r-
as follows: S/ multiplies (121) by 1{g2l=g2I-1+1}/2, removes the summation
E ■
jfl2!-1 =P+1
and replaces
with
C ■
ajk •••j1
jS1 j32 '•••'jS2r-1 j32r
/
jk-j1
(j32l jB2l-1 W0jg1 jS2 '•••'jS2r-1 jS2r
(22)
(23)
Note that we write
/
Cjk •••j1
= a
( jg 1 jS2 ) ^ ( •) 'js 1 = jS2
jk •••j1
( jg 1 jg 1) ^( •) ,jg 1 =jg2
/
ajk---j1
= a
jk •••j1
( jg 1 jg2) •),( jg3 jg4) ^( •) jg 1 =jg2 jg3 =jg4 ( jg 1 jg 1) ^( •)( jg3 jg3 ) ^( •) ,jg 1 =jg2 ,jg3 =jg4
Since (23) is again the Fourier coefficient, then the action of superposition S/Sm on (22) is obvious. For example, for r = 3
Q Q Q J n(p)
S3S2S1 <{ ^^ jk •••jq •••j1
q=01>02,-,g5,g6
2
1 3
2 s=1
( j32 j31 ) •) ( j34 j33 ) ^(•) ( j36 j35 ) •) 'j3 1 =j32 'jS3 =j34 'jS5 =j36
S3 S1 < c
(p)
jk -jq •••jl
q=gi,g2,-,g5,g6
00
22
J'as =P+1
( j32 jS 1 ) ^ (•) ( j36 j35 ) ^ (•) 'jS 1 =j32 'jS3 =j34 'jS5 = jS6
s^ C
(p)
jk-jq •••j1
q=gl,g2,•••,g5,g6
00 00
1
{54=53+1}
j's1 =p+1 j's5 =P+1
(jS4 jS3 ) ^(•) 'j31 = jS2 'jS3 = jS4 'jS5 = jS6
Theorem 3 [14], [49], [50]. Assume that the continuously differentiable functions ^¿(t) : [t, T] ^ R (/ = 1,..., k) and the complete orthonormal system of continuous functions {(f>o{x) = 1/VT — i) m i/ie space Lo{[t,T}) are such that the following conditions are satisfied:
1. The equality
S S t2
i J = J ^2{t2)<f>j{t2) J ^{h^.jihyihdto (24)
t j =0 t t
holds for all s £ (t,T], where the nonrandom functions $i(t), $2(t) are continuously differentiable on [t,T] and the series on the right-hand side of (24) converges absolutely.
2. The estimates
T
(t )$1(r )dr
<
tti(s)
jl/2+a '
(t)$2(t)dT
<
jl/2+a '
1
oo
J=p+1 t t
T
<
hold for all s E (t,T) and for some a,^ > 0, where (t), $2(t) are continuously differentiable nonrandom functions on [t,T], E N, and
T T
J ^!(t)dT < 00, J |^2(T)| dT < 00.
tt
3. The condition P
lim
p^TO
E
Sii Si„...
(p)
d 1 Jk •••Jq •••jl
9=31>32>...>32r-1>32r
= 0
q^gl^v^r-l^r-
holds for all possible g1, g2,..., g2r—1, g2r (see (U4)) and l1, l2,..., such that l1, l2,..., E {1, 2,... , r}, l1 > l2 > ... > d = 0,1, 2,... , r — 1, where r = 1, 2,..., [k/2] and
Q Q Q z' (p)
Sl1 Sl2 • • • S ^•••Jq-Jl
def (7(p)
CJk •••Jq •••J'l
q=g1,g2,-,g2r-1,g2r
for d = 0.
Then, for the iterated Stratonovich stochastic integral of arbitrary multiplicity k
T
t2
j wiTi
(il-ik) = / v*(tk)... / ^i(ii)dwi;l)...dwtkk)
(25)
the following expansion
J *
(il •••ik) T,t
p
k
= E ^n c
Jl ,•••,Jk=0 1=1
(ii ) 111J
that converges in the mean-square sense is valid, where
T t2
/ ^fc(tk)<fe(tk)... (ii)<fe (ii)dti.. .difc
(26)
(27)
s
2
is the Fourier coefficient, l.i.m. is a limit in the mean-square sense, i1;..., ik = 0,1,..., m,
T
j = / & (t)dw«
t
are independent standard Gaussian random variables for various i or j (in the case when i = 0), wT0) = t.
The results presented below in this section show that Conditions 1 and 2 of Theorem 3 are satisfied for complete orthonormal systems of Legendre polynomials and trigonometric functions in the space L2([t,T]).
In [13] (Sect. 2.1.2), the following equality is proved
T
1 « œ
- j MtJMtiWh^Cjj, (28)
where
2 „
t j=0
T t2
Cjj = J ^2(t2)0j (¿2) ^ (tl)dti dt2, tt
{ф,(x)}°=0 is a complete orthonormal system of Legendre polynomials or trigonometric functions in the space L2([t,T]), the functions (т), ^2(т) are continuously differentiable at the interval [t,T].
Note that the following estimate
j=p+i
< ^ (29)
P
holds under the above conditions, where constant C is independent of p [13 (Sect. 2.1.2).
The relations (28) and (29) have been modified as follows [13] (Sect. 2.7,
2.9)
1 s œ
- MtJMtJdh^Cjjis), (30)
t j=0
oo
E Cjj (s)
J=P+1
<
C
p (1 — z2(s))
1/4'
(31)
where (30) holds for the case of Legendre polynomials or trigonometric functions and (31) holds for the case of Legendre polynomials, s G (t,T) (s is fixed, the case s = T corresponds to (28) and (29)), constant C does not depend p, the functions ^1(t), ^2(t) are continuously differentiable at the interval [t,T],
S ¿2
Cj(s) = I fete)^te) /(ÎI)CMÎ2,
z(s) =
s
T +1
T-t
For the trigonometric case, the estimate (31) is replaced by
oo
X Cjj (s)
J =P+1
C < —.
p
(32) (Sect. 2.9)
(33)
where s E (t,T), constant C does not depend on p.
Note the well known estimate for the Legendre polynomials
|Pj(y)| <
K
v7+ï(i-y2)1/4'
y G ( —1, 1), j G N,
(34)
where Pj (y) is the Legendre polynomial, constant K is independent of y and j.
Using (34), we obtain the following useful estimates for the case of Legendre polynomials [13] (Sect. 2.2.5)
1(t )0j (t )dT
<
C
T
T )0j (t )dT
<
j (1 — (z(x))2)1/4'
C
(35)
(36)
j (1 — (z(x))2)1/4'
where j E N, z(x) E (—1,1) is defined by (32), x E (t, T), 1(t) is a continuously differentiable function at the interval [t,T], constant C does not depend on j.
2
2
x
For the case of trigonometric functions, we note the following obvious estimates
fa (t
C
< —,
J
T
1(t fa (t )dT
C
< —,
J
(37)
where j G N, x G (t,T), the function ) is continuously differentiable at the interval [t, T], constant C does not depend on j.
Remark 1. From (ESHEH), (33), (33)-(33) it follows that Conditions 1 and 2 of Theorem 3 are satisfied for complete orthonormal systems of Legendre polynomials and trigonometric functions in the space L2([t,T]) (a = 1/2, ^ = i).
x
2.4 Expansions of Iterated Stratonovich Stochastic Integrals of Multiplicities 1 to 5
The following three theorems were proved in [14] on the base of Theorem 3. Theorem 4 [13], [14], [49], [50]. Suppose that fa-(x)}°=0 is a complete or-
thonormal system of Legendre polynomials or trigonometric functions in the space L2([t,T]) and fa(t),^2(t),^3(t) are continuously differentiable nonran-dom functions on [t,T]. Then, for the iterated Stratonovich stochastic integral
J*fa3)]TT3) (ii, i2, i3 = 0,1,... ,m) defined by (3) the following relations
p
J*['<A(3)]Trs) = £ Cj,cfcfc':3), (38)
j1,j2,i3 =0
m j fa<3>]<!;«> - e ^cjl-'cfc^a 1 < J (39)
[ V jU2j3 =0 / J P
are fulfilled, where ii, i2, i3 = 0,1,..., m in (38) and ii, i2, i3 = 1,..., m in (39), constant C is independent of p; another notations are the same as in Theorem 1.
Theorem 5 [13], [14], [49], [50]. Let {fa- (x)}j=0 be a complete ortho-
normal system of Legendre polynomials or trigonometric functions in the
space L2([t,T]) and ^1(t),..., ^4(t) be continuously differentiable nonran-dom functions on [t,T]. Then, for the iterated Stratonovich stochastic integral
](n..
T,t
J*[^(4>(ii,...,i4 = 0,1,... ,m) defined by (3) the following relations
p
j.[V/4)].Tr4>=u.m. ^ Cjtj...cj:4), (40)
j 1 ,...J:=0
m | - £ Cu..,,Cf ■ ■ ■ cf j | < £ (41)
are fulfilled, where h,... ,i4 = 0,1,...,m in (40) and i\,... ,i4 = 1,...,m in (41), constant C does not depend on p, £ is an arbitrary small positive real number for the case of complete orthonormal system of Legendre polynomials in the space L2([t, T]) and £ = 0 for the case of complete orthonormal system of trigonometric functions in the space L2([t,T]); another notations are the same as in Theorem 1.
Theorem 6 [13], [14], [49], [50]. Assume that [fa (xj 0 is a complete or-
thonormal system of Legendre polynomials or trigonometric functions in the space L2([t,T]) and ^1(t),...,^5(t) are continuously differentiable nonran-dom functions on [t,T]. Then, for the iterated Stratonovich stochastic integral
J*[^(5>fc"5» (i1, ... ,i5 = 0,1,... ,m) defined by (3) the following relations
p
J'[V/5)]T'f:5> = lpi£- £ Cj5 j Cj:'>... j0, (42)
j1,...,j5=0
M Ej(43)
are fulfilled, where i\,... ,i5 = 0,1,...,m in (42) and i\,... ,i5 = 1,...,m in (43), constant C is independent of p, £ is an arbitrary small positive real number for the case of complete orthonormal system of Legendre polynomials in the space L2([t, T]) and £ = 0 for the case of complete orthonormal system of
p
trigonometric functions in the space L2([t,T]); another notations are the same as in Theorem 1.
Also we note the following theorem for the case k = 2.
Theorem 7 [13] (Sect. 2.8.1). Suppose that {fa(x)}°=0 is a complete orthonormal system of Legendre polynomials or trigonometric functions in the space L2([t,T]) and fa(r),fa(r) are continuously differentiable nonran-dom functions on [t,T]. Then, for the iterated Stratonovich stochastic integral
JfafaT^ (ii,i2 = 0,1,... ,m) defined by (3) the following relations
p
J• [/2)f/2) = lm £ Cjcfcf, (44)
j1j2=0
m I - J < £ (45)
are fulfilled, where i1,i2 = 0,1,...,m in (44) and i1,i2 = 1,...,m in (45), constant C is independent of p; another notations are the same as in Theorem 1.
Note that an analogue of Theorem 7 for the case k = 1 follows from (8).
3 Main Result
3.1 Theorem on Expansion of Iterated Stratonovich Stochastic Integrals of Multiplicity 6
In this section, we formulate the main result of the article.
Theorem 8 [13], [48]-[50]. Suppose that {fa(x)}°=0 is a complete orthonor-
mal system of Legendre polynomials or trigonometric functions in the space L2([t,T]). Then, for the iterated Stratonovich stochastic integral of sixth multiplicity
* T * ¿2
fa ... / dw<;i)...dw<:6) (46)
the following expansion
p
= £ ..j
that converges in the mean-square sense is valid, where i\, . . . ,i. = 0,1,... ,m,
t t2
Cj6..ji = j j (t.) --¡hi (ti№ ...di.
t t
and
T
j = / j(s)dw<:)
t
are independent standard Gaussian random variables for various i or j (in the case when i = 0), wT0) = t.
3.2 Proof of Theorem on Expansion of Iterated Stratonovich Stochastic Integrals of Multiplicity 6
This section is devoted to the proof of Theorem 8.
As noted in Remark 1, Conditions 1 and 2 of Theorem 3 are satisfied for complete orthonormal systems of Legendre polynomials and trigonometric functions in the space L2([t, T]). Let us verify Condition 3 of Theorem 3 for the iterated Stratonovich stochastic integral (46). Thus, we have to check the following conditions
2
p / œ
P™ £ £
jq1 ,jq2 0 P+1
= 0, (47)
jS1 =j32
^ 2
p / œ œ
' _ ) =0, (48)
jS1 _jS2 j93 _j34 /
2
Püs, £ £ £
jq1 ,jq2 _0 \jg1 _P+1 jg3 _P+1
pœ
* n.
j1
Pi?œ £ £ n-
jq1 ,jq2 _0 \jg1 _P+1
= 0, (49)
(jg4 jg3 W0,jg1 _jg2 ,jg3 _jg4 >04_g3+1
Püro E E E
= 0, (50)
jS1 = jS2 j33 = jS4 j35 = jS6
2
TO oo
v C
■6--J1
lim V V C
p^TO \
■ =P+1 ■% =P+1
= = = = . =0, (51)
(jS6 jS5 ) ^ (•) 1 = jS2 j33 = jS4 j35 =j36 ,g6=g5 +1
p^TO \ z-'
■ =p+1
= 0,
(jS4 jS3 (•) (jS6 jS5 (•)'jS1 = jS2 = jS4 j35 =j36 >04=g3 + 1 ,g6=g5 + 1
(52)
where the expressions Hg^ g2} fe g4}, fe g6}}) , ({gb g2} {g3, g4}, {q1 q2}}), ({g1, g2}, {q1,q2, q3, q4}) are partitions of the set {1, 2,..., 6} that is {g1, g2, g3,
g4,g5,g6} = {g1,g2,g3,g4,q1,q2} = {g1 ,g2,q1,q2,q3,q4} = {1, 2, ..., 6}; braces mean an unordered set, and parentheses mean an ordered set.
The equalities (47), (49) were proved earlier (see the proof of equalities (2.786), (2.792) in [13], pp. 541, 543 or equalities (194), (200) in [14], pp. 164, 166). The relation (52) follows from the estimate (29) for the polynomial case and its analogue for the trigonometric case. It is easy to see that the equalities (48) and (51) are proved in complete analogy with the proof of equalities (166), (200) in [14], pp. 152, 166 (also see the proof of equalities (2.758), (2.792) in [13], p. 528, 543).
Thus, we have to prove the relation (50). The equality (50) is equivalent to the following equalities
to to to
lim / v Zj /v Cj3j2j1j3j2j1 = 0, (53)
P—TO z-' z-' z-'
j 1 =p+1 j2 =P+1 j3 =P+1
TO TO TO
lim Y Y Y Cj1j3j2j3j2j1 = 0, (54)
P—TO z-' z-' z-'
j 1 =P+1 j2 =P+1 j3 =P+1
TO TO TO
lim Cj3j2j3 jl j2 jl = 0, (55)
P—>-TO z-' z-' z-'
j 1 =P+1 j2 =P+1 j3 =P+1
2
2
oo oo oo
llm S S S Cjlj2j3j3j2jl — 0, (56)
P—TO
jl=P+1 j2=P+1 j3=P+1
OO OO OO
lim S S S Cjlj2j2 j3j3jl — 0, (57)
P—TO
jl=P+1 j2=P+1 j3=P+1
TO TO TO
plimTO y^ Cj3j3j2j2jljl — 0, (58)
jl=p+1 j2=P+1 j3 =P+1
TO TO TO
l.im Cj2j3j3j2jljl — 0, (59)
P—TO
jl=p+1 j2=P+1 j3 =P+1
TO TO TO
lim (60)
P—TO
jl=p+1 j2 =P+1 j3 =P+1
TO TO TO
lim (61)
P—TO
jl=p+1 j2 =P+1 j3 =P+1
TO TO TO
lim
Cj3j3jl j2j2jl 0,
(62)
P—TO
jl=p+1 j2 =P+1 j3 =P+1
TO TO TO
lim S S S Cj2jlj3 j3j2jl — 0, (63)
P—TO
jl=p+1 j2 =P+1 j3 =P+1
TO TO TO
lim
Cj3jlj2 j3j2jl 0,
(64)
P—TO
jl=p+1 j2 =P+1 j3 =P+1
TO TO TO
lim S S S Cj2j3jlj3j2jl — 0, (65)
P—>-TO
jl=P+1 j2 =P+1 j3 =P+1
TO TO TO
lim S S S Cj3jlj3j2j2jl — 0 (66)
P—TO
jl=P+1 j2 =P+1 j3 =P+1
TO TO TO
lim
Cj2j3j3 jlj2jl 0 (67)
P—TO
jl=P+1 j2 =P+1 j3 =P+1
Consider in detail the case of Legendre polynomials (the case of trigonometric functions is considered in complete analogy).
First, we prove the following equality for the Fourier coefficients for the case ),... ) = 1
Cj6j5j4j3j2jl + Cjlj2j3j4j5j6 — Cj6 Cj5j4j3j2jl — Cj5j6 Cj4j3j2jl +
+ Cj4j5j6Cj3j2jl — Cj3j4j5j6Cj2jl + Cj2j3j4j5j6Cjl ' (68)
Using the integration order replacement, we have
C......—
Cj6 j5j4j3j2jl
T t6 t2
— J j (to) y j (t5)... J j (ti)dti • • • dt5dt6 —
t t t T T t5 t2
— J j M J j (t5) y j (t4) ...J j (ti)dti... dt4dt5dto-
t t t t T T t5 t2
- y j (to^ j (t5^ j (t4) ...J j (ti)dti... dÎ4dÎ5dÎ6 — t t6 t t
— C C.....—
Cj6 Cj5j4j3j2jl
T T T t4 t2
- y j (to^ j (t5^ j (t4) y j (t3) ...J j (ti)dti... dt3dt4dt5dto+
t t6 t t t T T T t4 t2
+ y j (to^ 0j5 (t5) y j (t4^ j (t3) ...J j (ti)dti... dt3dt4dt5dto —
t t6 t5 t t
— C C.....—
Cj6 Cj5j4j3j2jl
T T
-J j (t6) y j (t5)dt5dt6 Cj4j3j2jl + t t6
T T T t4 t2
+ J j Mj j (t5^ j (t4) y j (t3) ...J j (ti)dti... dt3dt4dt5dto —
t t6 t5 t t
= nj6 nj5j4j3j2j1 nj5j6 nj4j3j2j1 +
T T T t4 Î2
+ J j (to) ^ j (ts^ j j (t3) -J j (t1)dt1... dt3dt4dt5dt6 =
t t6 ^5 t t
= nj6 nj5j4j3j2j1 — nj5j6 nj4j3j2j1 + nj4j5j6 ^3.72.71 — nj3j4j5j6 ^2.71 + ^2.73.74.75.76 Cj1 —
TT T
- J j M J j (ts)... J j (t1)dt1... dtsdto =
t t6 t2
= nj6 nj5j4j3j2j1 — nj5j6 nj4j3j2j1 + ^4.75.76 ^3.72.71 —
— nj3j4j5j6 nj2j1 + nj2j3j4j5j6 Cj1 — .73.74.75.76 ' (69)
The equality (69) completes the proof of the relation ((
Under the Conditions 1 and 2 of Theorem 3 the following equalities are fulfilled [14] (pp. 108-112), [13] (pp. 485-489)
} ^ Cjk ...ji+ijiji-i ...js+ijijs-i...ji = 0 (1 — 1 — s + 1), (70) ji =0
1 œ
^ ^^jk-Jl+ljljljl-2-jl ^jk-jl + rjljljl-2—jll (^i")
ji =0 MO ji =P+1
where Cjfc...jl is defined by (0) and (5).
Obviously, the equality (70) is equivalent to the following relation
P TO
^ ^ Cjfc...jl+ljljl-l...js+ljijs-l...jl — ~ ^ ^ Cjfc...jl+ljljl-l...js+ljijs-l...jl • (72) ji =0 ji =P+1
Let us consider (53). From (72) we obtain
TO TO TO P P P
Cj3j2jlj3j2jl — ~ ^^ yi Cj3j2jlj3j2jl • (73)
jl=P+1 j2 =P+1 j3 =P+1 jl=0 j2 =0 j3=0
P
Applying (68), we get
p p p
Cj3j2jlj3j2j'l + ^ ^ Cj'lj2j3j'lj2j3 — 2 ^ ^ Cj3j2j'lj3j2j'l
j'l ,j2,j3=0 jl,j2,j3 =0 j'l,j2,j3=0
P (
y^ ( Cj3C
;„ n V
— r1 r1
CC j jo CC j
j2jlj3j2jl j2j3 jlj3j2jl jl j2j3 j3j2jl
j'l,j2,j3 =0
j j j C j j 1 + C
j3jlj2j3 j2 jl j2 j3jlj2j3 jl '
Cj
Note that
T-t
T t
Cj2jl ^y j (t^ j — tt
lM2ji + l)(2ji + 3) ifJ2 = Jl + 1, Ji = 0,1,2,..
I/V^i/T7! if ¿2 = ¿1-1, ii = 1,2,
0
T
if ji — j2 — 0 otherwise
V^i if il = 0
Cjl — J j (t)dT — <
t [ 0 if ji — 0
Moreover, the generalized Parseval equality gives
p
lim / v Cj'lj2j3 Cj3j2j'l j'l,j2,j3=0
T t3 t2
plim^ E J j(t3) j j(t2) j j(ti)dtidt2dt3x
jl,j2,j3=0 t t t
(74)
(75)
(76)
1
p
T t3 t2
j(t3^ j (t2) y j (t1)dt1 dt2dt3 —
t t t
P T T T
— P——TO E /^j'3 («/ j (t2) y j (t1 )dt1dt2dt3 X
jl,j2,j3 =0 t t3 t2
T t3 t2
j (*)/ j (t2^ j (t1)dt1 dt2dt3 — ttt
P » 3
— lim E / 1{t3<t2<tl^ n ^j'i (ti)dt1dt2dt3 X
jl,j2,j3=0 [t T]3 1=1
3
x / 1{tl<t2<t3} fa* (ti)dt1dt2dt3 —
[t,T ]3 1=1
— I 1{t3<t2<tl}1{tl<t2<t3} dt1dt2dt3 — 0. (77)
[t,T ]3
Using (E2HE3), (76) and (77), we get
TO TO TO P
lim / V / V /v Cj3j2jlj3j2jl — lim / v Cj3j2jlj3j2jl — P—TO z-' z-' z-' P—TO z-'
jl=P+1 j2=P+1 j3=P+1 jl ,j2,j3=0
1P
... _ C- . C- • • • —
j2 jl j3 j2 jl Cj2j3 Cjlj3j2jl
jl,j2,j3=0
— Cj3j'lj2j3 Cj2 jl + Cj2j3j'lj2j3 Cjl J —
P /
lim E Cj3 Cj
p—TO
CC
._. , y j2 jl j3 j2 jl Cj3jl j2 j3 Cj2jl
p—TO
j'l,j2 ,j3=0
lim / J Cj2jl0j2ji ~~ lim /J ( './:■,/./:./:■/ './:./ =
p—TO p—TO
jl j2=0 j'l ,j2,j3=0
P P TO
VTM lim V C II+ lim V V <" !/:,/ ./:./:/ 7:./ " (78)
P—TO z-' P—TO z-' z-'
jl,j2=0 jl,j2=0 j3=P+1
By analogy with the proof of equality (130) in [14] (see the proof of Theorem 12 [14]) or equality (2.722) in [13] (see the proof of Theorem 2.33 [13]) we
obtain
p TOTO
liP Y Cj2j10j2 j1 = Y Y Cj2j10j2j1 = 0, (79)
P—TO z-' P—TO z-' z-'
j1,j2=0 j'1=P+1 j2=P+1
where we used the following representation
Cj2j'10j2j'1 = T ¿5 ¿4 ¿3 ¿2
=-jJ== J (f)j2{h) J faiiU) J J (f)j2{to) J ^{t^dtidtodUdtdh = i i i i t T ¿5 ¿4 ¿2 ¿4
= -j==t J (Pnih) J (pjA+i) J (pj2{h) J (f)n{ti)dti J dt3dt2dUdt5 = ¿ ¿ ¿ ¿ ¿2 T ¿5 ¿4 ¿2
= ^—- j Oj..(I:,) J (¡>.h{U){U - t) J <f>j2{t2) J (f)j1{ti)dtidt2dt4dt5+ ¿ ¿ ¿ ¿
T ¿5 ¿4 ¿2
+ ^—-J (pj2{h) J (pjA+i) J (f>j2{t2){t-t2) J (f)n{ti)dtidt2dudt5 = ¿ ¿ ¿ ¿
dpf -
= Cj2 j1 j2 j1 + Cj2 j1j2 j1 . Further, we have (see (75))
P TO TO
lim Cj'3jij'2j'3Cj2ji = lim ^00j3+
?^to z—' z—' p^TO z—' V
■ji j2 =0 j3 =p+1 j3=p+1
p p-1 \
+ y^ Cji-1,jiCj3ji,ji-1,j3 + E Cji+1,jiCj3ji,ji+1,j3 + C10Cj301j3 J . (80) ■1=1 j1=1 '
Observe that
K
IQi-iJ + IQi+iJ <7- = (81)
|Cj300j3 1 + |Cj3 ii,ji-1,j31 + |Cj3ji,ji+1,j31 + |Cj301j31 <
C/3>P + 1), (82)
j3
where constants K, Ki do not depend on j, j3.
The estimate (81) follows from (75). At the same time, the estimate (82) can be obtained using the following reasoning. First note that the integration order replacement gives
T ¿4 is ¿2
Cjsjij2js = J j j ^ j y j (ti )dtidt2dt3dt4 =
t t t t
T ts / t2 \ / T
= f j to) J j (t2 ) y fas (ti )dti I dtj y fas (t4)dt4 ) dt3. (83)
t t V t /Vts
Further,
= j — 0,1,2,..., (84)
l^'l ^ 7—57-Ü74- (85)
where Pj (x) is the Legendre polynomial. From (34) and (84) we get
(1-^))J
where x £ (t,T), z(x) is defined by (32), constant C does not depend on j, j £ N. Applying the estimates (35), (36), and (85) to (83) gives the estimate (82).
Using (80), (81), and (82), we obtain
p to jl,j2=0 j3=P+1
£ ^P + E1' <
j3=P+1 j1 = 1 j1
CO / p
P \ 1
if p —y oo, where constant K is independent of p. Thus, the equality (53) is proved (see (78), (79), (86)).
The relation (54) is proved in complete analogy with the proof of equality
(53). For (54) we have (see (68))
p p \ p
Cj1jsj2jsj2j1
/ P P \ P
lim / v Cjlj3j2j3j2jl + / , Cjlj2j3j2j3jl I = 2 lim / v Cjlj3j2j3j2
P^œ \ z—* z—* j p^œ z—*
P /
lim E Cjl Cj
.... _ C • C • • + C■ ■ ■ C■ ■ ■ —
,_- , „- yj3j2j3j2jl Cj3jl Cj2j3j2jl + Cj2j3jl Cj3j2jl
P^œ ^—' 1 jl j2 j3=0
— Cj3j2j3jl Cj2jl + Cj2j3j2j3jl Cjl J =
PP
CC
j2,j3=0 jl,j2,j3 =0
P
2 Hm / v Cj2j1 Cj3j2j3j1 •
p^œ z—*
j1 J2,j3=0
To estimate the Fourier coefficient Cjjjj, we use the following (see the proof of (53) for more details)
T ¿4 t3 ¿2
Cj3j2j3ji = J j (^y j ^ j y j (t1 )dt1dt2dt3dt4 = t t t t
T t4 t3 t3
= / j(t4) y j (t3)J j (ti)y j(t2)dt2dtidt3dt4 =
t t t t1 T t4 / t3 \ t3
= / j (t4 )J j (t3M y j (¿2)^2) y^ji (ti)dti dMt4-t t t t T t4 t3 / ti
-J j (t4) y j (t3 ) J j (ti) ( y j (¿2)^2 ) dtidt3dt4 =
t t t t
T / t3 \ t3 / T
= J ■ (t3M J ■ (¿2)^2 I J ■ (t1 J ■ (¿4)^4 ) dt3-
t \t / t V3
T t3 / ti \ / T
— ^ j (t3) J ■ (¿1) ( J ■ (¿2)^2 I dtM J ■ (¿4)^4 ) dt3. t t t t3
Let us prove (55). From (72) we obtain
TO TO TO P P P
Cj3j2j3j1j2j1 •
E E E -
(87)
ji=p+1 j2=p+1 j3 =p+1 ji=0 j2=0 j3=0
Applying (68) and (87), we get (we replaced j3 by j4)
Cj4j2j4j1j2ji + ^ ^ Cjij2j1 j4j2j4 = 2 ^ ^ C ■ 1 ,j2,j4=0 j1,j2,j4 =0 j1,j2,j4=0
p
Cj4j2j4j1j2 ji
p ^
;„ n V
p
_ C ■ C ■ ■ + C- ■ ■ C- ■ ■ —
yj2j4j1j2j1 Cj2j4 Cj4j1 j2j1 + Cj4j2 j4 Cj1j2j1
j1,j2,j4 =0
Cj1j4j2j4 Cj2 ji + Cj2jij4j2j4 Cji J =
p / 2 E (Cj
p
j j1j4j2j4 Cji — Cj1j4 j2j4 Cj2j1 1 +
j1j2 ,j4 =0
p
+ y ^ Cj4j2j4 Cj1j2j1 • (88)
j1,j2,j4 =0
Further, we have (see (72))
p pip lim / v Cj4j2j4Cj1j2ji = lim / v I / , Cj1j2j1
p—TO z-' p—>-TO z-' \ z-'
j1,j2,j4=0 j2=0 \ji=0
p TO 2
=p—toE E j =0' (89)
j2=0 \ji =p+1 /
where we applied the equality (103) from [14], p. 134 (also see [13], p. 510).
p
p
Furthermore, by analogy with the proof of (53), we have
liniy3 ( Cj2j1j4j2 j4 Cj1 — Cj1j4j2j4 Cj2 jl ) = 0 (90)
p ?œ \ J
j1,j2 ,j4=0
To estimate the Fourier coefficient Cj1j4j2j4 in (90), we use the following (see
the proof of (53) for more details)
T t4 t3 / t2
Cj1j4j2j4 = J j (U) j j (t3^ j (t2M J j (£i )d£i ) d£2d£3 d£4 = t t t t
T t4 / t2 \ t4
= J j (£4) y j(Î2M y^j4(£i)d£i I y^j4(t3)dt3dt2dt4 = t t t t2 T / t4 \ t4 / t2
= y j (£4M y j(£3)^3 I y j (£2 M y j(ti)dt^ d£2d£4-t t t t T t4 / t2 \ / t2
-j j (£4 ) y j (Î2M y j (£3)^3 II y j (£i)d£i | d£2 ^£4.
t t t t
The relations (I87)-(I901) complete the proof of equality (155). Let us prove (56). Using (72), we get
œ œ œ p p œ
53 53 Cj1j2j3j3j2j1 = 53 Cj1j2j3j3j2j1 • (91)
j1=p+i j2=p+i j3=p+i j1=0 j2=0 j3=p+i
Applying (68) and (91), we obtain
pœ
2 53 Cj1j2j3j3j2j1 =
j1,j2 =0 j3=p+i
p œ x
53 53 ( Cj1 Cj2j3j3j2j1 — Cj2j1 Cj3j3j2j1 + (Cj3j2j1) —
. .;„—n —n_Li V
jl ,j2 =0 j3=P+1
Cj3j3j2jl Cj2 jl + Cj2j3j3j2jl Cjl ) _
pTO
2 £ £ faC
j2j3j3j2 ji Cj2 ji Cj3 j3j2 ji J +
ji j2=0 j3=p+1
pTO
+ E E (Cj3j2ji)2 • (92)
ji j2=0 j3=p+1
Using the estimate (1.217) from [13] (p. 158), we get
pTO
lim E E (Cji)2 = 0. (93)
—TO
ji j2=0 j3=p+1
By analogy with the proof of (53), we have
p TO , X
Hm E y v (Cji Cj2j3j3j2ji — Cj2j1 Cj3j3j2j1 j = 0, (94)
j1,j2=0 j3=p+1
where we applied the equality (131) in [14], p. 139 (also see the equality (2.723)
in
13], p. 516). To estimate the Fourier coefficient Cjjjj in (94), we used the
following (see the proof of (53) for more details)
T t4 t3 t2
jj2ji = J j (t4^ j w/ j (t2^ fai (t1 )dt1dt2dt3dt4 = tttt
T T T T
= J j j (t2)J j (*)/ j fa^MM^dt1 =
t ti t2 t3
T T / T \ 2
= \J fe(ii) J <f>h(t2) J o,JI,)dl, dtodh. (95)
t ti \t2 /
Combining the equalities (I9T1)-^I94), we obtain (156).
Let us prove (57) (we replace j2 by j4 and j3 by j2 in (57)). As noted in (p. 126), the sequential order of the series
TOTOTO
EE £
ji=p+1 j2=p+1 j4=p+1
is not important. This follows directly from the formulas (71) and (72). Applying the mentioned property and (72), we get
to to to p to to
E E E
Cj1j4j4j2j2j1 =
-E E E
Cj1j4j4j2 j2j1 .
(96)
j1=p+1 j2=p+1 j4=p+1 j1=0 j2=p+1 j4=p+1
Observe that (see the above reasoning)
to to to to
E E
Cj1j4j4j2j2j1 =
E E
Cj1j4j4j2j2 j1 .
(97)
j2 =p+1 j4 =p+1 j4 =p+1 j2 =p+1
Using (68) and (97), we obtain
p TO TO / \ p toto
53 53 53 (Cj1j4j4j2j2j1 + Cj1j2j2j4j4j1) =2 53 53 53
j1=0 j2=p+1 j4 =p+ 1 j1=0 j2=p+1 j4=p+1
p TO TO /
= y ^ 53 ( Cj1 Cj4j4j2j2j1 — Cj4j1 Cj4j2j2j1 + Cj4j4j1 Cj2j2j1 —
j1=0 j2=p+1 j4=p+1
— Cj2j4j4j1 Cj2j1 + Cj2j2j4j4j1 =
p TO TO / \
= ^ ^ ^3 53 I Cj1 Cj4j4j2j2j1 — Cj4j1 Cj4j2j2j1 — Cj2j4j4j1 Cj2j1 + Cj2j2j4j4j1 Cj1 j +
j1=0 j2=p+1 j4=p+1
p / TO \ 2
+ E E Cj2j2j1 . (98)
j1=0 \j2=p+1 /
c......
Cjlj4j4j2j2jl
The equality
P / to
Ji-E E j») =0 (99)
P^to
jl=0 \j2=P+1
follows from the relation (102) in [14], p. 134 (also see [13], p. 510)
By analogy with the proof of equality (53) we obtain
p TOTO
p to to /
¿»to£ E E (Cl c
—n .;„—nil ___nil \
._. ._. ._. , _ yj4j4j2j2jl Cj4jl Cj4j2j2jl
j^to z—' z—' z—' 1
jl=0 j2=P+1 j4=P+1
Cj2j4j4j'l Cj2j'l + Cj2j2j4j4j'l Cjl ) — 0, (100)
where we applied the equality (132) in [14], p. 140 (also see the equality (2.724)
in
13], p. 516). To estimate the Fourier coefficient Cj2j4j4jl in (100), we used
the following (see the proof of (53) for more details)
T ¿4 ¿3 ¿2
Cj2j4j4j1 = J j fa^ j W)J j fa) J j (t1 )dt1dt2dt3dt4 = ¿¿¿¿
T ¿4 ¿4 ¿4
= J j (^4^ j (t1 )J j (^2^ j (^MM^ dt4 = ¿ ¿ ¿1 ¿2
T ¿4 / ¿4 2
= 2 / fe^) / ( I (pjA^dto ) ofaofa =
T / t4 \ 2 t4
t t tl
= \\ ^hfa) I I (f>jA{t2)dt2 I I (f)j1{ti)dtidt4+ t t t T t4 / tl N 2
J <i>hfa) J fa (il) ( J (f>jA{t2)dt2 ) dtidt4-
t t t t / t4 \ t4 / tl
-J faj (t4 M y faj (t2)dt^ y faj (ti) ( y fa-4 (t2)dt^ dtidt4.
t t t t
The relation (57) follows from (96), «-(ITO.
Consider (58). Using the integration order replacement, we obtain
C—
Cj3j3j2j2jljl —
T t6 t5 t4 / t3 \ 2
= ^ j OjJIV,) j OjJI:,) J fa2fa) J <f>j2fa) I J <f>h{tl)dti\ dt.3dt4dt.5dt6 = t t t t t T / t3 \ 2 T T T
= ^ y <fe2(*3) i J (f>ji{ti)dti\ J fa2fa) j OjJI5) J <f>j3(t6)dt6dt5dt4dt3 =
t t t3 t4 t5
T /is \ 2 T / T \ 2
= \J <t>hfo) J ^n(ti)dti J <f>h(U) J Oj,( I:,)(!/:, dUdU. (101)
t \t /is \t4 /
Applying the estimates (35), (36), and (85) to (101) gives the following estimate
K
I^hhhkhh I < "2"2 til,h > 0, J> > 0), (102)
j1 j3
where constant K does not depend on j1,j2,j3. Further, we get (see (71))
to to to to to to
Cj3j3j2j2jljl
E E E
Cj3j3j2j2jljl EEEc
j l = J+1 j2 =J+1 j3 =P+1 j l =P+1 j3 =P+1 j2 =P+1
oo oo
1 ^ ^ P to to
9 53 53 ^ '/"■./:■,/: ./: ./ ~~ 53 53 53 ^ './"■./"■./: • (103)
(j2j2)^(') j2=0 jl=J+1 j3=J+1
2
jl=J+1 j3=P+1
where
C......
Cj3j3j2j2jljl
(j2j2)^(')
T ¿6 ¿5 ¿4 ¿2
= J j (h) J j (h) J J j (h) j j (¿OdMMMMio = t t t t t T t6 t5 t2 t5
t t t t t2 T t6 t5 t2
= J j (to)J j (t5)(t5 - t) J j (t2) y j (t1)dt1dt2dt5dto+ t t t t T t6 t5 t2
+ J j (to^ j (t5) y j (t2)(t - t2) J j (t1)dt1 dt2dt5dto =f
t t t t
=f C' + C'' (104)
j3j3jljl ^ j3j3jljl' V-1-^^/
Let us substitute (104) into (103)
TO TO TO 1 TO TO
53 53 53 ( = 2 53 53 ( './:■■./:',/ ./
j 1 =p+1 j2 =p+1 js =p+1 j 1 =p+1 js =p+1
1 TO TO p TO TO
+ 2 53 ( '/:■■/:■■/ / ~ T3 53 53 ^ './"■./:■,/: ./: ./ • (105)
j 1 =p+1 j3 =p+1 j2 =0 j 1 =p+1 j3 =p+1
The relation (132) from [14], p. 140 (also see the equality (2.724) from [13], p. 516) implies that
TO TO TO TO
pl—to £ £ =0 p—TO £ £ Cj =0 (106)
j1=p+1 js=p+1 j1=p+1 js=p+1
From the estimate (102) we get
p œ œ
^3 ^3 53 Cj3j3j2j2j1j1
j2=0 jl=P+1 j3=P+1
œœ
<*(P + D £ 72 £
jl=P+1 j1 j3=P+1 j3
, 2 œ x 2
I /;;:) <^-0 (107)
p
if p ^ œ, where constant K is independent of p.
The relations (105) (107) complete the proof of (158).
Let us prove (59). Using the integration order replacement, we get
C......=
Cj2 j3j3j2j1j1
T t6 t5 t4 / t3
= \J faW J (Pj-Ah) J (f>j3{U) f (f)j2{h) ( [ <t>h{ti)dti ) dhdudhdte = ttt T / t3 \2T T T
= 2 / ^'2(^3) I I (f>h{ti)dti ) I (f)j3{t4) I Oj.JI:,) I (f)j2{te)dt6dt5dudt3 =
t / t3 \ 2 t T t5
t \ t / ¿3 ¿4 ¿5
1J 0j2(i3) 17 I
t3 t5 t3
2 I ^2(^3) I j Oj (l\)(ll\ I j o/;. (/.-,) j (f)j2{t6)dte j (f)j3{t4)dt4dt5dt3 =
T
t3
T
T
t5
= ö I Öhfo) I I I J h-M) J OjAloVllv,
t3 t5
xdt5dt3—
(t4)dt4 x
T
t3
t3
T
T
-Ö / 3) /
2
j (t4)dt4 / j (t5M / j (t6)dt6 x
t t3
x dt5 dt3.
t5
(108)
Applying (72) and (71), we obtain
00 00 00
00 00 00
El El El Cj2j3j3j2j1ji = Cj2j3j3j2j1j1
j 1 =p+1 j2 =p+1 j3 =p+1 ji =p+1 j3 =p+1 j2 =p+1
p TO TO
pTO
2 ^ y ^hkhhhh
j2=0 ji=p+1
Cj2j3j3j2jiji =
j2 =0 ji =p+1 j3 =p+1
p p TO
— ^^ El Cj2j3j3j2jiji
(j3j3M0 j2=0 j3=0 ji=p+1
1 p TO
j2=0 ji=p+1 pTO
TO
53 C0000jiji
(j3j3W0 j1=p+1
pTO
E E
C0j3j30jiji
—
Cj200j2jiji
j3 = 1 ji=p+1 j2 = 1 ji=p+1
p p TO
53 53 53 Cj2j3j3j2j1 j1 •
j2 = 1 j3 = 1 ji=p+1
(109)
The equality
1
pTO
9 53 53 ^nhhhhh j2=0 ji=p+1
=0
(110)
follows from the inequality similar to (158) in [14], p. 149 (also see the relation (2.750) in [13], p. 526), where we used the following representation
C
j2j3j3j2jiji
2
2
T ¿6 ¿4 ¿3 ¿2
= y j (to^ y j (*)/ j (^2^ y j (fadt^^dMto = i i i i t
T ¿6 ¿3 ¿2 ¿6
= j j Mj j (t3)J j (t2^ j (t1)dt1dt^y dt4dt3dto =
¿ ¿ ¿ ¿ ¿3
T ¿6 ¿3 ¿2
+ y j (to)(t6 - t) y 0j2 (t3)J j (t2^ j (t1)dt1dt2dt3dto+ i i i t T ¿6 ¿3 ¿2
+ j j 06) / 0j2 (t3)(t - *)/ j (t2^ j (t1)dt1 dt2dt3dto = ¿ ¿ ¿ ¿
def c* i C** (111)
= C j2 j2 j1 j1 + C j2 j2 j1 j1 • (111)
The following estimate
s
I )0j (T)dT
c
¿
is proved in [14], p. 137 (also see [13], p. 513), where £ is an arbitrary small positive real number, far) is a continuously differentiable function at the interval [t,T], j E N, s E (t,T), z(s) is defined by (32), constant C does not depend on
Applying the estimates (35), (36), (85), and (112) (e = 1/2) to (108) gives the following estimates
K
I ^hhhhhh I — ,0 . .3/4 O'l'
j2,j3 > 0), (113)
jlj2j3
K
IQ2oo^ml <72- 0*i,i2 > 0), (114)
j1 j2
K
<" "./:,/,"./ ./ < -7- (./1 •./':: > 0), (115)
J1J3
K
|Coooo,m|<-2 (Jl>0). (116)
j1
Using the estimate (113), we have
p p œ
Cj2j3j3j2j1 j1
Cj
j2 = 1 j3 = 1 jl=P+1
œ 1 P 1 P 1
E ?£-£—<
jl =P+H1 j2 = 1 2 j3 = 1 j3
P
^ . dx I i dx
X2 \ / £
P
1 ■ / 4^1 —>■ 0
X3/4
P
1
P'
3/4
(117)
if p — to, where constants K, K1 do not depend on p. Similarly we get (see (114)-(116))
p TO
E C0000jljl
jl=P+1
+
Pœ
E E
C0j3j30jljl
j3 = 1 jl=P+1
+
E E
Cj200j2jljl
j2 = 1 jl=P+1
^ 0 (118)
if p ^ œ.
The relations (109), (110), (117), (118) prove (59). Consider (60). Using the integration order replacement, we get
C......—
Cj3j2j3j2jljl
T ¿6 ¿5 ¿4
t3
2
j(to) / j(tôW jM / j(t3M / j(¿1)^1 dt3dt4dt5dt6 —
t t t t
T
¿3
T T
T
j(t3M / j(Î1)dtU / j(t4M j(t5M j(to)dt6dt5dt4dt3 —
T
¿3
¿3 ¿4
T T
¿5
¿5
j(t3M / j(Î1)dtH / j(t5) / j(to)dto / j(t4)dt4dt5dt3 —
T
¿3
¿3 ¿5
T / ¿5
¿3
T
= ö / / / fete) / o;,(/,W/, / Oj.,{t{\)dtx
¿3 ¿
xdt5dt3-
¿5
2
1
2
1
2
2
1
2
2
T / t3 \ 2 / ¿3 \ T / T
\ J fete) J fete№ J Oj.Jh )(ll I J (f)j2{h) J (pj3{h)dt6 ) x
t / \t / ¿3 \Î5
x dt5dt3. (119)
Applying (72), we obtain
œ œ œ œ œ œ
1 C
j3j2j3j2jljl
53 53 Cj3j2j3j2j'lj'l— 53 53 53 C: j l =p+1 j2 =p+i j3 =p+i j l =p+i j3 =p+i j2 =p+i
p œ œ
EE E Cj3j2j3j2jljl • (120)
j2=0 jl=p+1 j3=P+1
Further proof of the equality (60) is based on the relations (119), (120) and is similar to the proof of the formula (59).
Let us prove (61). Applying the integration order replacement, we obtain
C......=
Cj3 j3j2jlj2jl
T ¿6 ¿5 ¿4 ¿3 ¿2
= J fe (te) y fe (t5) y fe (t^y fe (t3^ fe (^2^ y fe (ti)dtidÎ2dÎ3dÎ4dt5dt6 = î î î î î î
TTTTTT
= J fe (ti^ fe (t2) y fe(t3^ fe (t4^ fe(t5^ fe(te)dtedt5dt4dt3dt2dti =
t ¿l ¿2 ¿3 ¿4 ¿5
T T T T / T \ 2
= ^ y fe(*l) y fete) J (fijiih) J <f>j2{U) ( J Oj.JI:,)(!/:, I =
¿ ¿l ¿2 ¿3 ¿4
T / T \ 2 ¿4 ¿3 ¿2
= 7>J fete) (y Oj.Jh,)(lh, I y fa(Î3) y fete) y fafal^MMM^ =
¿ ¿4 ¿ ¿ ¿
T / T \ 2 ¿4 ¿2 ¿4
= \ J fe M (y fe te )di5 J y fe te ) y fe (il y fe {h)dt3dt2dt4 =
¿ ¿4 ¿ ¿ ¿2
T / T \ 2 / t4 \ t4 / t2
= U fete) J OjJh. Vlh, J <:>r(l,)dl, J <f>h(t2) J Mti)dti I X
t t4 t t t
xd£2d£4-
t / t \ 2 t4 / t2 \ 2
J fe(£4) J OjJh^lh, J (f)j2{to) J fete)d*i x
t t4 t t
x d£2d£4. (121)
Using (72), we get
œ œ œ œ œ œ
53 53 Cj3j3j2j1j2j1 = 53 53 53 Cj3j3j2j1j2j1 = j 1 =p+i j2 =p+i j3 =p+1 j 1 =p+1 j3 =p+1 j2 =p+i
p œ œ
= — ^3 53 Cj3j3j2j1j2j1 • (122)
j2=0 j1=p+1 j3=p+1
Further proof of the equality (61) is based on the relations (121), (122) and is similar to the proof of the relations (59), (60).
Consider (62). Using the integration order replacement, we have
C=
Cj3j3j1j2j2j1 =
T t6 t5 t4 t3 t2
= J fe (£o^ fe (£5^ fe (£4)/ j (£3^ fe (£2^ j (£ 1 )d£ 1 d£2^£3¿£4¿£5d£6 = tttttt TTTTTT
= J fe (£i)y fe (£2^ fe (£3^ fe (£4^ fe (£5^ fe(£o)d£6d£5d£4d£3d£2d£i =
t t1 t2 t3 t4 t5
T T T T T 2
= ^ /fete) /fete) /fe(*3) /fe(i4) I I Oj.,{!:,)(!!| ûMMMil =
T / T \ 2 ¿4 ¿3 ¿2
¿ ¿l ¿2 ¿3 ¿4
¿ ¿4 ¿ ¿ ¿
2 I fete) I I Oj.JI:,)(!/:, I I (t>j2{h) I fete) I (t)J1{ti)dtidtodhdt4 =
T / T \ 2 / t4 \ t4 / t2
T / T \ 2 ¿4 ¿2 ¿4
^ J fete) l J (/.-,)('//.-, I J (f)j2{t2) J J (f)j2{h)dhdt2dt4 =
¿ ¿4 ¿ ¿ ¿2
\j fete) [ I fete№ ) | I o,(i,)di,) I 4>J2{u) | I ^(t^dh ) x
¿4 ¿ ¿ ¿
xdt2dt4-
T / T \ 2 ¿4 / ¿2 \ / ¿2
^///fete) /fete№ I I [o,:{i,)di, | x
¿ ¿4 ¿ ¿
x dt2dt4. (123)
Applying (72) and (71), we obtain
TO TO TO TO TO TO
53 53 Cj3j3j'1j2j2j'1 = — 53 53 53 Cj2j3j'1j2j2j'1 =
j 1 =P+1 j2 =P+1 j3 =P+1 j2 =P+1 j3 =P+1 j 1 =P+1
p TO TO p TO TO
Cj2j3j1j2j2j1
EE E
C
ji=0 j2=p+1 j3=p+1 ji=0 j3=p+1 j2=p+1
pTO
2
ji=0 j3=p+1
p p TO
E E — Chh3i j2 j2j1 • (124)
(j2j2W • ) ^=0 j2=0 j3=p+1
The equality
1 p TO
j'1" 2 S 53 ^hhhhhh
ji=0 j3=p+1
= 0 (125)
(j2j2W • )
follows from the the inequality (158) in [14], p. 149 (also see the inequality (2.750) in [13], p. 526), where we proceed similarly to the proof of equality (110) (see (111)).
The relation
p p TO
p—TO
l——TOEE 53 Cj3j3jij2j2j1 =0 (126)
ji=0 j2=0 j3=p+1
is proved on the basis of (123) and similarly with the proof of (59). The equalities (H2HHH2EI) prove (162).
Let us prove (63). Using (72) and (71), we get
TO TO TO TO p
53 53 53 j1 jsjsj2j1 = y ^ 53 j1 jsjsj2 j1 =
j 1 =p+1 j2 =p+1 j's =p+1 j3 =p+1 j 1 ,j2 =0
p
2
j1,j2=0
p
(127)
fesMO j1,j2,js=0
Using the equality (130) in [14], p. 139 (also see the equality (2.722) in [13], p. 516) we have
1p lim - V C
p—to 2 ' j1 ,j2=0
j2j1jsjsj2j1
= 0, (128)
(j'sjsWO
where we proceed similarly to the proof of equality (110) (see (111)). Further, we will prove the following relation
p
lim / v Cj2j'1js jsj'2 j1 = 0 (129)
p—TO z-'
j1,j2,js =0
using the equality (68). From (68) we have
p 1 p
53 ^hhhhhh 9 53 ( ^hjihhhji ^hhhhhh J
j'1,j'2,j3 =0 j1 ,j2,j's=0
1 p (
2 53
2 j ,j ,j =0
... _ C • C • • + C■ ■ ■ C■ ■ ■ —
j1 jsjsj2 j1 Cj1j2 Cjsjsj2 j1 + Cjsj1j2 Cjsj2j1
j1,j2,js=0
— Cj3j3j'1j'2 Cj2j1 + ^^72 js jsj1 j2 Cj1^ =
53 \ js jsj1 j2 Cj1 — Cj3j3j'1 j2 Cj2j1 j + j1,j2,js=0
1p
+ 2 53 ^ './"■./ ./V^ './.",/:./ • (130)
j1 ,j2,j's=0
The generalized Parseval equality gives (by analogy with (77))
1p
Ö Z-^ ( ^ './:■,/:./ = 0- (131)
p—to 2 z—'
ji ,j2,j3=0
Let us prove the following equality
lim / V ( Cj2j3j3jij2 Cji Cj3j3jij2 Cj2j1 ) = (132)
p—TO z—' y /
ji,j2 ,j3=0
The relation
P
lim /v Cj2j3j3 j1 j2 Cj1 = 0 (133)
P—>-TO z-'
j'1,j'2,j3 =0
is proved by the same methods as in the proof of equality (53) and also using Theorem 12 from [14] (also see Theorem 2.33 from [13]) and (71).
Further, we have (see (71))
P 1 TO
y ^ ^ './:■,/:■,/•./:• = fa './:■,/ ■,/ ./: ~~ ^ ' (134)
2
j3=0
(j3j3W0 j3 =p+1
Moreover,
T t3 t2
C
= I I j (¿2) / j (¿1 )dt1dt2dt3 =
t t t
T t2 T T t2
: J j (¿2^ j dt3dt2 = J(T — ¿2)^1 (¿2) ^ j (¿1 )dM*2 =
t t t2 t t TT TT
J j (¿1 y(T — ^fai (¿2)di2di1 = J j fa) J (T — fafai fafaM^ = t t1 t t2
= J (T — ¿1 )1{t2<ti}0ji(i1)j(¿2)di1di2 = j• (135)
[t,T ]2
Using (134), (135), and the generalized Parseval equality, we obtain
p i p lim / ( '/■,/:■,/ ^ './:./ = o / v
p^to z—» 2 p^to z—»
j1 J2,j3=0 j1,j2=0
p to p to
lim / V /v Cj3j3jlj2 Cj2jl — lim 7 v ), Cj3j3jlj2 Cj2jl • (136) «—VOO ' ' ' * «—vno ' » ' »
p—TO ' ^ ' ^ p—TO
jl,j2=0 j3=P+1 jl,j2=0 j3=P+1
We have (see (95))
T t / t \ 2
<" !/:,/:,/ ./, = \J OjAI\ ) J o,.(/■>) I j (pj3{h)dt3 ) J/W/,. (137) t tl \i2
By analogy with (86) and also using (137), we get
p TO
P^TO
lim >J /J Cj3j3jlj2Cj2jl = 0 (138)
jl ,j2=0 j3=P+1
Combining (136) and (138), we obtain
p
p—TO
lim Cj3j3jlj2 Cj2jl - (139)
)—vto ' *
jl,j2,j3=0
The relation (I32) follows from (133) and (139). From (I3n)-(132) we get (129). The equalities (I27)-(I29) complete the proof of (63).
For the proof of (64)-(67) we will use a new idea. More precisely, we will consider the sums of expressions (64)-(67) with the expressions already studied throughout this proof.
Let us begin from (64). Applying the integration order replacement, we obtain
Cj3j1j2j3j2j1 + Cj3j1j2j3j1 j2 =
T t6 t5 t4 / t3 \ / t3
= j (toW j (t5 ) / j M / j (t3M / j (t2)dt2 / j (t1)dt1 I X
xdt3dt4dt5 dt6 =
T ¿6 ¿5 / ¿3 \ / ¿3 \ ¿5
J j (t6) J j (t5^ j (t3M J j (t2)dt2 J j (t1 )dt1 J j (t4)dt4 x
¿ ¿ ¿ ¿ ¿ ¿3
xdt3dt5 dt6 =
T ¿6 / ¿5 \ ¿5 / ¿3
= J j (t6) J j (t5M J j (t4)dt4 J j (t3 ) I J j (t2 )dt2 ) x ¿ ¿ ¿ ¿ ¿
r; \
x / j(t1)dtM dt3dt5dt6-
T ¿6 ¿5 / ¿3 \ 2 / ¿3
-J j (t6 )J j fa) f j (t3H J j (t2)dtJ J j (t1)dt^ x
i i i \i / \t
xdt3dt5 dt6 =
T / ¿5 \ ¿5 / ¿3
= J j (t5) I J j (t4)dt4 I J j (t3M J j (t2)dt2 ) x
i \i / i \i
r; WT
x / j (t1)dtM dt3 / j (t6)dt^ dt5-
T ¿5 / ¿3 \ 2 / ¿3
- J j ^ J j (t3H J j (t2)dt2 J j (t1)dt1 ) dt3x
i i \i / \t
x (/j| (140)
t5
Using (72), we get
TO TO TO ✓ \
53 53 ( Cj3j1j2j3j2j'1 + Cj3j'1j2j3j'1j2 j = j1 =P+1 j2 =P+1 j3=P+1
P P TO , x
= ^ ^ 53 I Cj3j1j2j3j2j1 + Cj3j'1j2j3j'1 j2 j • (141)
j1=0 j3=0 j2 =P+1 ^
Further, by analogy with the proof of equality (59) and using (140), we obtain
p p œ
p^œ
^ V ^ y y y jlj2j3j2j'l + Cj3j'lj2j3j'lj2 j — (142)
j'l =0 j3=0 j2=p+i
From (141) and (142) we get
oo oo oo
1im ^^ y ^ 'y ^ ( Cj3j'lj2j3j2j'l + Cj3j'lj2j3j'lj2 ) — 0- (143)
p^œ z—* z—* z—* \ j
j l =p+i j2 =p+i j3 =p+i
Moreover (see (53)),
oo oo oo
1im E E E
Cj3j'lj2 j3j'lj2 (144)
p^œ
j l =p+i j2 =p+i j3 =p+i
Combining (143) and (144), we have
oo oo oo
pim y^ y^ y Cj3j'lj2j3j2j'l — 0-
p^œ
j l =p+i j2 =p+i j3 =p+i
The equality (64) is proved.
Consider (65). Using the integration order rep lacement, we have
Cj2j3j'lj3j2j'l + Cj2j3j'lj3j'l j2 —
T t6 t5 t4 / t3 \ / t3
— J j M y j (t5) J j M y j (t3M y j (t2)dt2 y 0jl (ti)dti ) X t t t t \t ) \t
xdt3dt4dt5 —
T t6 t5 / t3 \ / t3 \ t5
— J j y j (t5^ j (t3M y j (t2)dt2l I y j (ti )dtil y j (t4)dt4 X t t t t t t3
xdt3dt5 —
T t6 / t5 \ t5 / t3 \
— J j (t6) y j (t5H y j (t4)dt4 y j (t3 H y 0j2 (t2)dt2 X
t t t t t
is
x j J j (ti)dt^ dt3dt5
T t6 t5 /is \ / is ' 2
- J j (t6) J j ^ J j (t3M J j (t2)dt2 J j (t1)dt1 ) x i i i i i
xdt3dt5dt6 =
T / i5 \ i5 /is
= J j (t5 W J j (t4 )dt4 J j (t3M J j (t2)dt2 ) x i i i i
x (/ (ti)dti^ ^t/ j te)dto ) dt5-
T t5 /is \ / is X 2
- J j № J j ten J j (t2)dt2 J j (t1)dt1 | dt3x i i i i
x J j te)dto ) dt5. (145)
i5
Using (72), we obtain
oo oo oo
53 53 ( Cj2jsjljsj2 j1 + Cj2jsj1jsj1 J2 j = jl=p+1 j2=P+1 js =P+1
P TO TO / \
= ^ ^ ^3 53 ( Cj2jsj1jsj2j1 + Cj2jsj1jsj1j2 j • (146)
js =0 j1=P+1 j2=P+1
By analogy with the proof of (59) and applying (145), we get
P TO TO / \
Hm): ): 53 (Cj2j'sj1js j2j1 + Cj2jsj1jsj1 j2 j =0. (147)
js=0 j1=P+1 j2 =P+1
From (146) and (147) we have
oo oo oo
pli|"TO ^3 y V 53 ( Cj2jsj1jsj2j1 + Cj2jsj1jsj1j2 j = (148)
j1=p+1 j2=P+1 js=P+1
Moreover (see (54)),
oo oo oo
1im E E E Cj2j3jlj3jlj2 — 0- (149)
p^œ z—* z—* z—*
jl=p+i j2=p+i j3=p+i
Combining (148) and (149), we finally obtain
oo oo oo
„1im- E E E Cj2j3j'lj3j2j'l — 0-
p^œ
jl=p+1 j2=P+1 j3=P+1
The equality (65) is proved.
Now consider (66). Using the integration order replacement, we obtain
Cj3jlj3j2j2jl + Cj3jlj3j2jl j2 =
T ¿6 ¿5 ¿4 / ¿3 \ / ¿3 \
= J j M J j (t5) J j M y j (t3H y j (t2)dt2 I I y (t1)dt1 I x ¿ ¿ ¿ ¿ ¿ ¿
xdt3dt4dt5 dt6 =
T ¿6 ¿5 / ¿3 \ / ¿3 \ ¿5
= y j (t6^ j y j (t3H y j (t2)dt2 y j (t1)dt 1 y j (t4)dt4 x
¿ ¿ ¿ ¿ ¿ ¿3
xdt3dt5 dt6 =
T ¿6 / ¿5 \ ¿5 / ¿3
= y j M y j (t5H y j (t4)dt4 y j (t3H y j (t2)dt2 | x ¿ ¿ ¿ ¿ ¿
T/ \
x / (t1)dtfa dt3dt5dt6-
T ¿6 ¿5 / ¿3 \ / ¿3
-f j fa) J fa (t5)J j (t3) y j 02^2 y j (t1)dt^ x ¿ ¿ ¿ ¿ ¿
H
x / fa3 (t4)dt4 J dt3dt5dt6 =
T / i5 \ i5 /is
= J j (t5H J j (t4)dt4 I J j (t3M J j (t2)dt2 ) x i i i i
x (/ j (t1)dt1 ) ^i/ j(te)dt^ ^5-
T i5 /is \ / is
-J j / j (¿3) J j (¿2)^2 J j (t1)dt^ x
i i \i / \t x (/j* (/j
Applying (72) and (71), we obtain
TO TO TO / \
53 53 ( Cjsj'1jsj2j2j1 + Cjsj1jsj2j1j2 j = j1 =p+1 j2 =p+1 js=P+1
P TO TO / \
53 53 53 ( Cjsj1 jsj2j2j1 + Cjsj1jsj2j1j2 j j1 =0 js=P+1 j2=P+1
p p TO
53 53 53 ( Cj3jlj3j2j2jl + Cj3jlj3j2jlj2
¿1=0 ¿2=0 j3=p+1
1 p TO
2 53 53 ^ './."■./'./"■./:'./:./'
¿1=0 ¿3 =p+1
The equality
1 p TO
j'1" 2 53 53 ^hhhhhh
¿1=0 ¿3=p+1
(151)
= 0 (152)
follows from the equality (130) in [14], p. 139 (also see the equality (2.722) in [13], p. 516), where we proceed similarly to the proof of equality (110) (see
(111)).
By analogy with the proof of (59) and applying (150), we get
P P TO , N
P—m>:>: 5-y Cj3j1 j3j2j2j'1 + Cj3j'1j3j2j'1j2 j = 0 (153)
j1 =0 j2=0 j3=P+1
From (EH-flES) we have
TO TO TO f \
TO ^ > ^ ( Cj3j1j3j2j2j'1 + Cj3j'1j3j2j'1j2 j = 0 (154)
j1=P+1 j2=P+1 j3=P+1
Moreover (see (55)),
oo oo oo
E 53 Cj3j'lj3 j'2j'lj2 — 0- (155)
p^œ
j l =p+i j2 =p+i j3 =p+i
Combining (154) and (155), we finally obtain
oo oo oo
plimœ 53 53 53 C ^^^l =o.
j l =p+1 j2 =p+1 j3 =p+1
The equality (66) is proved.
Finally consider (67). Using the integration order replacement, we have
Cj2j3j3jl j2jl + Cj2j3j3jljl j2 =
T ¿6 ¿5 ¿4 / ¿3 \ / ¿3 \
= J j W J j (t5) J j ^ J j (t3M J j (t2)dt2 I I J j (t1)dt H x ¿ ¿ ¿ ¿ ¿ ¿
xdt3dt4dt5 dt6 =
T ¿6 ¿5 / ¿3 \ / ¿3 \ ¿5
= J j M y j (t5^ j (t3M y j (t2)dt2 y j (t1)dt H y j (t4)dt4 x
¿ ¿ ¿ ¿ ¿ ¿3
xdt3dt5 dt6 =
T ¿6 / ¿5 \ ¿5 / ¿3
= J j fa) f j 05) y j 04)dt4 y j 03 H y fa (t2)dt2 ) x ¿ ¿ ¿ ¿ ¿
t3
x ( J (t1)dt^ dt3dt5dta-
T t6 t5 / t3 \ / t3
-f fe(¿a)/ ^¿3(t5) J fa(¿3) J fa(¿2)^2 J fa(ti)dti I x t t t \t / \t
("f \
x / ^¿3 (t4)dt4 dt3dt5dta =
T / t5 \ t5 / t3
= J ^¿3 (t5 W J fa (t4 )dt4 J fa (t3M J fa (t2)dt2 | X t t t t
x (/ ^¿i (ti)dti^ dt3 ^¿2 (ta)dt^ ^5-
T t5 / t3 \ / t3
-f fa (t5^ fe (¿3) J ^¿2 (¿2^2 J (tl)dti ) X t t t t
x ^ J ^¿3 (£4)^4^ dt3 ^ J (ta)dta^ dt5. (156)
Using (72) and (71), we get
TOTOTO
53 53 ( C¿2¿3¿3¿1¿2¿1 + ^'¿^¿^¿^¿^¿^¿2 j = ¿1 =p+i ¿2 =p+i ¿3=p+i
1 TO TO
2 53 53 ( ^ './:'./."■./."■./'./:./' ¿1=p+i ¿2 =p+i \
p TO TO
¿373 WO
+ C¿2¿3¿3¿1¿1¿2
¿3 ¿3W')
53 53 53 I C¿2¿3¿зjl¿2jl + C¿2¿3¿3¿1¿1¿2 )
¿3=0 ¿l=p+i ¿2=p+i
1 TO TO
2 53 53 ( ^ './: ./:■,/:■,/ ¿l=p+i ¿2=p+i \
+ C
¿373 WO
¿2¿3¿3¿1¿1¿2
(¿373 W-),
+
p p TO
+ y ^ Yl 5-y ( Cj2j3j3jij2ji + Cj2j3j3j'ijij2 j
ji=0 j3=0 j2=p+1
1 p TO
2 y v 53 ^hhhhhh
j3=0 j2 =p+1
(157)
(j1jiW0
The equalities
1
OO OO
ji=p+1 j2=p+1
+ C
(j3j3W0
j2j3j3jijij2
= 0, (158)
(j3j3 W 0
1
pTO
j3=0 j2=p+1
(j1jiW • )
1TO
7 / v ^ './: ./:',/:',/ ./ ./V
p—to 4 z—' j2=p+1
1
OO oo
PlS) 2 53 53 ^ './: ./:■,/:■,/ ./ ./•/
j3=p+1 j2=p+1
(j1jiW • )(j3j3W • )
=0
(j1jiw • )
(159)
follows from the equalities (130), (131) in [14], p. 139 (also see the equalities (2.722), (2.723) in [13], p. 516), where we used the same technique as in (111). When proving (159), we also applied (71) and (29).
By analogy with the proof of (59) and applying (156), we obtain
p p TO
p1—roES 53 (Cj
ji=0 j3=0 j2=p+1
j2j3j3jij2ji + Cj2j3j3j'ijij2 j 0
(160)
From (157)-(160) we have
TOTOTO
p1—?ro ZEE ■
j i =p+1 j2 =p+1 j3 =p+1
j2 j3j3 ji j2 ji + Cj2j3j3j'ijij2 J 0
(161)
Furthermore (see (57)),
oo oo oo
53 53 53 Cj2j3j3j1j1j2 = 0
p—TO
ji=p+1 j2=p+1 j3=p+1
(162)
Combining (161) and (162), we finally obtain
oo oo
pirn E E E Cj2j3j3jlj2jl —
jl=p+1 j2=P+1 j3=P+1
The equality (67) is proved. Theorem 8 is proved. 4 Conclusion
In the first part [14] of this work, expansions of iterated Stratonovich stochastic integrals of multiplicities 1 to 5 with respect to components of the multidimensional Wiener process were obtained. The second part of the work (this article) is devoted to the development of the approach from [14]. More precisely, in
this article we have obtained the expansion of iterated Stratonovich stochastic integrals of multiplicity 6 with respect to components of the multidimensional Wiener process. The noted results make it possible to construct efficient procedures for the mean-square approximation of iterated Stratonovich stochastic integrals that appear in strong methods with orders 1.0, 1.5, 2.0, 2.5, and 3.0 of convergence for Ito SDEs with multidimensional non-commutative noise (in the framework of the approach based on the Taylor-Stratonovich expansion). The above procedures based on multiple Fourier-Legendre series have been successfully implemented as part of the software package in the Python programming language in [52 .
References
[1] Gihman, I.I., Skorohod, A.V. Stochastic Differential Equations and its Applications. Kiev, Naukova Dumka, 1982, 612 pp.
[2] Kloeden, P.E., Platen, E. Numerical Solution of Stochastic Differential Equations. Berlin, Springer, 1992, 632 pp.
[3] Milstein, G.N. Numerical Integration of Stochastic Differential Equations. Sverdlovsk, Ural University Press, 1988, 225 pp.
[4] Milstein, G.N., Tretyakov M.V. Stochastic Numerics for Mathematical Physics. Berlin, Springer, 2004, 616 pp.
[5] Kloeden, P.E., Platen, E., Schurz, H. Numerical Solution of SDE Through Computer Experiments. Berlin, Springer, 1994, 292 pp.
[6] Platen, E., Wagner, W. On a Taylor formula for a class of Ito processes. Prob. Math. Stat. 3 (1982), 37-51.
[7] Kloeden, P.E., Platen, E. The Stratonovich and Ito-Taylor expansions. Math. Nachr. 151 (1991), 33-50.
[8] Averina, T.A. Statistical Modeling of Solutions of Stochastic Differential Equations and Systems with Random Structure. Siberian Branch of the Russian Academy of Sciences, Novosibirsk, 2019, 350 pp.
[9] Kulchitskiy, O.Yu., Kuznetsov, D.F. The unified Taylor-Ito expansion. J. Math. Sci. (N. Y.), 99, 2 (2000), 1130-1140.
DOI: http://doi.org/10.1007/BF02673635
[10] Kuznetsov, D.F. New representations of the Taylor-Stratonovich expansion. J. Math. Sci. (N. Y.), 118, 6 (2003), 5586-5596.
DOI: http://doi.org/10.1023/A:1026138522239
[11] Kuznetsov, D.F. Numerical Integration of Stochastic Differential Equations. 2. [In Russian]. Polytechnical University Publishing House, Saint-Petersburg. 2006, 764 pp.
DOI: http://doi.org/10.18720/SPBPU/2/s17-227
[12] Kuznetsov, D.F. Stochastic Differential Equations: Theory and Practice of Numerical Solution. With MatLab Programs, 3rd Edition. [In Russian].
Polytechnical University Publishing House, Saint-Petersburg. 2009, 768
pp. DOI: http://doi.org/10.18720/SPBPU/2/s17-230
[13] Kuznetsov D.F. Strong Approximation of Iterated Ito and Stratonovich Stochastic Integrals Based on Generalized Multiple Fourier Series. Application to Numerical Solution of Ito SDEs and Semilinear SPDEs.
arXiv:2003.14184v34 [math.PR], 2022, 928 pp.
DOI: http://doi.org/10.48550/arXiv.2003.14184
[14] Kuznetsov, D.F. A new approach to the series expansion of iterated Stratonovich stochastic integrals of arbitrary multiplicity with respect to components of the multidimensional Wiener process. [In English]. Differen-cialnie Uravnenia i Protsesy Upravlenia, 2 (2022), 83-186. Available at:
http://diffjournal.spbu.ru/EN/numbers/2022.2/article.1.6.html
[15] Allen, E. Approximation of triple stochastic integrals through region subdivision. Commun. Appl. Anal. (Special Tribute Issue to Professor V. Lakshmikantham), 17 (2013), 355-366.
[16] Li, C.W., Liu, X.Q. Approximation of multiple stochastic integrals and its application to stochastic differential equations. Nonlinear Anal. Theor. Meth. Appl., 30, 2 (1997), 697-708.
[17] Tang, X., Xiao, A. Asymptotically optimal approximation of some stochastic integrals and its applications to the strong second-order methods. Adv. Comp. Math., 45 (2019), 813-846.
[18] Gaines, J.G., Lyons, T.J. Random generation of stochastic area integrals. SIAM J. Appl. Math., 54 (1994), 1132-1146.
[19] Wiktorsson, M. Joint characteristic function and simultaneous simulation of iterated Ito integrals for multiple independent Brownian motions. Ann. Appl. Prob., 11, 2 (2001), 470-487.
[20] Ryden, T., Wiktorsson, M. On the simulation of iterated Ito integrals. Stoch. Proc. and their Appl., 91, 1 (2001), 151-168.
[21] Averina, T.A., Prigarin, S.M. Calculation of stochastic integrals of Wiener processes. [In Russian]. Preprint 1048. Novosibirsk, Inst. Comp. Math. Math. Geophys. Siberian Branch Russ. Acad. Sci., 1995, 15 pp.
[22] Prigarin, S.M., Belov, S.M. One application of series expansions of Wiener process. [In Russian]. Preprint 1107. Novosibirsk, Inst. Comp. Math. Math. Geophys. Siberian Branch Russ. Acad. Sci., 1998, 16 pp.
[23] Kloeden, P.E., Platen, E., Wright, I.W. The approximation of multiple stochastic integrals. Stoch. Anal. Appl., 10, 4 (1992), 431-441.
[24] Rybakov, K.A. Applying spectral form of mathematical description for representation of iterated stochastic integrals. Differencialnie Uravnenia i Protsesy Upravlenia, 4 (2019), 1-31. Available at:
http: / / diffjournal.spbu.ru/EN / numbers/2019.4/article.1. 1 .html
[25] Rybakov, K.A. Using spectral form of mathematical description to represent Stratonovich iterated stochastic integrals. Smart Innovation, Systems and Technologies, vol. 217. Springer, 2021, pp. 287-304.
DOI: http://doi.org/10.1007/978-981-33-4826-4_20
[26] Rybakov, K.A. Using spectral form of mathematical description to represent Ito iterated stochastic integrals. Smart Innovation, Systems and Technologies, vol. 274. Springer, 2022, pp. 331-344.
DOI: http://doi.org/10.1007/978-981-16-8926-0_22
[27] Kuznetsov, D.F. Approximation of iterated Ito stochastic integrals of the second multiplicity based on the Wiener process expansion using Legen-dre polynomials and trigonometric functions. [In Russian]. Differencialnie Uravnenia i Protsesy Upravlenia, 4 (2019), 32-52. Available at:
http://diffjournal.spbu.ru/EN/numbers/2019.4/article.1.2.html
[28] Foster, J., Habermann, K. Brownian bridge expansions for Levy area approximations and particular values of the Riemann zeta function. Combin. Prob. Comp. (2022), 1-28.
DOI: http://doi.org/10.1017/S096354832200030X
[29] Kastner, F., Roßler, A. An analysis of approximation algorithms for iterated stochastic integrals and a Julia and MATLAB simulation toolbox. arXiv:2201.08424v1 [math.NA], 2022, 43 pp.
DOI: http://doi.org/10.48550/arXiv.2201.08424
[30] Malham, S.J.A., Wiese A. Efficient almost-exact Levy area sampling. Stat. Prob. Letters, 88 (2014), 50-55.
[31] Stump, D.M., Hill J.M. On an infinite integral arising in the numerical integration of stochastic differential equations. Proc. Royal Soc. Series A. Math. Phys. Eng. Sci. 461, 2054 (2005), 397-413.
[32] Platen, E., Bruti-Liberati, N. Numerical Solution of Stochastic Differential Equations with Jumps in Finance. Springer, Berlin, Heidelberg. 2010,
pp.
[33] Kuznetsov, D.F. Multiple Ito and Stratonovich Stochastic Integrals: Approximations, Properties, Formulas. Polytechnical University Publishing House, Saint-Petersburg. 2013, 382 pp.
DOI: http://doi.org/10.18720/SPBPU/2/s17-234
[34] Kuznetsov, D.F. Stochastic Differential Equations: Theory and Practice of Numerical Solution. With MATLAB Programs, 6th Edition. [In Russian]. Differencialnie Uravnenia i Protsesy Upravlenia, 4 (2018), A.1-A.1073. Available at:
http: //diffjournal.spbu.ru/EN/numbers/2018.4/article.2.1.html
[35] Kuznetsov, D.F. Development and application of the Fourier method for the numerical solution of Ito stochastic differential equations. Comp.
Math. Math. Phys., 58, 7 (2018), 1058-1070.
DOI: http://doi.org/10.1134/S0965542518070096
[36] Kuznetsov, D.F. On numerical modeling of the multidimensional dynamic systems under random perturbations with the 1.5 and 2.0 orders of strong convergence. Autom. Remote Control, 79, 7 (2018), 1240-1254. DOI: http://doi.org/10.1134/S0005117918070056
[37] Kuznetsov, D.F. On numerical modeling of the multidimensional dynamic systems under random perturbations with the 2.5 order of strong convergence. Autom. Remote Control, 80, 5 (2019), 867-881. DOI: http://doi.org/10.1134/S0005117919050060
[38] Kuznetsov, D.F. Explicit one-step mumerical method with the strong convergence order of 2.5 for Ito stochastic differential equations with a multidimensional nonadditive noise based on the Taylor-Stratonovich expansion. Comp. Math. Math. Phys., 60, 3 (2020), 379-389. DOI: http://doi.org/10.1134/S0965542520030100
[39] Kuznetsov, D.F. A comparative analysis of efficiency of using the Legendre polynomials and trigonometric functions for the numerical solution of Ito stochastic differential equations. [In English]. Comp. Math. Math. Phys., 59, 8 (2019), 1236-1250. DOI: http://doi.org/10.1134/S0965542519080116
[40] Kuznetsov, D.F. Application of the method of approximation of iterated Ito stochastic integrals based on generalized multiple Fourier series to the high-order strong numerical methods for non-commutative semilinear stochastic partial differential equations. arXiv:1905.03724v15 [math.GM],
2022, 41 pp. DOI: http://doi.org/10.48550/arXiv.1905.03724
[41] Kuznetsov, D.F. Application of multiple Fourier-Legendre series to implementation of strong exponential Milstein and Wagner-Platen methods for non-commutative semilinear stochastic partial differential equations.
arXiv:1912.02612v6 [math.PR], 2022, 32 pp.
DOI: http://doi.org/10.48550/arXiv.1912.02612
[42] Kuznetsov, D.F. A method of expansion and approximation of repeated stochastic Stratonovich integrals based on multiple Fourier series on full orthonormal systems. [In Russian]. Differencialnie Uravnenia i Protsesy Upravlenia, 1 (1997), 18-77. Available at:
http: / / diffjournal.spbu.ru/EN / numbers /1997.1 / article.1.2.html
[43] Kuznetsov, D.F. Mean square approximation of solutions of stochastic differential equations using Legendres polynomials. J. Autom. Inform. Sci., 32, 12 (2000), 69-86.
DOI: http://doi.org/10.1615/JAutomatInfScien.v32.i12.80
[44] Kuznetsov, D.F. Expansion of iterated Stratonovich stochastic integrals based on generalized multiple Fourier series. Ufa Math. J., 11, 4 (2019),
49-77. DOI: http://doi.org/10.13108/2019-11-4-49
[45] Kuznetsov, D.F. Expansion of iterated Ito stochastic integrals of arbitrary multiplicity based on generalized multiple Fourier series converging in the mean. arXiv:1712.09746v24 [math.PR]. 2022, 111 pp.
DOI: http://doi.org/10.48550/arXiv.1712.09746
[46] Kuznetsov, D.F. Exact calculation of the mean-square error in the method of approximation of iterated Ito stochastic integrals based on generalized multiple Fourier series. arXiv:1801.01079v14 [math.PR]. 2022, 68 pp.
DOI: http://doi.org/10.48550/arXiv.1801.01079
[47] Kuznetsov, D.F. Mean-square approximation of iterated Ito and Stratonovich stochastic integrals of multiplicities 1 to 6 from the Taylor-Ito and Taylor-Stratonovich expansions using Legendre polynomials. arXiv:1801.00231v19 [math.PR]. 2022, 106 pp.
DOI: http://doi.org/10.48550/arXiv.1801.00231
[48] Kuznetsov, D.F. The hypotheses on expansions of iterated Stratonovich stochastic integrals of arbitrary multiplicity and their partial proof.
arXiv:1801.03195v18 [math.PR]. 2022, 138 pp.
DOI: http://doi.org/10.48550/arXiv.1801.03195
[49] Kuznetsov, D.F. Expansions of iterated Stratonovich stochastic integrals based on generalized multiple Fourier series: multiplicities 1 to 6 and beyond. arXiv:1712.09516v21 [math.PR]. 2022, 204 pp.
DOI: http://doi.org/10.48550/arXiv.1712.09516
[50] Kuznetsov, D.F. Expansion of iterated Stratonovich stochastic integrals of fifth and sixth multiplicity based on generalized multiple Fourier series.
arXiv:1802.00643v14 [math.PR]. 2022, 129 pp.
DOI: http://doi.org/10.48550/arXiv.1802.00643
[51] Kuznetsov, D.F. The proof of convergence with probability 1 in the method of expansion of iterated Ito stochastic integrals based on generalized multiple Fourier series. Differencialnie Uravnenia i Protsesy Upravlenia, 2 (2020), 89-117. Available at:
http://diffjournal.spbu.ru/RU/numbers/2020.2/article.1.6.html
[52] Kuznetsov, M.D., Kuznetsov, D.F. SDE-MATH: A software package for the implementation of strong high-order numerical methods for Ito SDEs with multidimensional non-commutative noise based on multiple Fourier-Legendre series. Differencialnie Uravnenia i Protsesy Upravlenia, 1 (2021), 93-422. Available at:
http://diffjournal.spbu.ru/EN/numbers/2021.1/article.1.5.html
[53] Kuznetsov, D.F., Kuznetsov, M.D. Mean-square approximation of iterated stochastic integrals from strong exponential Milstein and Wagner-Platen methods for non-commutative semilinear SPDEs based on multiple Fourier-Legendre series. Recent Developments in Stochastic Methods
and Applications. ICSM-5 2020. Springer Proceedings in Mathematics & Statistics, vol. 371, Eds. Shiryaev, A.N., Samouylov, K.E., Kozyrev, D.V. Springer, Cham, 2021, pp. 17-32.
DOI: http://doi.org/10.1007/978-3-030-83266-7_2
[54] Rybakov, K.A. Orthogonal expansion of multiple Ito stochastic integrals. Differencialnie Uravnenia i Protsesy Upravlenia, 3 (2021), 109-140. Avail-
able at: http://diffjournal.spbu.ru/EN/numbers/2021.3/article.1.8.html
[55] Stratonovich, R.L. Conditional Markov Processes and Their Application to the Theory of Optimal Control. Elsevier, N. Y., 1968, 350 pp.
[56] Johnson, G.W., Kallianpur, G. Homogeneous chaos, p-forms, scaling and the Feynman integral. Trans. Amer. Math. Soc., 340 (1993), 503-548.
[57] Rybakov, K.A. Orthogonal expansion of multiple Stratonovich stochastic integrals. Differencialnie Uravnenia i Protsesy Upravlenia, 4 (2021), 81115. Available at:
http: //diffjournal.spbu.ru/EN/numbers/2021.4/article.1.5.html
