CC BY
5
dx
dt
DIFFERENTIAL EQUATIONS AND
CONTROL PROCESSES N. 2, 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
Abstract. The article is devoted to 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-Legendre series and multiple trigonometric Fourier series. The theorem on the mean-square convergent expansion for the iterated Stratonovich stochastic integrals of arbitrary multiplicity is formulated and proved under the condition of convergence of trace series. This condition has been verified for integrals of multiplicities 1 to 5 and complete orthonormal systems of Legendre polynomials and trigonometric functions in Hilbert space. The Hu-Meyer formula and multiple Wiener stochastic integral were used in the proof of the mentioned theorem. The rate of mean-square convergence of the obtained expansions is found. The results of the article can be applied to the numerical integration of 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
Dmitriy F. Kuznetsov
Peter the Great Saint-Petersburg Polytechnic University e-mail: sde_kuznetsov@inbox.ru
integral, Ito stochastic differential equation, generalized multiple Fourier series, multiple Fourier-Legendre series, multiple trigonometric Fourier series, mean-square convergence, expansion.
Contents
1 Introduction 84
2 Expansion of Iterated Ito Stochastic Integrals of Arbitrary Multiplicity k (k £ N) Based on Generalized Multiple Fourier Series Converging in the
Mean 86
3 Expansions of Iterated Stratonovich Stochastc Integrals of Multiplicities 1
to 4. Some Old Results 92
4 Expansion of Iterated Stratonovich Stochastic Integrals of Arbitrary Multi-
plicity k (k £ N) 94
5 Expansion of Iterated Stratonovich Stochastic Integrals of Multiplicity 3 131
6 Expansion of Iterated Stratonovich Stochastic Integrals of Multiplicity 4 138
7 Expansion of Iterated Stratonovich Stochastic Integrals of Multiplicity 5 151
8 Estimates for the Mean-Square Approximation Error of Iterated Stratono
vich Stochastic Integrals of Multiplicity k (k £ N) 168
9 Rate of the Mean-Square Convergence for Expansions of Iterated Stratono
vich Stochastic Integrals of Multiplicities 1 to 5 172
10 Theorems 4—6, 11—13, 15—18 from Point of View of the Wong—Zakai Approx^
imation 174
References 177
1 Introduction
Let 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 + J a(xT,t)dr + J B(xT,t)dwT, x0 = x(0,u), 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. Let x0 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.
It is well known [2]-[5] that Ito SDEs are adequate mathematical models of dynamic systems of various physical nature under the influence of random disturbances. One of the effective approaches to the numerical integration of Ito SDEs is an approach based on the Taylor-Ito and Taylor-Stratonovich expansions [2]-[13]. The most important feature of such expansions is a presence in
them of the so-called iterated Ito and Stratonovich stochastic integrals, which play the key role for solving the problem of numerical integration of Ito SDEs and have the following form
J [^ ) = I ^k (tk) ...I ^i(ti)dwt;i)... dwt:k), (2)
* T * ¿2
( k) } = / ^ (tk )... i ^ (ti)dwt( ... dwi:k(3)
where ^1(r),... (t) are nonrandom functions on [t,T], wT: (i = 1,... ,m) are m,
are independent standard Wiener processes and w[0) = t, i1,... ,ik = 0, 1, ...,
and
denote Ito and Stratonovich stochastic integrals, respectively.
Note that ^(t) = 1 (I = 1,... ,k) and i1,... ,ik = 0,1,...,m in the classical Taylor-Ito and Taylor-Stratonovich expansions [2]-[8]. At the same time
(t) = (t — t)qi (l = 1,..., k, qi,..., qk = 0,1,...) and ii,..., ik = 1,..., m in the unified Taylor-Ito and Taylor-Stratonovich expansions [9], [10] (also see [11]-[13]).
Effective solution of the problem of mean-square approximation of iterated Stratonovich stochastic integrals (3) based on multiple Fourier-Legendre series and multiple trigonometric Fourier series composes the subject of the article.
Note that another approaches to the mean-square approximation of the iterated Ito and Stratonovich stochastic integrals (2) and (3) can be found in [14]-[31].
2 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 7 below).
Suppose that ^ (t),..., (t) E L2([t, T]). Define the following function on the hypercube [t,T]k
!^i(ti) ...^k (tk), for ti <...<tk
, (4)
0, otherwise
where t1,..., tk E [t,T] (k > 2), and K(ti) = ^i(ti) for ti E [t,T].
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
pi,-,ph ^^
pi pk
K (ti,...,tk) ..^Cjk jiH j (ti) ji=0 jk =o i=i
= 0,
L2([t,T ]k)
where
C =
Cjk ...ji =
/k
K (ti,...,tk j (ti )dti ...dtk (5)
[t,T ]k l=i
k
is the Fourier coefficient,
m
k\ —
/
\
1/2
f2(ti,... )dti.. .dtk
/
Consider the partition {rj}N=o °f [t, T] such that
t — To < ... < tn — T, AN — max ATj ^ 0 if N ^ œ, ATj — Tj+1 — Tj.
o<j<N—1
(6)
Theorem 1 [11] (2006), [12]-[13], [32]-[52]. Suppose that every ^(t) (l = 1, ..., k) is a continuous nonrandom function on [t,T] and (x)}°=0 is a complete orthonormal system of continuous functions in the space L2([t,T]). Then
j [^e
(¿l-ifc )
— l.i.m. J
pi ,...,Pfc ^œ
T,t ,
where
Pi Pk / k
j [^tp*..........4;,)
ji =0 jfc=0
J=1
l.i.m. £ j (Ti:)AwT;:)... j(Tik)Aw(;fckM, (i:.....ik№ 1 k )
(7)
¿i,..., ik = 0,1,..., m, l.i.m. is a limit in the mean-square sense, J is defined by (2),
(il-ifc ) T,t
Gk — Hk\Lk, Hk — {(/1,...,W : /1,...,/k — 0, 1,..., N — 1},
Lk — {(l1,...,1k ): h ,...,1k — 0, 1,..., N — 1; lg — (g — r); g,r — 1,...,k},
T
zj;) — J j(T)dw
t
(;) T
are independent standard Gaussian random variables for various i or j (in the case when i = 0), Cjkis the Fourier coefficient (5), AwTj = wj+1 — wTj (i = 0, 1, ..., m), {Tj}=0 is a partition of the interval [t, T], which satisfies the condition (6).
Let us consider the transformed particular cases of Theorem 1 (see (7)) for k = 1,..., 5 [11] (the cases k = 6, 7 and the case of an arbitrary k (k G N) can be found in [12]-[13], [32]-[52])
pi
JIV/1']?,1' = l.i.m. £Cji<<;■>, (8)
' pi—^OO ' * J1
j 1=0
Pi P2 /
J[V(s)]^y2)=Ä £ Z j 1 j 1 j -1{, 1=,2=0}1{J1=j2}), (9)
j 1=0 j2=0 V
P1 P2 P3
j^^r1 = u.m ££ £ j cfW -
j1=0 j2=0 j:=0
-1{;1=;2=0}1{j1=j2}Cj:;:) - 1{i2=i:=0}1{j2=j:}Cj;1) - 1{;1=;:=0}1{j1=j:}Cj22) ) , (10)
P1 P4 /4
J[V<4)tr*4) = 1-iPn- E•••E^, ncf
P1 ,...,P4 l
j1=0 j4=0 \/=1
1 1 A;:)A;4) n 1 /-(;2M;4)
-1{;1=;2=0}1{j1=j2}zj: Zj4 - 1{;1=;:=0}1{j1=j:}zj2 j -
-1{;1=;4=0}1{j1=j4}zj2 j - 1{;2=;:=0} 1{j2=j:}Sj1 j -
-1{;2=;4=0}1{j2=j4}Zj1 j - 1{;:=;4=0}1{j:=j4}zj1 j + + 1{i1=i2=0}1{j1=j2}1{i:=i4=0}1{j:=j4} + 1{;1=;:=0}1{j1=j:}1{;2=;4=0}1{j2=j4} +
+ 1{;1=;4=0}1{j1=j4}1{;2=;:=0}1{j2=j:} , (11)
P1 P5 /5
J= P1l:^ Z••■£Mil4*
j1=0 j5=0 \/=1
1 1 Z(;:)Z(;4)Z(;5) _ 1 1 Z(;2)Z(;4)Z(;5) —
1{;1 = ;2=0} 1{j1=j2}Zj: z j4 zj5 1{;1=;:=0} 1{j1=j:}Zj2 Zj4 Zj5
1 z(;2)Z(;:)Z(;5) _ 1 1 Z(;2)Z(;:)Z(;4)_
1{;1 = ;4=0}1{j1=j4}Sj2 Zj: zj5 1{;1=;5=0}1{j1=j5}Zj2 Zj: Zj4
1 Z (;1)Z (i4) Z (;5 ) _ 1 1 Z (;1)Z (;:)Z (;5) —
1{;2 = ;:=0} 1{j2=j:}Zj1 z j4 zj5 1{;2=;4=0}1{j2=j4}Zj1 j j
1 z(;1)Z(;:)Z(;4) _ 1 1 Z(;1)Z(;2)Z(;5)_
1{;2=;5=0}1{j2=j5}zj1 j zj4 1{;:=;4=0}1{j:=j4}Sj1 j j
— 1 {¿3= ¿5=0} !{j 3=j5}Z (¿1)Z j1 Z fe )Z (¿4) j2 Zj4 — 1 {¿4 = =¿5=0} =j5 } Z (¿1) j1 Z(^¿2 ) Z(¿3) 1 Zj2 Zj3 +
+I{i1= =¿2=0} j =j2} {¿3= ¿4=0} {j3 = . xc(¿5) + 1 {¿1 = = ¿2=0} j =j2} {¿3 =¿5=0} {j3 = ■ iZ (¿4
+I{i1= =¿2=0} j =j2} {¿4= ¿5=0} {j4 = ■ iZ (¿3) + 1 {¿1 = = ¿3=0} j =j3} {¿2 = ¿4=0} {j2 = ■ iZ (¿5 =j4}Z j5
+I{i1= =¿3=0} j =j3} {¿2= ¿5=0} {j2 = Z (¿4) =j5}Zj4 + 1 {¿1 = = ¿3=0} j =j3} {¿4 = = ¿5=0} { j4 = Z (¿2 =j5 } Z j2
+I{i1= =¿4=0} j =j4} {¿2= ¿3=0} {j2 = ■ XZ (¿5) + 1 {¿1 = = ¿4=0} j =j4} {¿2 = ¿5=0} {j2 = ■ xZ (¿3
+I{i1= =¿4=0} j =j4} {¿3= ¿5=0} {j3 = ■ iZ (¿2) + 1 {¿1 = = ¿5=0} j =j5} {¿2 = ¿3=0} {j2 = ■ iZ (¿4
+ I{i1 = =¿5=0} j =j5} {¿2= ¿4=0} {j2 = Z ^ =j4}Zj3 + 1 {¿1 = = ¿5=0} =j5} {¿3 = ¿4=0} {j3 = Z (¿2 =j4}Zj2
+ I{i2 = =¿3=0} {j2 = =j3} {¿4= ¿5=0} {j4 = ■ iZ (¿l) + 1 {¿2 = = ¿4=0} {j2 = =j4} {¿3 = ¿5=0} {j3 = ■ xZ (¿1
+ 1{i2 = i5=0}l{j2=j5}1{i3=i4=0}l{j3 =
(il)
=j4}S jl
(12)
where 1A is the indicator of the set A.
The convergence with probability 1 (the cases of Legendre polynomials and trigonometric functions) [50] and convergence in the mean of degree 2n (n G N) [13] are proved for the approximations J[^M]^--^-"^ and J[^^V^)pi''"'pfc, correspondingly (see (7)). As it turned out, Theorem 1 remains valid for the systems of Haar and Rademacher-Walsh functions in the space L2([t,T]) [11], 13]. Versions of Theorem 1 for iterated stochastic integrals with respect to martingale Poisson measures and for iterated stochastic integrals with respect to martingales are obtained in [13]. Another modification of Theorem 1 can be found in [13], where complete orthonormal with weight r(x) > 0 systems of functions in the space L2([t,T]) were considered. Application of Theorem 1 and Theorem 2 (see below) for the mean-square approximation of iterated stochastic integrals with respect to the infinite-dimensional Q-Wiener process is given in [39], [40].
Note that in
an analogue of Theorem 1 for the case of an arbitrary complete orthonormal system of functions in the space L2([t,T]) and ^i(t),... , ^(t) E L2([t,T]) was considered. Another proof of the mentioned analogue of Theorem 1 [53] can be found in [13] (Sect. 1.11), [44] (Sect. 15).
Let us consider the generalization of Theorem 1 for the case of an arbitrary complete orthonormal system of functions in the space L2([t,T]) and ^i(t ),..., ^(t ) E L2([t,T]) using our notations [13] (Sect. 1.11), [44] (Sect. 15).
In order to do this, let us consider the unordered set {1, 2,..., k} and separate it into two parts: the first part consists of r unordered pairs (sequence order of these pairs is also unimportant) and the second one consists of the remaining k — 2r numbers. So, we have
({{gi, g2},..., {g2r-1, g2r}}, {qi,..., qk-2r}), (13)
v V ' -V-
part 1 part 2
where {g1, g2,..., g2r-1, g2r, q1,..., qk-2r} = {1, 2,..., k}, braces mean an unordered set, and parentheses mean an ordered set.
We will say that (13) is a partition and consider the sum with respect to all possible partitions
y ^ agig2,---,g2r-ig2r,qi---qk-2r. (14)
({{si,S2},...,{S2r-l,S2r }},(91,...,9fc-2r }) {si,32,...,S2r-1,S2r,91,...,9fc-2r }={i,2,.",k}
Below there are several examples of sums of the form (14)
y^ agig2 = «12,
({fli,S2}) {fli,S2}={i,2}
y^ agig2g3g4 = «1234 + «1324 + «2314,
({{Si,S2},{S3,S4}}) {3i,S2,S3,S4}={i,2,3,4}
y^ agig2,qiq2 = «12,34 + «13,24 + «14,23 + «23,14 + «24,13 + «34,12,
({Si,S2},{9i,92}) {fli,S2,9i,92}={i,2,3,4}
y^ «gig2,qiq2q3 = «12,345 + «13,245 + «14,235 + «15,234 + «23,145 + «24,135 +
({31,32},{91,92,9:}) {31,32,91,92,9:} = {1,2,:,4,5}
+«25,134 + «34,125 + «35,124 + «45,123,
y^ «g1g2,g:g4,q1 = «12,34,5 +«13,24,5 + «14,23,5 +«12,35,4+«13,25,4 + «15,23,4 +
({{S1,S2 },{s:,34}},{91})
{S1,S2,S:,S4,91}={1,2,:,4,5}
+«12,54,3 + «15,24,3 + «14,25,3 + «15,34,2 + «13,54,2 + «14,53,2 + «52,34,1 + «53,24,1 + «54,23,1.
Thus, we can formulate the following theorem.
Theorem 2 [13] (Sect. 1.11), [44] (Sect. 15). Suppose that (t),..., ^(t) G 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 Pfc / k [k/2]
J[^]Trk) = p l.i.m. £... £ Cjk..ji n +B-1)
ji=0 jk=0 \/=1 r=1
r k—2r
X
S n 1{:g2s-i = Ss =0}1{jS2s-i = ¿«2. } II j (15)
({{Si,S2},...,{S2r-i,S2r }},{9i,...,9fc-2r}) S = 1 1=1 /
{Si,S2,...,S2r-i,S2r,9i,...,9fc-2r } = {i,2,.",k}
that converges in the mean-square sense is valid, where [x] is an integer part of a real number x; another notations are the same as in Theorem 1.
The connection of the expression in parentheses on the right-hand side of (15) with Hermite polynomials is discussed in [13] (Sect. 1.10, 1.11), [44] (Sect. 14, 15). A similar result can be found in [53].
Note that Theorems 1 and 2 allow us to calculate exactly the mean-square approximation error for the approximations of iterated Ito stochastic integrals (2) of arbitrary multiplicity k.
Theorem 3 [13] (Sect. 1.12), [45] (Sect. 6). Suppose that {^(x)}=0 is an arbitrary complete orthonormal system of functions in the space L2([t,T]) and -01 (t),... ,^k(T) G L2([t,T]), ¿1,... ,ik = 1,... ,m. Then
M j J [^k)fr;fc) - J [^T^* 2} = J K 2(t1,...,tk )dt1 ...dtk -
[t,T ]k
p ( T t2
- E Cjk...jimI J[^(k)]T,^ (tk)...Jj(t1)dwt;i)...dwt:k
where J[^(k)]T;it";k)p,'",p is the expression on the right-hand side of (13) before passing to the limit l.i.m. for the case p1 = ... = pk = p; i1,...,ik =
pi,---,pfc ^to
1,..., m; the expression
£
(j'iv-Jfc)
means the sum with respect to all possible permutations (ji,..., ). At the same time if jr swapped with in the permutation ji ..., ), then ir swapped with iq in the permutation (ii,..., ik); another notations are the same as in Theorems 1, 2.
3 Expansions of Iterated Stratonovich Stochastc Integrals of Multiplicities 1 to 4. Some Old Results
def
Let M2([t,T]) (0 < t < T < to) be the class of random functions £(t, w) = £T : [t, T] x ^ — R, which satisfy the following conditions: £(t, w) is measurable with respect to the pair of variables (t, w), £t is FT-measurable for all t G [t, T], £T is independent with increments ws+A — ws for s > t, A > 0, and
T
i M {(£T)2} dT < to, M {(£t)2} < to for all t G [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
nT° = nt(i) + J asds + J Mw? w. p. 1, (16)
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—
second integral on the right-hand side of (16) 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(—to, to) consisting of twice continuously differentiable in x functions on the interval (—to, to) such that the first two derivatives are bounded.
The mean-square limit
N-1 /1 \ f
LLm T,F{\('/.•; •)/ ^<0,r)dw?> m
n—to j=0 ^ / */
is called [54] the Stratonovich stochastic integral with respect to the component
j }j=0
wT^ (l = 1,... ,m) of the multidimensional Wiener process wT, where {rj}N
is a partition of the interval [t, T], which satisfies the condition (6).
It is known [54] (also see [2]) that under proper conditions, the following relation between Stratonovich and Ito stochastic integrals holds
* T t T
I F(tf\T)dw? = J F(tf\T)dw? + ±l{i=l} J^(i]T,T)bTdT (18) 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 (18) is correct, for example, consisits of the conditions n^ £ Q4([t,T]), F(ni^T) £ M2([t,T]), F(x, t) £ C2(-to, to) (for fixed t), where i = 1,..., m.
As it turned out, approximations of the iterated Stratonovich stochastic integrals (3) are essentially simpler than the appropriate approximations of the iterated Ito stochastic integrals (2) based on Theorems 1 and 2. For the first time this fact was mentioned in [11] (2006).
According to the standard connection (18) between Ito and Stratonovich stochastic integrals, the iterated Ito and Stratonovich stochastic integrals (2) and (3) of first multiplicity are equal to each other w. p. 1. So, we begin the consideration from the multiplicity k = 2 (the case k = 1 is given by (8)).
The following three theorems adapt Theorems 1, 2 for the integrals (3) of multiplicities 2 to 4.
Theorem 4 [13], [32], [33], [48]. Suppose that ^i(t),^2(t) are continuously differentiate functions on [t,T] and (x)}°==0 is a complete orthonormal system of Legendre polynomials or trigonometric functions in the space L2([t,T]). Then, for the iterated Stratonovich stochastic integral of second multiplicity
* T * t2
J*['<A(2)]T,f' = / / *i(ii)dw<;'»dw«:2>
the following expansion
Pi P2
= Ä L £ jcjr'cj2'
ji=0 j2=0
that converges in the mean-square sense is valid, where i1,i2 = 0,1,...,m; another notations are the same as in Theorems 1, 2.
Theorem 5 [13], [32], [33], [48]. Suppose that {0(x)}TO 0 is a complete or
thonormal system of Legendre polynomials or trigonomertic functions in the
space L2([t,T]). Furthermore, let the function ^2(t) is continuously differentiate at the interval [t,T] and the functions ^i(t), ^3(t) are twice continuously differentiate at the interval [t, T]. Then, for the iterated Stratonovich stochastic
integral of third multiplicity
* T * is * ¿2
J*№(a)If3' = / «ta)/ V'2(î2)| ^1(i1)dw<;')dwi;2»dw<ss)
t t t
the following expansion
p
j*[V'(a)]r,r*s) = E Cjs,2,1 zjijzjss) (19)
jl j2 ,jS=0
that converges in the mean-square sense is correct, where ii, i2, i3 = 0,1,..., m; another notations are the same as in Theorems 1,2.
Theorem 6 [13], [32], [33], [48]. Suppose that (x)}™ 0 is a complete orthonormal system of Legendre polynomials or trigonometric functions in the space L2([t, T]). Then, for the iterated Stratonovich stochastic integral of fourth multiplicity
>jc T ^ ¿4 ^ ¿3 ^ ¿2
/T!;,!2W4) = J J J J dw<;')dwi;2»dw<33)dw<;4)
t t t t
the following expansion
p
7T,t p1^ ^ Cj4j3j2jl j zj2 Zj3 zj4
j1 J2,j3,j4=0
that converges in the mean-square sense is valid, where ii, i2, i3, i4 = 0,1, ..., m; another notations are the same as in Theorems 1, 2.
4 Expansion of Iterated Stratonovich Stochastic Integrals of Arbitrary Multiplicity k (k G N)
In this section, we prove the expansion of iterated Stratonovich stochastic integrals (3) of arbitrary multiplicity k (k G N) under the condition of convergence of trace series.
Let us introduce some notations and formulate some auxiliary results. Consider the Fourier coefficient
t t2
Cjk ...j ! = J fa (tk ) j (tk ) ..J ! (ti)dti.. .dtk (20)
t t
corresponding to the function (0), where (x)}°=0 is a complete orthonormal system of functions in the space L2([t,T]). At that we suppose 0o(x) = 1 /y/T=t.
Denote
с......
Cjfc •••Л+1Л ЛЛ-2-Л 1
def
(ЛЛ WO
T tl+2 ti + 1
= j ^k (tk (tk ) ...J (tl+i) ^ ^l (t )x
t t t tl t2
x J ^i-2(ti-2)0j _2 (ti-2) •. .y ^i(ti)j (ti)dti.. .dti-2dti ti+1.. .dtk = (21)
tt
T ti+2 ti+i
= VTW j c/Ali/jOj, (I I..) ... J ф1+1(и+1)фл+1(и+1) J Ф1Ш1- 1Шо(и)х t t t tl t2
X J^i-2(ti-2)0ji_2(ti-2).. ^^i(ti)0ji(ti)dti.. .dti-2dtiti+i.. .dtk =
tt
= л/î1 — tCjk„jl+1ojl_2...j1,
i.e. \/T /С '/;.../г. (|/г ....... is again the Fourier coefficient of type (120) but with
a new shorter multi-index jk ... jl+i0jl-2 ... j and new weight functions ^i(r), ..., -0г_2(т), \/T - Î'0/_i(t)'0/(t), -0/+i(r), ..., (also we suppose that
{/, I — 1} is one of the pairs {gi, g2},..., {g2r-i, g2r} (see (13))).
Let
с......
-Лг+i Л Л Л-2-Л
def
(Л Л W m
T
tl+2
tl+1
def
= J ^k (tk )0jfc (tk) ■ J ^Z+i^Z+03 (tZ+0 j ^Z (tl№—iM 3 (tz )X t t t tl t2
x/^z-2(iz-2)0ji-2 (tz—2) ...J ^i(ii)0ji (ti)dti... dtz-2dti tz+i ...dtk = (22)
tt
= ä.....
i.e. C
3k•••3l+l3m3l-2•••3l
is again the Fourier coefficient of type (20) but with a new shorter multi-index ... j/+1jmj/-2 ... j and new weight functions (t), ...,
^/-2(t), ^/—1(t)^/(t), ^/+1(t), ..., (t) (also we suppose that {/, I — 1} is one of the pairs {01,02},..., {g2r- 1,g2r} (see (13))). Denote
ä
(p)
3k-Jq •••3l
def
00 00
def
^ S ••• S S ä3k ••
3s2r-1 =P+i 3'fl2r-8 =P+i 3'flS =P+i 3'fll =P+i
31
(23)
3S1 3S2 '•••'3S2r-1 3S2r
Introduce the following notation
SA C.
(p)
3k •••3q •••31
q=gl,g2,•••,g2r-l,g2r
œ œ
def J- n V^
~ fei^i-l + l} / y / y
2
OO
OO
00 oo
y^ y^ • • • ^ X] ä3k •31
3s2l+l =p+i 3s2l-3 =P+i 3'flS =P+i 3s 1 =p+i
3s2r-1 =P+i 3s2r-3 =P+i
(3S2l3S2l — 1 M^'fll 3S2 '•••'3S2r-1 3S2r
Note that the operation S/ (l = 1, 2,..., r) acts on the value
ä (p)
ä3k •••3q •••31
q=gl,g2,-,g2r-l,g2r
(24)
as follows: S/ multiplies (24) by 1 {g2l=g2I-1+1}/2, removes the summation
œ
,
3fl2l-1 =P+1
and replaces
with
ä
ä3k •••31
3S1 3S2 '•••'3S2r-1 3S2r
C
3k•••3l
(3S2l3S2l — 1 MO^Sl 3S2 '•••'3S2r-1 3S2r
(25)
(26)
Note that we write
ä
ä3k •••31
ä
ä3k •••31
=ä
(3S 13S2 ) ^ ( •) >3S1 3S2
3k •••31
=ä
(3sl 3s2 W m ?3sl 3S2
3k •••31
(3s 13's l) ^( •) 3 1 =3S2
(3sl 3 s l )^3' m ,3Sl 3S2
ä
=ä
3k •••31
(3S 13S2 ) •), (3S3 3S4 ) ^( •) >3's 1 3S2 >3S3 =3S4 (3S 13S1 ) ^ ( • )(3S3 3S3 ) ^ ( •) >3S1 =3S2 >3S3 =3S4
Since (26) is again the Fourier coefficient, then the action of superposition S/Sm on (25) is obvious. For example, for r = 3
Q Q Q ) 7(p)
S3S2S^ <C3k-3q -31
q=gl>g2,-,g5,g6
3
1 3
23 n —1^1} ^jk •••jl
s=i
(3S2 3S1 ) •) (3S43S3 ) ^( •) (3S6 3S5 ) •) 3S1 =3S2 '3S3 =3S4 '3S5 =3S6
S3 S^ (7,
(p)
3k •••3q •••31
OO
22
3S3 =p+i
q=gl>g2,-,g5,g6
(3S2 3S1 ) ^ ( •) (3S6 3S5 ) ^ ( •) 31 3S2 >3S3 =3S4 '3S5 =3S6
S^ ä
(p)
3k •••3q •••31
q=gl,g2,-,g5,g6
oo oo
-1
2
{g4=g3+1}
jgi =P+1 jS5 =P+1
( j34 j33 ) ^ ( • ) 1 =j32 'j33 =j34 >j35 =j36
Theorem 7 [48], [49]. Assume that the continuously differentiable functions fa(t) : [t,T] ^ R (l = 1,...,k) and the complete orthonormal system of continuous functions {(f>o{x) = 1/y/T — t) in the space Lo{[t,T}) are such that the following conditions are satisfied:
1. The equality
oo
t2
- I = E I $2{to)(f)j{to) I ^{h^jit^dhdto
t j=0t t
(27)
holds for all s £ (t, T], where the nonrandom functions $i(t), $2(t) are continuously differentiable on [t,T] and the series on the right-hand side of (27) converges absolutely.
2. The estimates
T
(t )$i(t )dr
<
(s)
J
_
,'1/2+a'
(t )$2(t )dt
<
^l(s)
J
_
,'1/2+a'
e j ^2(t)0j (t) ^ $i(0)0j (0)d0dt
j=P+1 t t
<
Pß
hold for all s £ (t,T) and for some a,^ > 0, where (t), $2(t) are continuously differentiable nonrandom functions on [t,T], £ N, and
T T
J ^2(t)dT < oo, J |^2(t)| dT < oo.
tt
3. The condition
lim
£
sii si2... SiA C
(p)
jk •••jq •••jl
J1,...,Jq,...,Jfc = U 9=Sl,S2,...,S2r-1,S2r
= 0
q=ff1,ff2v,g2r-1,g2r
s
s
s
2
holds for all possible gi,g2,... ,g2r-i,g2r (see (131)) and 1i, 12,..., such that 1i, 12,..., G {1, 2, ..., r}, /i > /2 > ... > d = 0,1, 2,... ,r - 1, where r = 1, 2,..., [k/2] and
SZi SZ2 • • • Sz,< ä
(p)
d ) 3k •••3q •••31
=f /7(p)
q=gl,g2v>g2r-l,g2r
3k ...3q...31
q=gl,g2,-,g2r-l,g2r
for d = 0.
Then, for the iterated Stratonovich stochastic integral of arbitrary multiplicity k
T
t2
j *[#*<
(—k) = i,k(tk)... / ^(ti)dw«;l)...dw<;*»
(28)
the following expansion
J *
(il •••ik) T,t
k
^ £ ä3k II Z
(il ) 3l
z=i
j1 ,---,j'fe =0
that converges in the mean-square sense is valid, where
T t2
Cjfc ---jl = J 1 (tk ) j (tk) ...J 1 (ti)0j1 (ti)dti ...dtk tt
(29)
(30)
is the Fourier coefficient, l.i.m. is a limit in the mean-square sense, i1,..., = 0,1,..., m,
T
3 = 3 (T )dw
T
jj t
are independent standard Gaussian random variables for various i or j (in the case when i = 0), w[0) = t.
Proof. The proof of Theorem 7 will consist of several steps.
Step 1. Let us find a representation of the random variable
Y^ ä3k -IIZ3(
3l,-,3k=0
(il ) 3l
z=i
p
k
p
that will be convenient for further consideration.
Let us consider the following multiple stochastic integral
N-1 k
J'[$]Tr'' ' = U-m. £ *(ta ,---Tjt ^Aw<j; >, (31)
Jlv-j =0 1 = 1
jq =jp; q=p; q,p=i,...,fc
where we assume that $(t1,... ,tk) : [t,T]k ^ R is a continuous nonrandom
function on [t,T]k. Moreover, AwTj = w«+i — wTj (i = 0,1,... ,m), (Tj}n
Tj+1 vv Tj — "V, l j }j=0
is
a partition of the interval [t,T], which satisfies the condition (6), ¿i,... ,ik = 0,1,..., m.
The stochastic integral with respect to the scalar standard Wiener process (ii = ... = ik = 0) and similar to (31) was considered in [55] (1951) and is called the multiple Wiener stochastic integral [55]. Note that the following well known estimate
m{ 'H))2} < Ck J $2(ti,...,tk)dti ...dtk (32)
[t,T ]k
is correct for the multiple Wiener stochastic integral, where J7^]^' ' ' is defined by (31) and Ck is a constant.
From the proof of Theorem 1 (see the proof of Theorem 5.1 in the monograph [11] (2006) in Russian or proof of Theorem 1.1 in the monograph [13] in English) it follows that (7) can be written as
pi pk
J [fa(k>]% ' ) = l . i-m. £ ' ' ji J'[j ... j ]& ' ), (33)
Pi, ' ' ' ,Pfc^œ ^—' ^—' ji =0 jfc=0
where J'[ j... j' ' H) is the multiple Wiener stochatic integral defined by (31) and J' ' 'k) is the iterated Itô stochastic integral (2).
Let us consider the following multiple stochastic integral
N—1 k
J [$]<irk)= ^ £ $(T;i ,..-.Tjk ^Aw« (34)
j1,'",jfc=0 l = 1
where we assume that ... , t*) : [t,T]* ^ R is a continuous nonrandom function on [t,T]*. Another notations are the same as in (31).
The stochastic integral with respect to the scalar standard Wiener process (i1 = ... = i* = 0) and similar to (34) (the function ... , t*) is assumed to be symmetric on the hypercube [t,T]*) has been considered in the literature (see, for example, Remark 1.5.7 [56]). The integral (34) is sometimes called the multiple Stratonovich stochastic integral. This is due to the fact that the following rule of the classical integral calculus holds for this integral
J} = J...J[v*© w. p. 1,
where ..., t*) = ^i(ti)... v*(t*) and
T
J = i VI (T )dwTil) (l = 1,...,k).
Theorem 8 [13] (Sect. 1.9), [44] (Sect. 13). Suppose that $(t1,...,t*) : [t,T]* ^ R is a continuous nonrandom function on [t,T]*. Furthermore, let {fy (x)}°=0 be a complete orthonormal system of functions in the space L2([t, T ]), each function fy (x) of which for finite j is continuous at the interval [t,T] except may be for the finite number of points of the finite discontinuity as well as fy (x) right-continuous at the interval [t,T]. Then the following expansion
Pi Pfc / k [k/2]
J'mSr> =,,, i.i.m. E • ■• ■■ E C*J n j> ^(-1)
ji=0 j =0 V=1 r=1
r *—2r \
x E n1 = =0}1{j„,._i=}II j ' <35»
({{gl ,52},...,{S2r-1,S2r >},(91,...,9fc-2r}) S=1 1 = 1 /
{si,32,...,S2r-1,S2r,91,...,9fe-2r }={1,2,.",k}
converging in the mean-square sense is valid, where J'^]^'"^ is the multiple Wiener stochatic integral defined by (31),
C — •••ji —
/k
[t,T ]k 1-1
is the Fourier coefficient. Another notations are the same as in Theorems 1,2. Introduce the following notations
i
J[^p)[«,,...,.,I = JJ 1{ijp=ijp+i=0! x
p=1
T +2
X J (tk ) ...J ^s;+2 (tsj +2) J ^ s, (ts; + 1 + 1 (tS + 1) X t t t
^s; + 1 ts1+3 tS1 + 2
X j ^s, 1 (ts;-1) ...y ^si+2(tsi+2^ ^si (tsi+1)^si+1(tsi+1)x t t t
tsi + i t2
(H) J,Tr(isi+2)
x y ^s1-i(tsi-i)... y fa(ii)dwj^...dtsi+idw;
î i
...dw'^V,,...dw<kk >, (36)
where (si,... ,si) e At,;,
Ak,i = {(si,... ,si) : si > s;-i + 1,... ,S2 > si + 1; si,... ,si = 1,... ,k - 1},
(37)
l = 1, 2,..., [k/2] , ii,..., = 0,1,... ,m, [x] is an integer part of a real number x, 1a is the indicator of the set A.
Let us formulate the statement on connection between iterated Stratonovich and Ito stochastic integrals (3) and (2) of arbitrary multiplicity k.
Theorem 9 [41] (1997) (also see [11]-[13], [32], [33]). Suppose that every
fa(t) (l = 1,... ) is a continuous nonrandom function at the interval [t,T]. Then, the following relation between iterated Stratonovich and Ito stochastic integrals (3) and (2) is correct
[k/2] 1
= J[lß(+ £ £ Jr^/r ".....* w. p. 1,
r=l ( sr ,...,si)GÄfcir
(38)
where is supposed to be equal to zero.
0
Consider (35) for $(t1,..., t*) = KP1'„Pk(t1,..., t*) and without passing to the limit l.i.m. (see the proof of Theorem 8 in [13] (Theorem 1.13) or (Theorem 9))
[k/2]
J [Kpi. ••Pfc ]T,t ••Pfc ]T,t k) — £ (-1)
r=1
r
X E II ^ = ig2s =0}X
({{31,32},...,{32r-1,32r }},{91,...,9fc-2r }) S=1 {31,32,...,32r-1,32r,91,...,9fc-2r }={1,2,-.,k}
X J[K^1;;;^,q1'''qk-2r^ '''iqfc-2r} (39)
w. p. 1, where J'[KP1 . . .Pk^''ik) is the multiple Wiener stochastic integral (131), J[Kp1'„pk]Ti1tis the multiple Stratonovich stochastic integral (34),
P1 pk *
Kp1„'pfc(t1,... ,t*) = £ ... £ Cjk'"j1 J]fyji(ti), (40)
j1=0 jk =0 1=1
Pi Pk r k—2r
^••jr91 qk 2r (tqi, • • •, tqk-2r) — E • • • E Cj'k"j'i Ü 1{j32s-i = js2s} Ü j).
ji=0 jk =0 s=1 1=1
(41)
Passing to the limit l.i.m. (p1 = ... = p* = p) in (39), we get w. p. 1 (see Theorems 1, 2)
P k [k/2]
l.i.m. £ Cik„ji I] Ciil) — J[^Tif"} — ^ E(—1) ji, •••jk=0 1=1 r=1
X E niii32s-i = i32s =0}X
({{3i,32},...,{32r-i,32r }},(9i,...,9k-2r }) S = 1 {3i,32,...,32r-i,32r,9i,...,9k-2r } = {i,2,...,k}
X J[K^ • >9i-9k-2r •••iqk-2r ) —
r
P [k/2]
J [^(k)frik) — l-i-m^ Cjk •••ji£ (—1)
' rt—vno < * < *
P—TO
ji^^^^jik=0 r=1
r k—2r
X £ ni(.32,-i = .3. =0}1{j32s-i = ,32, }II £ 1 , (42)
-i,32r }},{9i,...,9k-2r}) s = 1 1=1 /
({{3i,32},...,{32r-i,32r }},{9i,...,9k-2r }) s = 1 {3i,32,...,32r-i,32r,9i,...,9k-2r } = {i,2,...,k}
where J[^(*)]Ti1t'ik) is the iterated Ito stochastic integral (2). If we prove that w. p. 1
[k/2] 1 P [k/2]
r=1 (sr ^^s^eA^r jl,•••,jk=0 r=1
r k—2r
X
E n 1{.32s-i = .32, =0}1{j32s-i = ,32s } H j'') ) , (43)
({{3 i,32 },..., {32r-i,32r }},{?i,...,9k-2r }) s = 1 1=1
{3i,32,...,32r-i,32r,9i,...,9k-2r } = {i,2,...,k}
then (see (42), (43), and Theorem 9)
Pk
I.——to. e c^IIz
ji,-jk=0 1=1
[k/2]
= J[^k)%"ik) + ^ "T E J№(Ä)]^"ifc)[ar,-,ai] = J*№k)%"ik) (44) , 2 , ,
r=1 (sr v^S^eA^r
w. p. 1, where notations in (44) are the same as in Theorem 9. Thus Theorem 7 will be proved.
From (39) we have that the multiple Stratonovich stochastic integral J[KP1.''Pkfc"'^ of multiplicity k is expressed as a finite linear combination of the multiple Wiener stochastic integral J'[KP1 'Pk]!1t'''ik) of multiplicity k and multiple Stratonovich stochastic integrals J[KP^'Pf 'q1 'qk-2rj!^1 'iqk-2r) of multiplicities k — 2, k — 4, ..., k — 2[k/2]. By iteratively applying the formula (39) (also see
(9)-(I2)), we obtain a representation of the multiple Stratonovich stochastic integral of multiplicity k as the sum of some constant value and multiple Wiener stochastic integrals of multiplicities not exceeding k
j [k ](«i--«fc) = j ' [ k ](ii--ifc) + J [KPi...Pk]T,t = J [KPi-"Pfc]T,t +
[k/2] r
+E E II1iig2s-i = ^=oix
r=i ({{si,32},...,{S2r-1,S2r }},(91,...,9fc-2r}) s=i
{Si,S2>...>S2r-1>S2r>91>...>9fc-2r }={1>2>...>k}
x J/[Kpgi;;;pgfc2r'qi-qk-2r]^"'iqfc-2r) (45)
w. p. 1, where Kpi...pfc (t i,..., ) and Kp^^.'.'.'p^2" 'qi'"qfc 2r (t^, ...,tqfc_2r) are defined by the equalities (40), (41).
From (1451) we have the following generalization of (I471)-(l5fll) (see below) for the case of an arbitrary k (k e N)
Pi Pfc k pi pfc
TTZ (ij) = V^ Z (ii) Z (ik) =
... Cjfc'''ji H j = Z^ ... Z^ '''jij ... j =
ji=0 jfc =0 i=i ji=0 jfc=0
pi pk
= £ --.Ec. ..j J '[j... j ]Trk >+
ji=0 jfc=0
Pi Pfc [k/2] r
+E... E ^wi E E n 1{iS2S-i = ^=0}x
ji=0 jfc =0 r=i ({{Si,S2},...,{S2r-i,S2r }},{?i,...,?fc-2r}) s=i
{Si,S2,...,S2r-i,S2r,9i,...,9fc-2r }={i,2,.",k}
X j.-! = } J'[ j . . . "■iqk-2r' w. p. 1. (46)
The formulas (45), (46) can be considered as new representations of the Hu-Meyer formula for the case of a multidimensional Wiener process [57] (also see [56], [58]) and kernel (ti,... ,tk) (see (40)).
For example, for k = 2, 3, 4, 5 we have from (46) w. p. 1
Pi P2 Pi P2
j j = J' [KPiP2 ]T,f ECj2ji 1{ii=i2=0}1{j1=j2}, (47)
ji=0 j2=0 ji=0 j2=0
Pi P2 P3
V^ V^ V^ C Z(!1)Z(i2)z(i3) = Tf[K ](!i!2!3) I
Cj3j2ji j Zj2 zj3 = J [KP1P2P3]T,t +
ji=0 j2=0 j3=0 Pi P2 P3 /
+ E E E Cj3j2jj 1{ii=!2=0}1{j1=j2} J' [ j ^ + 1{'2='3=0}1{j2=j3} J' [ j ]T,i + ji=0 j2=0 j3=0 \
+ 1{ii=!3=0}1{ji=j3} J' [ j , (48)
Pi P4
EV^ C Z (!1)Z (i2) Z (!3)Z (i4) = T' [K ] (!1!2!3!4) I
. . . / v Cj4j3j2j1 j j Zj3 Zj4 = J [KPiP2P3P4]T,t +
ji=0 j4=0
Pi P4 /
. . . E Cj4j3j2j\[ 1{ii=i2=0} 1{ji=j2} J [j j 4 + ji=0 j4=0 V
+ 1{!i = !3=0}1{ji=j3}J' [ j j ]?:/4) + 1{!i = !4=0}1{ji=j4}J' [ j j ]T!,f) + + 1{!2=!3=0}1{j2=j3}J' [ j j ]T!,1t!4) + 1{!2 = !4=0}1{j2=j4}J' [ j j] T^^t'3 ) + + 1{!3=!4=0} 1{ j3=j4} J' [ j j] Tit2 ) + + 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} ), (49)
Pi P5
EV^ C Z (!1)Z (i2 ) Z (!3)Z (i4) Z (i5 ) = T' [K ](i1i2i3i4i5) +
. . . / y Cj5j4j3j2j1 j Zj2 Zj3 Zj4 Zj5 = J [KPiP2P3P4P5 ]T,t +
ji=0 j5=0
Pi P5
+ E . . . E Cj5j4j3j2ji ( 1{«1 = «2=0} 1{ji=j2} J' [j j j] Tt 4 5 +
ji=0 j5=0 V
+ 1{! 1 = !3=0}1{ji=j3}J' [ j j j ]TT"4!5) + 1{' 1 = '4=0}1{ji=j4}J' [ j j j ^^ + 1{! 1 = !5=0} 1{ji=j5} J' [ j j j] Tt3! 4) + 1{'2 = '3=0} 1{j2=j3} J' [ j j j ]T,i^ 5) +
+ 1{i2=i4=0} 1{j2=j4} J [ j j j] T,i ^ + 1{i2=i5=0} 1{j2=j5} [ j j j] T,'/^ +
(¿1«3i4)
+ 1{i3 = i4=0} 1{j3=j4} J j j ^^ + 1{i3 = i5=0} 1{j3=j5} [ j j j ]T,i " ' +
+ 1{i4 = i5=0} 1{j4=j5} J j j]Ti1ti2i3) +
1(¿1*2*4)
+ 1{H = =¿2=0} j =j2} {¿3 = = ¿4=0} {j3 = =j4}J
+1{ü= =¿2=0} j =j2} {¿3 = = ¿5=0} {j3 = =j5}J
+1{ü= =¿2=0} j =j2} {¿4 = = ¿5=0} j =j5}J
+1{ü= =¿3=0} j =j3} {¿2 = = ¿4=0} {j2 = =j4}J
+1{ü= =¿3=0} j =j3} {¿2 = = ¿5=0} {j2 = =j5}J
+ 1{H = =¿3=0} j =j3} {¿4 = = ¿5=0} { j4 = =j5}J
+1{ü= =¿4=0} j =j4} {¿2 = = ¿3=0} {j2 = =j3} J
+1{ü= =¿4=0} j =j4} {¿2 = = ¿5=0} {j2 = =j5}J
+1{ü= =¿4=0} j =j4} {¿3 = = ¿5=0} {j3 = =j5}J
+1{ü= =¿5=0} j =j5} {¿2 = = ¿3=0} {j2 = =j'3}J
+ 1{H = =¿5=0} j =j5} {¿2 = = ¿4=0} {j2 = =j4}J
+1{ü= =¿5=0} j =j5} {¿3 = = ¿4=0} {j3 = =j4}J
+ 1{^2 = =¿3=0} {j2 = =j3} {¿4 = = ¿5=0} { j4 = =j5}J
+ 1{^2 = =¿4=0} {j2 = =j4} {¿3 = = ¿5=0} {j3 = =j5}J
+ 1{»2 = = ¿5=0} {j2 = =j5} {¿3 = = ¿4=0} {j3 = =j4}J'
j fe' +
(¿5)
T,t ■
j ]Ti4t)+
(¿5)
T,t ■
(¿4)
T,t "
(¿2) T,t ■
j ]T,i) +
j fe+
j ]T,t/ +
j ]T,t +
j + j ]T;/+
(¿1)
(¿1)
r
j fe+
j ](,1)
T,i
(50)
Further, we will use the representation (46) for p = ... = pk = p, i.e.
Z Cfk^n z
j1,---,jfc=0
k p (¿1) =
1=1 j1,-Jfc=0
P [k/2] r
+ E E il1^- = ¿«2» =°}
j1v,jfc =0 r=1 ({{S1,S2},...,{S2r-1,S2r }},{?1,...,9fc-2r}) S=1
X
{S1,S2,...,S2r-1,S2r,91,...,9fc-2r }={1,2,.",k}
p
X 1{jS2s-i = j32s } J [^j91 . . . ^j9fc-2r ]T,t w. p. 1. (51)
Step 2. Let us prove that
y ^ Cjfc ."ji+i jiji-1 -"js + iji js-i.-ji = 0 (52)
j=0
or
p
^ ^ Cjfc '''ji+lji ji-i -''js + lji js-i-ji Cjk '''ji+ljiji-1 -''js + lji js-i-ji , (53)
ji =0 ji =p+i
where l — 1 > s + 1, p e N.
Our further proof will not fundamentally depend on the weight functions ^i(r),...,^k(r). Therefore, sometimes in subsequent consideration we write ^i(r),... ,^k(r) = 1 for simplicity. Using the integration order replacement, we have
c=
Cjfc "Ji+iji ji-i "Js + iji js-!-.!
T ti+2 t;+i ti
= J j (tk) ...J ji (t;+i^ y 0ji(t; ^ 0ji-i (ti—1)... t t t t
¿s+2 ts+1 ts
...J 0js + 1 ^ / 0ji 0js-1 (ts—i) ...
t t t
t2
ft)^ ...dts—idtsdts+i ...dtl—idtl dtl+i =
t
T ts + 1 ts t2
= j ^js+1 (tS+i^ ^ji 0js-1 (tS —i) j (ti)dti ...dts —idts X
t t t t
/ T T T T
x ( y ^js+2 (ts+2) ...J ji (ti—i) J j(t; ^ j (t;+i)...
\ts + 1 ti-2 ti-i ti
. . .J j (tk )dtk ... dt;+idt; dt;—i . . . dts+2 I dts+i =
tfc-1 /
T is + 1 ts t2
— j 0js+1 0js-1 (ts-1) ...J j (t1)dt1 ...dts-1 dtsx
t t t t
Gjs-1...j1 (ts)
TT T
x J (ti ^ j (t1+1) - J j (tk )dtk X
ts+1 t; tfc_1
Hjfc...j;+1(t;)
x
{ \
t; ts+3
[ 0j;_1 (ti-i).. .J 0js+2 (ts+2)dts+2 ... dti-1 dti
ts+1
\
ts+1
Qj;_1...js+2 (t;'ts
dts+1 —
/
T ts + 1
T
X / (t1 )Hjk. --ji + 1 (t1 )Qji-1---js+2 (t1, ts+1)dt1 dts + 1-
ts+1
Applying the additive property of the integral, we obtain
t ¿s+3
Qjl-1 ...js+2 (t1, ts+1) = J 0j"i-1 (t1-1) 'J 0js+2 (ts+2)dts+2 . . . dt1-1
ts+1 ts+1
ti ts+4 ts+3
= J j (t1-1) ... J 0js+3 (ts+3^ y 0js+2 (ts+2)dts+2dts+3 . . . dt1-1
ts + 1 ts+1 t
ti ts+4 ts+1
- y j (t1-1) ...J 0js+3 (ts+3)dts+3 . . . dt1-1 y 0js+2 (ts+2)dts+2
ts + 1 ts+1 t
(54)
Eh;tj,+2 (tij j (ts+i), d<
(m)
(55)
d
Combining (54) and (55), we have
P
Cj
jk •••ji+i jiji-1 •••js + iji js-1-ji
jl=0
d / T p ts+1
J2 I / ^j's + 1 (tS + 1)qjmL.js+2 ^E [ j (tS )Gjs-1...ji (tS )dtS X
jl =0 {
T
(m)
x j (t )Hjk ...ji+i (t )h;-m-;..jS+2 (t )dtidt s+1 . (56)
jl — 1 •••js+2
ts + 1
Using the generalized Parseval equality, we obtain
ts + i T
oo
J j (ts )Gjs-i...ji (ts )dt^ (ti )Hjk ...ji + i (ti 1...js+2 (ti )dtl =
jl=0 t ts+i
T
= 1{T<ts + i}Gjs-1 •••ji (t ) • 1{T>ts + i} Hjk •••jl + 1 (t ) hj'm1 •••js+2 (T )dT = 0. (57)
is + 1}w js-i "JiV 7 {T>ts + 1} jk •••ji + iV / ji-1 ...js+2
t
From (56) and (57) we get
P
•ji+1 jljl-1 •••js + iji js-1...j1
Cjfc ...j
jl=0
d / T ^ ts + 1
E / ^j's+1 (tS+1)qjm!...js+2 (t-+0 £ J j (ts )Gjs-ij (ts )dts X m=1 V t ji =p+1 t
T
(m)
x / (ti )Hjk ••j+i (ti )jJ., (ti )dti dts+J . (58)
jl — 1 ...js+2
ts + 1
Combining Condition 2 of Theorem 7 and (I541)-(l5fil), (58), we have
P
Cjk •••j+1jj-1 •••js + 1jjs-1...ji =
j i =0
œ d / T ts+1
EE/ (wi)qjm!...jS+2(is+iW ^(ts)GjS_i...jix
j; =p+1 m=1 \ t t
OO
T
x y (t1 )Hjk...j;+i (t1 )hj-I-1...js+2 (t )dt1dts+1
ts + 1
T t;+2 t; + i t;
E y j (tk ) ^j; (t1 ^ ^j;-i (t1-1) ••• j; =p+1 t t t t ts+2 ts+i ts
■ j + i fc + J / 0js_i (tS-1) ...
t t t
t2
(t1)dt1...di.-1dt.di^1...dil-1dil ^ ...«ft, =
t
œ
•••j;+ij;j;-i...js+ij;js-i•••ji • (59)
j; =p+1
The equality (59) implies (52), (53).
Step 3. Using Conditions 1 and 2 of Theorem 7, we obtain
p T tl+2
ECjfc-Ji+UijUi-2-ji = (tk )0jfc (tk ) ... ^l (tl+1)0ji+i (tl+i)x ji=0 t t
p ^i+i ti
xE / ^(ti(ti) / ^i-i(ti-i)^ji(ti-i)x ji=o i {
ti-1 t2
x y ^i(t/-2)0ji-2 (ti-2).. .J ^i(ti)0ji (ti)dti... dti-2dti-idti dti+1... dtk
tt
T ti+2
= / ^k (tk ) j (tk ) ...J (tl+i)0ji+i (tl+i)x tt
to ti+1
xE / (ti)^ji(ti) / fai-1(ti-1)^ji(ti-1)x j =0tt
t-1 t2
x J fai(ti-2)0ji-2(ti-2).. .y fa1(t1) j(t1)dt1...dti-2dti-1dtidti+1.. .dtk-
tt
to
— Cjfc...j'l+1j'lj'lj'i-2 •••ji = j i =P+1
- v .
Cjk...ji +1ji ji jl-2 •••ji . (60) (j'ij'i WO j =p+1
Step 4. Passing to the limit l.i.m. in (51), we have (see (33))
P^to
P
lpifn. £ Cjkj Zj;1»... Zjkk' = J'+
P^to
j'ivJfc=0
[k/2] r
^E E II 1{;S2s-1 = 's2s =0}x
r=1 ({{31,32},...,{32r-1>32r }}>{91>...>9fc-2r}) S = 1
{31,32,...,32r-1>32r>91 v">;fc-2r- } = {1>2>...>k}
x l i m \ a c■ 1 T1 J'[S ■ S ]('qi•••i,k-2r) w 1
p_400' -Z—/ jk •••j1 XX {j'32s-1 = j'32s } [sj;1 . . . S jqfc-2r
j'ivjfc=0 s=1
(61)
Taking into account (53) and (60), we obtain for r = 1
P
Cjk...ji 1{j3i = j32 } J [ j . . . ^-2 ]Tqr' 9k-2 =
ji vjfc=0
1{'3i = '32^¿m .E E Cjk...ji
j31 =P+1 ji >...jqv..jfc=0
9=31 >32
xj' j ... v2 2)+
p 1
1{g2>g1 + 1} x
j3i= j32
+ 1{¿з1 = ¿32 =0}
1
U\m- E tJ'j.-J
p^to z—* 2
Ji,...,jq,...,Jfc=0
9=31,32
1
(j32 j31 )^0);j31 = j32
{g2=g1 + 1}X
^^ •••¿9k_2 ) j91 ' ' ' 0j9fc_2 ]T,t
OO
1
.m
*31- ^"J p^to
{¿3i = ¿32 =0}J
^m- E E Cjk•••j1
j31 =P+1 j1 ,...,j9,...,jfc = 0
9=31,32
1{g2=g1 + 1}X
j31 j32
xj ..j fe ••^)
1
^ = ¿32 =0}
liS' E E Cjk-j1
j31 =P+1 j1,j9 ,...,jk = 0
9=31,32
X
j31 j32
XJ [j .j • J9'_2 ' +
p
1
j1,...,j9,...,jfc=0
9=31,32
(j32 j31 )^0);j31 = j32
1{g2=g1 + 1}X
X J [0j91 . . . 0j9k_2 ]T,t
n^r • • ¿9k_2 )
(62)
1 -1
2
{g2=g1+i}J ^TjT^^ + ^31 = ¿32 JlT,t
¿1 • • • ¿k)[g1]
l.i.m. RTPtg1;g2(i9l• • • ¿9k_2) (63)
P^to
w. p. 1, where J [^j^""'*)[g1] (g = 1, 2,..., k - 1) is defined by (36),
R
^51,52 ^91 •••¿9k_2 )
T;t
Ec (p) c jk •••j9•••j1
j1,...,j9,...,Jfc =0
9=31,32
J [0j91 . . . 0
q=g1;g2
j91 j9k_2 T,t
](% •••¿9k_2 )
Let us explain the transition from (62) to (63). We have for g2 = g1 + 1
p
1
{;31 = ;32 =0}
1
Liim- £ -/'.n-J P^TO z—* 2
j1,...,j;,...,jfc=0
;=3i,32
x
(j32 j31 = j32
xJ' [ j ...Sj;k-2 ]T,t
(;;i •••;;fc-2)
-1
2 {»fli-
LLm. Cjk.ji
j1,...,j;,...,jfc=0 ;=3i>32
x
(j32 j31 )^0,j31 = j32
xz00)j ' [j ...Sj.k-2 ]Tr,!,k-2'
PP
1
j1,...,j;,...,jfc = 0 jmi =0 ;=3i>32
^ E E Cjk . j
x
(j32 j31 )^jmi ,j3i = j32
j J' [ j ...Sj;k-2 ]T,t
(;;i •••';fc-2)
PP
-1
2 is-
lim- E E Cjk . ji
j1,...,j;,...,jfc=0 jmi =0 ;=3i,32
x
(j32 j31 ) ^jm i ,j31 j32
x J' [j j ...Sj;k-2 ]T°t;i )
(64)
w. p. 1,
2
(65)
where
C
Cjk ...ji
(j32 j31 )^jm1 ,j31 = j32 ,32=ffi + 1
T
r31+3
r31+2
= (tk )Sjk (tk ) ... (tffi+2)0j31+2 (t3i+2M fagi+1(tgi (tgi j (tgi)x
j31+2 '
t t t
^i t2
x r (tgi-1)0jsi-i (tgi-1)... ^1(^1) j (t1)dt1.. .dtgi-1dtgi dtgi+2 ...dtk,
T T
-(°) _
jmi I 0jmi '
tt
j) = I j (t)dw(°) = I j (t)dT =<!
if jm =0
0 if jmi = 0
T ) =
y/T^t'
The transition from (64) to (65) is based on (33).
By Condition 3 of Theorem 7 we have (also see the property (32) of multiple Wiener stochastic integral)
lim M< i Rp)gi'g2(iqi"^-2)x2'
piœ
T,t
£ C
jfc-jq ...ji
piœ
Ji,...,jq,...,jfc=0
q=si,32
= 0,
q=gi,g2,
where constant K does not depend on p.
Thus
1
{ifli = ^ ^ E Cjk-ji j = j»2}J' [ j . . . ^j'qfc-2J
(iqi •••iqk-2 1
T,t
j'i,.-,j'fc=°
2'
1
{g2=gi+1}J rr 'JT,t
(ii •••ife ) [gi]
w. p. 1.
Involving into consideration the second pair {g3, } (the first pair is {gi,g2}), we obtain by analogy with (62) for r = 2
n 1{ig2s-i = ^ =°}lî;i;m- E Cjk...ji n 1{j's2S-i = ^}X
s=1
jiv-Jfc =° s=1
](i9i •••i?fc-4 1
jqi Tjqfc-4J T,t
1
2
p
1
2
2
p
II 1{'32s-i = '32s =0}x
s=1
1
j1,...J;,...Jk=0 ;=31>32>33>34
1
{g2s=g2s-1 + 1}
(j32 j31 M-)(j34 j33 MOj3i = j32 ,j33 = j34 s=1
1 to
2 53 ^ ••••'
j31 =P+1
1
(j34 j33 M-)j3i = j32 ,j33 = j34
{g4=g3 + 1}
1 to
2 53 ^ ••••'
j33=p+1
(j32 j31 ) ^ (•) ,j31 = j32 ,j33 = j34
1{g2=g1 + 1} +
OO OO
+ £ £ j.
j'33 =p+1 j'31 =p+1
ji
J' [ j . . . S
1 (;;i •••';k-4)
j;i rj;k-4J T,t
= (66)
j3i j32 ,j33 j34 ,
7 II M92s=92s-1 + 1}J
ii...ik)[s2,si] + T,t +
s=1
^ 1{'32s-i = '32s =0}lPi-m- RT,t
(P)g1
s=1
P^to
(67)
w. p. 1, where g3 d=f s2, g1 d=f s1? (s2,s1) G Ak,2, J[fa^]^"^)[s2'si] is defined by (36) and Ak,2 is defined by (37),
R
';i •••';k-4 )
T,t
£
C
(P)
jk ...j; ...ji
j1 ,...,j;,...,jk =0
;=3i>32>33 >34
-SW C
(P)
jk ...j; ...ji
- S2 ^ C
(P)
jk ...j; ...ji
x
q=ffi>32>33,34
xJ' [ j — jk-4 ]T,t
(';i •••';k-4)
2
P
2
Let us explain the transition from (66) to (67). We have for g2
g4 = ga + 1
= g1 + 1,
1
£ 7 <"./-,/ P^TO z—» 4
j1 >... >j; >... >jk =0
;=31>32>33>34
X
(j32 j31 M-)(j34 j33 M0j3i = j32 ,j33 = j34
X
n 1{'32s-i = '32s =0} J'[ j . . . ^j;k-4]T,t
(';i •••';k-4)
s=1
1
P
= i'A»' £
j1 >.. >j; >... >jk=0 ;=31,32,33,34
X
(j32 j31 )^0(j34 j33 )^0,j31 = j32 ,j33 = j34
2
X
■{i32s-1 = '32s =0}Z00)Z00)J' [ j . . . 0j;k-4 ] T,* ;k 4 )
s=1
Ä £ £ ^
j 1 ,...,j; ,...,jk=0 jm1 ,jm3 =0 ;=31>32>33>34
X
(j32 j31 )^jmi (j34 j33 )^jm3 ,j3i = j32 J33 = j34
2
x n 1{'32s-i = '32s =0j j* J' [ j . . . ^j;k-4 ]T,t
s=1
(';i •••';k-4)
Ä £ £
j 1 ,...,j; ,...,jk=0 jm1 ,jm3 =0
;=31,32,33,34
X
(j32 j31 )^jmi (j34 j33 )^jm3 ,j3i = j32 J33 = j34
X
n 1{'32s-1 = '32s =0} J' [ j j ^;i
, ](00';i •••';k-4)
. . S j;k-4 ]T,t
(68)
s=1
±J[^*>]^"ifc)[a2'ai] w. p. 1.
(69)
The transition from (68) to (69) is based on (33).
Note that
C
Cjfc...ji
= C
( j32 jg i1 ^jm i Jjs i = jS2
jk ...ji
( jg i jg i 1 ^jm i ,jg i = jg2
is the Fourier coefficient, where g2 = g1 + 1. Therefore, the value
C
Cjk ...ji
(j32 jSi )^jmi (j34 j33 )A,jm3 !jSi = jS2 'j33 = j34
= C
= Cifc •••j1
(jS1 j31 )Ajm1 (jS3 jS3 )Aim3 = ^ 'jS3 = jS4
is determined recursively using (22) in an obvious way for g2 = g1 + 1 and
g4 = g3 + 1
By Condition 3 of Theorem 7 we have (also see the property (32) of multiple Wiener stochastic integral)
lim M < i R
piœ
)(P)gi,g2,g3,g4(iqi •••iqk-4 )
T,t
<
< K lim V
piœ ^—'
C
(p)
jk ...jq ...ji
+ SW Cj
(p)
jk ...jq ...ji
ji,jq ,...,jk =0 q=Si,S2,S3,S4
q=gi,g2,g3,g4
+
q=gi,g2,g3,g4/
+ S^ C
(p)
jk ...jq ...ji
= 0,
q=gi,g2,g3,g4
where constant K is independent of p. Thus
1{
s=1
{iS2s-i = Ss =°}
l-i-m- E Ck.ji A 1
piœ ^ z—' -1- -1-
jill {jS2s-i = j»2s }
X
jivJk =° s = 1
(' ' ) 1 2
X ./[(^ . . . (f)Jqi_ Jy®1 9fc-4 = ^ n 1{g2S=g2S-l + l} j
s=1
T,t w- p- 1,
where g3 = S2, g1 = S1, (^2,^1) G A^, J [^TV^^2'511 is defined by (36) and Ak,2 is defined by (37).
2
2
2
Involving into consideration the third pair {g6,gs} ({gi, is the first pair and {g4,g3} is the second pair), we obtain by analogy with (66) for r = 3
n 1{ig2s-1 = ig2sS ...jin 1{jg2s-1 = jg2s}x
s=1
p^œ
j'ivj'fc =0
s=1
l(i9i •••i?fc-6 )
j9i Vj9fc_6J T,t
II1
s=1
{iS2s_1 = »32« =0}
X
^ £ (
j1,...,jq,...,jfc = 0 9=31,32,33,34,35,36
X
(j32 j31 W0(j34 j33 W)(j36 j35 M0j3i = j32 ,j33 = j34 j35 = j36
3
X n 1{g2s=g2s_1 + 1}
s=1
OO
22 ^'fc-ii j31 =p+1
(j34j33 W0(j36 j35 M0j3i = j32 j33 = j34 ,j35 = j36
1{g4=g3 + 1}1{g6=g5 + 1}
1 œ
j'33=p+1
1
(j32 j31 W0(j36 j35 M-)j3i = j32 'j33 = j34 ,j35 = j36
{g2=gi + 1} 1{g6=g5 + 1}
OO
22 53 ^
j'35 =P+1
(j32 j31 )^0)(j34j33 W0j3i = j32 'j33 = j34 ,j35 = j36
1{g2=gi + 1}1{g4 =
1
OO OO
+ 2 S S ^ifc-ii
j'33 =p+1 j'31 =p+1
(j36 j35 W0j3i = j32 'j33 = j34 ,j35 = j36
1{g6=g5 + 1} +
3
3
1 œ œ
+ 2 S S ( '■" '"•'
j'35 =p+1 j'31 =p+1
(j34 j33 W0j3i = j32 ,j33 = j34 j35 = j36
1{g4=g3 + 1} +
OO
OO
+5 E E c
jk ...ji
j'35 =P+1 j'33 =P+1
1
(j32 j31 M0j3i = j32 'j33 = j34 ,j35 = j3(
{g2=gi + 1}
oo oo
oo
E E E Cjk...j'i
jfl5 =p+1 jfl3 =p+1 jgi =p+1
X
jSi j$2 'jfl3 j^4 'jfl5 jfl6
j (iqi •••iqk-6)
jqi V^jqk-^ T,t
1 3
23 II ■'■{328=328-1 + 1}^
s=1
(ii ...ik )[s3,s2,si] T,t
+
+ n 1{ig2s-i = ig2s =°}l-i.m- RT,t
(P)gi,g2,...,g5,g6(iqi •••iqk-6 )
s=1
piœ
w. p. 1, where g2i-i = Si; i = 1, 2,3, (s3,s2,si) G AM, J(k)]k)[S3'S2'Si] is
defined by (136) and Ak,3 is defined by (137),
p
R
iqi •••iqk-6)
T,t
E
-C
(p)
jk...jq...ji
ji,...,jq,...,j'k=0 q=Si,S2,...,S5,S6
+
q=gi,g2,...,g5,g6
(p)
jk ...jq ...ji
+ S^ C
(p)
q=gi,g2,...,g5,g6
jk ...jq ...ji
+
q=gi,g2,...,g5,g6
C
(p)
jk ...jq ...ji
q=gi>g2v,g5,g6
—S3S1 < C
(p)
jk ...jq ...ji
q=gi,g2,...,g5,g6
c(p)
— <Cjk...jq ...ji
—S2S1 < C
(p)
jk ...jq ...ji
q=gi,g2,...,g5,g6 (iqi •••iqk-6)
q=gi,g2,...,g5,g6
*qi ••• "qk-6 yjqi ' V/jqk-^ T,t
By Condition 3 of Theorem 7 we have (also see the property (32) of multiple Wiener stochastic integral)
lim M R
piœ
,(p)gi,g2,...,g5,g6(iqi •••iqk-6 )
T,t
<
3
2
< K lim V
P^to ^—'
c (p)
Cjfc •••j'9 •••jl
+ I Si < Cj(p)
jk •••j9 •••j1
J1,...,j9,...,Jfc=0 9=31,32 ,...,35,36
q=01>02,-,05,06
+
q=gl,g2v,g5,g6y
+ S^ C.
(p)
+ S^ /7j
(p)
+ S3S1 < Cj
(p)
jk •••j9 •••j1
jk •••j9 •••j1
2
jk •••j9 •••j1
2
+
q=01>02,-,05,06 +
q=gl,g2,-,g5,g6
(p)
q=gl,g2v,g5,g6
+ I S3S^ Cjk •••j9 •••jl
q=gl>g2v,g5,g6
+ S2S^ Cj
(p)
jk •••j9 •••jl
— 0,
q=gl,g2v,g5,g6
where constant K does not depend on p. Thus
+
lim- n = ¿32s =0}^^ Cj'k-j-1 n 1{j32s_1 = j32s }X
s=1
2s_1 32s ' ^ p^TO
j'lv-Jk =0
s=1
XJ' [0j9l •••0.
-i^l •••¿9k_6 )
1
j91 j9k_6 T,t
.J [^(k) ]
(¿^•¿k )[S3,S2 ,Sl]
9311 -{g2S=g2S-i+iH ir 'JT,i ..... w. p. L
2 s=1
2
2
2
2
2
3
3
3
where g2i-1 = s*; i = 1, 2, 3, (53,-82,-81) G AM, J[^W]^"**'52'811 is defined by (136) and Ak,3 is defined by (1371).
Repeating the previous steps, we obtain for an arbitrary r (r = 1, 2, ..., [k/2])
r p r
n 1{ig2s-1 = ig2s =°}li;i;m. E Cik ..1 n 1iiS2s-1 = i»2s }X
5=1 i1'...'ik =0 s=1
r
X J' . . . 0iqfc-2r ]T,t •••гqfc-2r ) = 11 1iiS2s-1 = i*2s =0}X
s=1
P
xl.i.m.
p^TO ^—^ 2
j1 ,...,j;,...,jk=0 ;=31,32 ,...,32r-1,32r
1
nrCjk-3i
X
X
(j32 j31 )^(^)...(j32r j32r-1 = j32 '•••'j32r-1 = j32r
(';i •••';k-2r)
Ii 1{g2s=g2s-1 + 1} J' [ j . . . Sj;k-2r ] T^t
+
s=1
r
+ II 1{.32,-1 = '32, =0)^ rt?)91'92'"'92'-1'92'<"« ) = (70)
s=1
1r
s=1
+ 1! 1 = .32. =0}l;iim. RTf'"-1'92'-2'1 (71)
"^s-i = '32s =0}
s=1
def
w. p. 1, where g2'—1 = S'; i = 1, 2,..., r; r = 1, 2,..., [k/2], (sr,..., s1) G Ak,r, J[fa^;^)[sr'•••'si] is defined by (36) and A^ is defined by (37),
r
R
(P)9i'92'...'92r-1'92r ('
T,t
-2r )
E
(—1)r C
(P) jk •••j; •••ji
Ji ,...,J;,...Jk =0 ;=31,32,...,32r-1,32r
+
q=9i'92'...'92r-1'92r
il=1
+
q=9i'92'...'92r-1'92r
r
+(—1)r—2 £ Siisj (7jp^.j,...ji
11 ,i 2 = 1 I i1>i2
+
q=9i'92'...'92r-1,92r
+(—1)^X3 SiiSi2 ...Si
(7 (P) i'-1 ] Cjk •••j; •••ji
i1,i2,...,ir-1 = 1 11 >i2 >... > i'_i
x J [Sj;i . . . ]
q=9i'92'...'92r-1'92r (';i •••;;k-2')
/
X
;r-t;k-2r
j;k-2r ^ T,t
(72)
r
r
Let us explain the transition from (70) to (71). We have for g2 = g1 + 1,
, g2r = g2r-1 + 1
p
l.i.m.
p^TO ^—' 2
j1,...,j9,...,jk = 0 9=31,32,...,32r_1,32r
1
^ v 9,.Cjk—ji
X
(j32 j31 0 ••• ( j32r j32r_1 M0j3l = j32 '•••'j32r_1 = j32r
X
n ^^l = ¿32s =0} J [0j91 . . . 0j9k_2r ft
^91 •••¿9k_2r )
s=1
—l.i.m.
2r p^TO
Cjk •••jl
j1 ,...,j9 ,...,jk =0 9=31,32,...,32r_1,32r
X
X
(j32 j31 ^^-C^r j32r _ 1 )^0,j31 = j32 '•••'j32r_1 = j32r
(«91 •••¿9k_2r )
Ii 1{i32s_1 = ¿32s =0} (Z00J J [0j91 . . . 0j9k_2r ] T91
s=1
1,. = —l.i.m.
2r p^TO
1{
j1,...,j9,...,jk =0 jm1 ,jm3 •••jm2r_1 =0 s = 1 9=31,32 ,...,32r_ 1,32r
Ks^ = ¿32s =0}
X
xC,
jk...j1
X
(j32 j31 )Ajml •••(j32r j32r_ 1 )^jm2r_1 'j31 = j32 '•••'j32r_1 = j32r
XC?(0) C7(0) ...Cr J' [0j .^j ^ -¿9k_2r)
jm2r_ 1 LTj9l r j9k_2r J T ,t
—l.i.m.
2r p^TO
1{
j1,...,j9,...,jk =0 jm1 ,jm3 •••jm2r_1 =0 S = 1 9=31,32 ,...,32r_ 1,32r
Ks^ = ¿32s =0}
X
C
jk...j1
X
(j32 j31 )Ajml •••(j32r j32r_ 1 )^jm2r_1 'j31 = j32 '•••'j32r_1 = j32r
X J [0jml 0jm3 . . . 0jm2r_l 0j91 ^ ^ ^ 0j9k_2r
«»••«¿.l )
(73)
Ij 2r
^••¿k)[sr'•••'sl] w. p. 1.
(74)
r
The transition from (73) to (74) is based on (33). Note that
C
=C
(j32 j31 ) ^jm i ,j31 = j32
jk ...ji
(j31 j31 ) ^jm 1 ,j31 = j32
is the Fourier coefficient, where g2 = g1 + 1. Therefore, the value
C
jk-ji
(j32 j31 )^jmi •••(j32dj32d-1 )^jm2d-1 'j3i = j32 '•••'j32d-1 = j32d
=C
= Cjk ...ji
(j31 j31 )^jmi •••(j32d-1 j32d-1 )^jm2d-1 'j3i = j32 '•••'j32d-1 = j32d
is determined recursively using (22) in an obvious way for g2 = g1 + 1, ..., g2d = g2d-i + 1 and d = 2,..., r.
By Condition 3 of Theorem 7 we have (also see the property (32) of multiple Wiener stochastic integral)
lim M ^ i R,
P^to [
(P)9i'92'...'92r-1'92r (';1 •••;;k-2r A2 . ^ •T't 1 > <
P
< K lim V
P^to ^—'
(7 (P) Cjk ...j; ...ji
j1 ,...,J;,...,Jk = 0 ;=31,32,...,32r-1,32r
+
+ E ( Cj(p)..j;...ji
ii=1
q=9i'92'...'92r-1'92r 2
+
q=9i'92'...'92r-1'92r
+ E ( Sii"M Cj(p)..j;...ji
i1 ,i2 = 1 i1>i2
+
q=9i'92'...'92r-1'92r
+ E (Si1 Si2 ...Si'-^ jj; ...ji
i1,i2,...,i'-1 = 1 ii >i2 >.. .>i'-1
q=9i'92'...'92r-1'92r
/
= 0,
where constant K does not depend on p.
2
r
2
r
2
r
So we have
n 1{i32 _1 = i32s .E Cjk ...j! II 1{j32s_1 = j32s }X
s=1
ji,...,j'fc =0
s=1
X J [0jqi • • • 0j9fc_2r ]T,t
(:91 "^q^r )
2>- n ^-{92s=g2s-l + l}J
(i 1 ...»k ) [sr ,...,Si] -1
s=1
(75)
def
where g2i-1 = s«; i = 1,2,...,r; r = 1,2,..., [k/2], (sr ,...,s1) G Ak,r J[^(k)]Ti1t-ifc)[Sr'-'si] is defined by (36) and Ak,r is defined by (33).
Note that
E
({{31,32>,...,{32r_1,32r }},(91,...,9k_2r }) {31,32 ,32r_ 1,32r ,91 ,...,9k_2r} = {1,2,...,k}
A
g1,g3,...,g2r_1
g2=g1 + 1,g3=g2 + 1,...,g2r=g2r_1 + 1
AS1,S2,...,Sr ,
(sr ,...,s1)eAk,r
(76)
where A
g1,g3,--.,g2r_15 ^ XS1
, ASl;S2v..;Sr, are scalar values, g2i-1 = s«; i = 1, 2,...,r; r =
1, 2,..., [k/2] , Ak,r is defined by (33).
Using (61), (75), (76), and Theorem 9, we finally get
p
k
Ipl-m. £ 6V..j1lI j > = 1^ E Cj(:1)...Cj(
p^œ
j1,...,j'k =0 1=1 [k/2]
= l.i.m.
p^œ
(:1) X:k) jk
j1,...,j'k=0
= J
(:1...:k) T,t
1
53 o»' 53
1 2
=1 (sr ,...,s1)eAk,
J
(:1...:k)[sr,...,S1] = j*[^(k)jC:1...:k) (77)
T,t
k,r
w. p. 1, where J[^^V"**)[sr'-'si] is defined by (36). Theorem 7 is proved.
Let us make a number of remarks about Theorem 7. An expansion similar to (29) was obtained in [57], where the author used a definition of the Stratonovich stochastic integral, which differs from (17). The proof from [57] is somewhat
p
r
r
r
p
simpler than the proof proposed in this article. However, in our proof, we essentially use the structure of the Fourier coefficients (30) corresponding to the kernel K(ti,... ) of the form (4). This circumstance actually made it possible to prove Theorem 7 using not the condition of finiteness of trace series, but using the condition of convergence to zero of explicit expressions for the remainders of the mentioned series. This leaves hope that it is possible to estimate the rate of convergence in Theorem 7 (see Theorems 14-17 below).
Note that under the conditions of Theorem 7 the sequential order of the series
œ œ œ œ
...
J'fl2r-1 =P+i jS2r-3 =P+i J'fls =P+i j'si =P+i
in (23) is not important. We also note that the first and second conditions of Theorem 7 are satisfied for complete orthonormal systems of Legendre polynomials and trigonometric functions in the space L2([t, T]) (see the proofs of Theorems 11-13 below). Note that in the proofs of Theorems 4-6, 11-13 the conditions of Theorem 7 are verified for various special cases of iterated Stratonovich stochastic integrals of multiplicities 2 to 5 with respect to components of the multidimensional Wiener process.
Taking into account the modification of Theorem 1 for the case of integration interval [t, s] (s G (t,T]) of iterated Ito stochastic integrals (see Theorem 1.11 in [13]), we can formulate an analogue of Theorem 7 for the case of integration interval [t, s] (s G (t,T); the case s = T is considered in Theorem 7) of iterated Stratonovich stochastic integrals of multiplicity k (k G N).
Denote
j.j (s)
def
def
£ £ ■■• £ £ eu..*(s)
jS2r-1 =P+1 jS2r-3 =P+1 J'fls =P+1 j'si =P+1
jS1 j32 '•"'j32r-1 j32r
and introduce the following notation
M (s)
def 1
1
q=g1,g2,...,g2r-1,g2r
{g2i=g2i-1 + 1|
oo oo
E E
jS2r-1 =P+1 jS2r-3 =P+1
••• E E ••• E E (s)
jg2I+1 =p+1 jg2l-3 =p+1 j'ss =p+1 js1 =p+1 where l = 1, 2,..., r,
Cjfc...j1 (S) is defined by analogy with (21),
(jS2i jS2i-1 WO
S Î2
C** (s) = J ^(tk)* (tk) • • J 01(t1)0j1 (t1)dt1 • • .dtk• (78)
t t
Theorem 10 [13], [48], [49]. Assume that the continuously differentiate functions ^¿(t) : [t, T] ^ R (l = 1,..., k) and the complete orthonormal system of continuous functions {4>o{x) = 1/y/T — t) in the space Lo{[t,T}) are such that the following conditions are satisfied:
1. The equality
s OO s t"2
i J = E J fyfoWjfo) J ^lih^.jihyihdto (79)
t j =0 t t
holds for all s £ (t, T], where the nonrandom functions $i(r), $2(r) are continuously differentiate on [t,T] and the series on the right-hand side of (79) converges absolutely.
2. The estimates
(t )$1(r )dr
<
U/2+a
(tf)^)^
<
7'1/2+a '
2
s
s
oo
<
*3(s)
pß
^ J $2(T)0j (r)J (tf)dtfdT j=p+1 t t hold for all s,t such that t < t < s < T and for some a,^ > 0, where $1(r),
$2(r) are continuously differentiate nonrandom functions on [t,T], j,p G N,
and
s s
/ (t)^2(s,t)| dT < oo, / |^3(t)| dT < oo
for all s G (t,T).
3. The condition
p
lim
E
Sh Sl2 <Cj(;p)..jq ...j1 (s)
Ji,...,jq,...,jk =0 9=31,32,...,32r_1,32r
= 0
q=g1,g2,...,g2r_1,g2r
holds for all possible g1 ,g2,... ,g2r-1,g2r (see (H3^)) and l1, l2,..., such that l1, l2,..., G {1, 2, ..., r}, /1 > l2 > ... > d = 0,1, 2,... ,r - 1, where r = 1, 2,..., [k/2] and
ShSh •••£/„ 4 <Cj(:;)..jq..j(s)
= jj... (s)
q=g1,g2,...,g2r_1,g2r
for d = 0.
Then, for the iterated Stratonovich stochastic integral of arbitrary multiplicity k
<Î2
J *
srk) = ^(tk)• • / ^1(t1)dwt(:1)•••dwt(:k),
(80)
the following expansion
J vtJ
(h-H )
=]às- E Cjk- wii zj:
(:i )
j1,...,jk=0
1=1
that converges in the mean-square sense is valid, where s G (t,T), Cjk...j(s) is the Fourier coefficient (78), l.i.m. is a limit in the mean-square sense,
s
2
s
k
p
il,..., ik = 0,1, • • •, m,
T
C,<:) = j (T )dw
T
are independent standard Gaussian random variables for various i or j (in the case when i = 0), wt0) = t.
In Sect. 2.1.2 of the monograpth [13], the following formula is proved
t
1
2
(t1 )^2(t1)dt1 = ^ Cjj,
(81)
j=0
where
T t2
Cjj = J fa^j (¿2^ ^i(ti)0j (ti)dti dt2, t t
(x)}°==0 is a complete orthonormal system of Legendre polynomials or trigonometric functions in the space L2([t,T]), the functions (t), ^2(t) are continuously differentiable at the interval [t,T].
Moreover [13] (Sect. 2.1.2), the following estimate
oo
E Cjj j'=p+1
<
c p
(82)
holds under the above assumptions, where constant C does not depend on p. The relations (81) and (82) have been modified as follows [13] (Sect. 2.7,
2.9)
1
2
oo
2 j =0
(83)
oo
E Cjj (s) j =p+1
<
C
1
p 1(1 - z2(s))
1/4
+ 1
(84)
where (83) holds for the case of Legendre polynomials or trigonometric functions and (84) holds for the case of Legendre polynomials, s G (t, T) (s is fixed, the
case s = T corresponds to (81) and (82)), constant C does not depend p, the functions ^1(t), ^2(t) are continuously differentiable at the interval [t,T],
t2
j (s) = 02(t2)0j (t2M 01(t1)0j (t 1 ) dt 1 dt2,
z(s) =
T + A 2
s
2 / T-t
(85)
For the trigonometric case, the estimate (84) is replaced by [13
oo
E Cjj (s) j =p+1
C < —.
P '
(86)
where s £ (t,T), constant C does not depend on p.
Note the well known estimate for the Legendre polynomials
|Pj(y)l <
K
v7+T(i-y2)1/4
, y £ (-1, 1), j £ N,
(87)
where Pj (y) is the Legendre polynomial, constant K does not depend on y and j •
Using (87), we obtain the following useful estimates for the case of Legendre polynomials [13] (Sect. 1.7.2, 2.2.5)
t )0j (t )dT
<
C
1
T
0(t )0j (t )dT
<
j \ (1 - (z(x))2)1/4
C7 1
+1 ,
0(t )0j (t )dT
<
C
1
+
+ 1 ,
1
j I (1 - (z(x))2)1/4 (1 - (z(v))2)1/4
+1
(88)
(89)
(90)
where j £ N, z(x),z(v) £ (—1,1), x,v £ (t,T), v < x, ^(r) is a continuously differentiable function at the interval [t,T], constant C does not depend on j.
s
x
x
For the case of trigonometric functions, we note the following obvious esti mates
x
C
T
T(t
t (t )dT
t (t )dT
<
<
J '
C J '
C
< —,
J
(91)
(92)
(93)
where j £ N, x, v £ (t, T), v < x, the function ^(t) is continuously differentiable at the interval [t, T], constant C does not depend on j.
x
5 Expansion of Iterated Stratonovich Stochastic Integrals of Multiplicity 3
In this section, we present a simple proof of an analogue of Theorem 5 based on Theorem 7. In this case, the conditions of Theorem 5 will be weakened.
First, we show that the equalities
t2 œ t2 T
\ [ (r)$2(r)dr = [ $2(r)^-(r) [ m^dedr, (94)
2
ti j=0 ti ti
t2 oo
I $!(r)$2(r)dr = 53 y J $2(r)^(r)drd05 (95)
ti j=0 ti e
hold for all ti,t2 such that t < ti < t2 < T, where (x)}°=0 is a complete orthonormal system of Legendre polynomials or trigonomertic functions in the space L2([t,T]), the nonrandom functions (t), $2(t) are continuously differentiable on [t,T] and the series on the right-hand sides of (94), (95) converge absolutely.
From (83) we get
tl tl T
^J $i(T)$2(r)dT = ]r J $2(t)^(t) J $I(6%(6>)g№, (96) t j=0 t t t2 to T
2 oo 2
^J (r)$2(r)dr = E J $2(r)^(r) J $i(6%(0)g№. (97) t j=0 t t
Subtracting (96) from (97), we obtain
t2 oo t2
/W rt rt
$i(r)$2(t)dT = ^ / $2(t(t) / (9)d9dT
i j Qiyi )v>2vi = Z^ J ^'i mv) j t1 j=011 t
t2 t1 W i i
= E / ^2(t(t) / $i(0)&(0)d0dr+
j=0 tl t
t2 T
OO 2
+ E / ^2(t(T) / $i(0)<fe(9)d9dT. (98)
j=0 ti ti
Generalized Parseval's equality gives
t2 t1 W 2 1
E / ^M^ (T )dW $i(0)& (0)d0
oo
j=0 tl t T T
= E I l{t1<T<t2|^2(T (T )dW l{0<t1}$l(0)0j (0)d0 = j =0 t t T
= J 1{t1<T<t2}^2(T )1{T<t1}$i(T )dT = 0. (99)
t
Combining (98) and (99), we obtain (94). The equality
t2 T t2 t2
y $2(t )0j (t ) y $i(0)0j (0)d0dT = j (9) j $2(t )0j (t )dTd0
t1 t1 t1 e
completes the proof of (95).
Theorem 11 [48], [49]. Suppose that (x)}°=0 is a complete orthonormal system of Legendre polynomials or trigonometric functions in the space L2([t,T]). Furthermore, let ^i(t),^2(t),^3(t) are continuously differentiable nonrandom functions on [t,T]. Then, for the iterated Stratonovich stochastic integral of third multiplicity
* T * is * ¿2
j*[^3)]Tf3) = y ^y ^y ^i(t1)dwi;i)dwi22)dw(ss) (100) t t t
the following expansion
p
j-i^'Ist'E Cj.cw
jl ,j2 ,j3=0
that converges in the mean-square sense is valid, where i1, i2, i3 = 0,1,... ,m,
T ¿3 ¿2
Cjsj2ji = y ^(^j(*)/^(^j(^2^ y (ti)dtidt2dt3 t t t
and
T
V
j = j (T )dw«
t
are independent standard Gaussian random variables for various i or j (in the case when i = 0); another notations are the same as in Theorem 1.
Proof. As follows from Sect. 4 (see (E1HI81), (86), (ESI), (89), (21), (92)), Conditions 1 and 2 of Theorem 7 are satisfied for complete orthonormal systems of Legendre polynomials and trigonometric functions in the space L2([t,T]) (a = 1/2, ß = 1). Let us verify Condition 3 of Theorem 7 for the iterated Stratonovich stochastic integral (100). Thus, we have to check the following conditions
p / œ \ 2
^E E j =0, (101)
j3=0 \ji=p+1 /
We have
p / W \ 2
p^DE cj =0, (102)
j1=0 \j3=p+i /
p W 2
,l™E E Cj.j2j1 = °- (103)
j2=0 \j1=p+i '
2
p W 2
E E
j3=0 \j1=p+i
p I w T ^ ^
^(ta) j(taW ^2) j fe) / 0i(ti) j (ii)dMMt3
j3=^j1=p+i 1 1 1
(104)
2
p / T W t3 t2
E J ^(ta) j(ta) E J ^2) j fe) y 0i(ti)0j1 (ti)dtidt2dt^ <
j3=0 \ t j1=p+i t t
(105)
, t t3 t2
32
< E J ^(ta) j(ta) X) J ^2) j (t2^ 0i(ti)0j1 (ti)dtidt2dta
j3=0 \ t j1=p+i t t
(106)
t / t3 t2
' oo 3 2
= J 032(ta) E J ^Mj My 0i(ti)0j1 (ti)dtidtJ dt3 < (107) t \ji=p+i t t /
K
< — 0 (108) p2
if p ^ oo, where constant K does not depend on p.
Note that the transition from (104) to (105) is based on the estimate (84) for the polynomial case and its analogue (86) for the trigonometric case, the transition from (106) to (107) is based on the Parseval equality, and the transition from (107) to (108) is also based on the estimate (84) and its analogue (86) for the trigonometric case.
By analogy with the previous case we have
2
p / oo
53 ( 53 C333331 31=0 \33=p+1
T t3 t2
53 I 53 /^3) 3(ts^^2) 3^l(tl)03i(tl)dtidt2dt3
31=0 \j3=p+1 t t t
po
po
T
T
T
53 53 J ^l(tl)^3i (ti^ ^2) 3(t2)y ^s(ts)033(ts)dtsdt2dti
31 =0 \33 =p+l t t1 t2
(109)
T T T
p I T o T T
53 / ^l(tl)03i (tl) X J Mt2(t2^s(ts(ts)dtsdt2dtu <
31=0 \ t 33=p+l t1 t2
(110)
T T T
oo / „ oo „ „
< E J ^ (tl) j (tl) J ^2(t2)033 (t2^ ^3(ts)033 (ts)dtsdt2dtl
31=0 \t j3=p+l t! t2
2
T / _ T
oo „
T
/ ^2(tl) E / ^2(t2)0j3(t2M ^l(t3)0j3(t3)dt3dt2 I dtl < (111)
t V33 =p+l t1 t2
K
< — o
p2
(112)
if p ^ oo, where constant K is independent of p.
The transition from (109) to (110) is based on analogues of the estimates (84), (86) for the value
™ T T
53 J ^2) j(¿2) y j(t3)dt3dt2
j3 =p+i ti t2
for the polynomial and trigonometric cases, the transition from (111) to (112) is also based on the mentioned analogues of the estimates (84), (86).
Further, we have
p / o
53 ( 53
j2=0 \ji=p+i
2
2
2
(113)
p / to T ^ H
= E E J ^3) j (h)J Mt2)j (t2)J (t1)dt1dt2dt3
32=0 \j1 =p+l t t t
pi TO T ^ T
= E E / ^2) j (t2) WO j (h)dh ^jfe) j (ts)dts dt2
32 =^31=P+1 { { t2
p / T TO ^ T
= E /^j (t2) E / 01(t1 )031 (t1 )dt1 03(t3)03i (t3)dt3dt2 I < J'2=A t 31 =p+1 t t2
2(114)
TO / T TO T
<E / 02fe) j(t2) E / (t1)031 (t1)dt1 03(t3(t3)dt3dt2
32=0 \ t 31 =p+1 t t2
T f TO t2 T \2
= 02(t2) E / (t1)031 (t1)dt1 / 03(t3)031 (t3)dt3 dt2- (115)
t V31=p+1 t t2 /
The transition from (113) to (114) is based on the estimates (88), (89) and its obvious analogues (91), (92) for the trigonometric case. However, the estimates (88), (89) cannot be used to estimate the right-hand side of (115), since we get the divergent integral. For this reason, we will obtain a new estimate based on the relation [13] (Sect. 2.2.5)
J ^{s)Us)ds=V^tfTTI J Pj{y)iKu{y))dy = t -1
./71 _ f
• (pj+Mx)) - Pj-Mxmw-
2V2J+1
z(x)
T f [ ((PJ+i(y) - Pj-i{y))inu{y))dy I , (116)
2 J
-1
where x £ (t,T), j > p +1, z(x) is defined by (85), Pj(x) is the Legendre polynomial, is a derivative of the continuously differentiable function ^(s)
with respect to the variable u(y),
, , T - t T + t
u(y) = +
From (87) and the estimate |Pj (y)| < 1, y £ [-1,1] we obtain
IP(y)l = (v)l£ • IP3(y)ll_'° < IP3(y)!l-e <
C
jl/2-£/2(l _ y2)l/4-e/4
-74. (117)
where y £ (-1,1), j £ N, and £ is an arbitrary small positive real number.
Combining (116) and (117), we have the following estimate
^l(r )03 (t )dr
<
C
jW2 \ (1 _ z2(s))l/4_£/4
+ 1
(118)
where s £ (t,T), z(s) is defined by (85), constant C does not depend on j. Similarly to (118) we obtain
T
^3(t )03 (t )dt
<
C
1
jW2 I (1 _ z2(s))l/4_£/4
+ 1 ,
(119)
where s £ (t,T), constant C is independent of j. Combining (88) and (119), we have
s T
I ^i(t)03(t)dW ^3(t)03(t)dT
<
<
L
1
j2_/2 I (1 _ z2(s))l/4_e/4
+1
1
(1 _ z2(s))1/4
+ 1 ,
(120)
where s £ (t,T), z(s) is defined by (85), constant L does not depend on j. Observe that
00
Et
1
<
dx
1
3 =p+l
jW2 " J x2_e/2 (1 _ £/2)pl_e/2'
(121)
s
1
Applying (120) and (121) to estimate the right-hand side of (115) gives
p i o V K
E E Cnnn < > 0 (122)
¿2=0 \ji=p+1 / P
if p ^ oo, where £ is an arbitrary small positive real number, constant K is independent of p.
The estimation of the right-hand side of (115) for the trigonometric case is carried out using the estimates (91), (92). At that we obtain the estimate (122) with £ = 0. Theorem 11 is proved.
6 Expansion of Iterated Stratonovich Stochastic Integrals of Multiplicity 4
Theorem 12 [48], [49]. Suppose that {0j(x)}0=0 is a complete orthonormal sys-
'jV^ J}j =0
tem of Legendre polynomials or trigonometric functions in the space L2([t,T]). Furthermore, let ^1(r),..., ^4(r) are continuously differentiable nonrandom functions on [t,T]. Then, for the iterated Stratonovich stochastic integral of fourth multiplicity
* T * ¿2
(4)]Tr4) = / ^4(t4)... i ^i(ti)dwt( ;i}... dwt(44) (123)
tl t4
t t
the following expansion
p
J*№(4,iir4) = 1^- Z Cj4..jij1'...j4'
ji ,...,¿4=0
that converges in the mean-square sense is valid, where i1,..., i4 = 0,1,..., m,
T ¿2
CjWi =j (*)■../* (t1 )j (tl№ ...dt4
t t
and
T
j = / <k (T )dw<->
t
are independent standard Gaussian random variables for various i or j (in the case when i = 0); another notations are the same as in Theorem 1.
Proof. As follows from Sect. 4 (see (ISI])-(IH4), (86), (SB), (89), (91), (92)), Conditions 1 and 2 of Theorem 7 are satisfied for complete orthonormal systems of Legendre polynomials and trigonometric functions in the space L2([t,T]) (a = 1/2, 3 = 1). Let us verify Condition 3 of Theorem 7 for the iterated Stratonovich stochastic integral (123). Thus, we have to check the following conditions
oo
Ji^ E E jj = 0,
p^œ
j3,j4=0 \j'i=P+1
p
p^œ
j2
p
Jim E E jjj ) =
j2 ,j4=0 \j1=P+1
J™ E E Cjij3j2ji = 0,
p^œ
j2 ,j3=0 \j'l=P+1
p
Jim E E Cj4j2j2ji = 0,
p^œ
j 1 ,j4=0 \j2=p+1
p
pJim E E Cj2j3j2ji ) =o,
jl ,j3=0 \j2=p+1 pœ
Am E E Cj3j3j2ji ) =o,
ji ,j2=0 \j3=p+1 œœ
pim ( E E Cj2jij2ji ) =o, j2=p+1 ji=p+1
œœ
Am ( E E Cjijjji) =0, j2=p+1 ji=p+1
124)
125)
126)
127)
128)
129)
130)
131)
p
2
2
2
2
oo oo
üm 53 53 Cj3j3jiji =
p^œ
j3=p+1 ji=p+1
lim ( V C
p^œ
j3j3j1ji
j3=p+1
oo
lim V C \ ^-'
p^œ
j3j3jiji
ji=p+1
lim C
I^rvi \ -^
p^œ
jij2j2ji
ji=p+1
Ü'ij'iWO,
(j3j3W-),
(j2j2W0,
= 0,
= 0,
= 0,
(132)
(133)
(134)
(135)
where we use the notation (121) in (I133l)-(IT351).
Applying arguments similar to those we used in the proof of Theorem 11, we obtain for (124)
pœ
2 / T
2 p I œ
t4
E E Cj4j3jiji I = 53 I 53 J ^4M j (t4^ ^3(t3)0j3 (t3)x
j3 ,j4=0 \ji=p+1 / j3 ,j4=0 \ ji=p+1 t t
t3
t2
x / ^2^2) j (t2M ^i(ti) j (ti)dtidt2dt3dt4 =
(136)
T
t4
I J M^j (t4^ ^3(t3)0j3 (t3) x
j3,j4=0 V t t
OO
t3
t2
x J ^2(t2)0ji(Î2^ (ti) j(ti)dtidt2dt3dt4
ji=p+11 t
oo ( T t4
< 53 / ^4&) j (t4^ ^3fo) j (t3)x j3 ,j4 =0 \ t t
t3
t2
x 53 J WÎ2) j (t2) J ^i(ti)0ji (ti )dtidt2dt3dt4
ji=p+11 t
< (137)
(138)
2
2
2
2
2
2
2
= J I{i3<i4|042(t4)032(t3 )X [t,T ]2
/ 0 t3 t2 \ 2
x E /02(t3)0j1 (t2 )i 0i(ti (ti dt3 dt4 < (139)
\ji=p+1t t /
K
< — 0 (140)
p2
if p ^ oo, where constant K is independent of p.
Note that the transition from (136) to (137) is based on the estimate (84) for the polynomial case and its analogue for the trigonometric case, the transition from (138) to (6) is based on the Parseval equality, and the transition from (6) to (140) is also based on the estimate (84) and its analogue for the trigonometric case.
Further, we have for (125)
p f o \ 2 P i o T t4
E E = E E / ^4(0^4 (t4M 03(t3)0ji (t3)x
j2 ,j4=0 \j1=P+1 ' j2 ,j4=0 yi=P+1 t t
t3 t2 \ 2
x J ^Mj (t2)J 0i(ti)0ji (ti)dtidt2dt3dtj = (141)
tt
P / o T ^
= E E J Mt4)j (t4)J Mt2)j (t2)x
j2,j4=0 \j1=P+i t t
t2 t4 \ 2
xy 0i(ti)0j1 (ti)dt^ 03(t3)0j1 (t3)dt3dt2dtj = (142)
t t2
P / T t4
= E y 04(t4)0j4 (t4^ y 02(t2)0j2 (t2) x
j2j4=0 \ t t
2
oo t2 t4
x E / (ti)dtW ^3(^3) j(^dMM^
<
ji=p+1 t
t2
< E I (^¿4 ^2 (i2)j (t2)x
¿2 ¿4=0 \ t t
x E / W^Ofe (¿1)^1 / ^3(t3)0ji (^dMM^ ¿'l=P+1 t t2
= J !{t2<t4>^4(t4M (t2) x
[t,T ]2
/ o * ^ \2
x E / (t1)dtw ^3(¿3)^1(£3)^3 dt2dt4 < \j1=P+1 t t2 /
K
< - 0 (143)
if p ^ oo, where £ is an arbitrary small positive real number for the polynomial case and £ = 0 for the trigonometric case, constant K does not depend on p.
The relation (143) was obtained by the same method as (140). Note that in obtaining (143) we used the estimates (90), (118) for the polynomial case and (91), (93) for the trigonometric case. We also used the integration order replacement in the iterated Riemann integrals (see (141), (142)).
Repeating the previous steps for (126) and (127), we get
p f o \ 2 P i o T t4
E E = E E / ^(toj (t4M ^3 (t3)0j3 (t3)x
¿2 ,¿3=0 \jl=P+1 ' ¿2 ,¿3=0 y'l=P+1 t t
t3 t2 \ 2
xj ^(^j (t2)y ^(¿Oj (¿OdMMM^ I = tt
P / o T r
= E E I WtOj (i3W «^j (t2)x
¿2 ¿3=0 \j'l=P+1 t t
t2 T \ 2
xj (ti)0ji (t1)dt^y ^4(t4)0ji (t^dMM^ I =
t t3
2
P I T t3
= E / ^(Oj (*)/ WOj (t2)x
j2,jS=0 \ t t
oo t2 T \ 2
x E J (ti)0ji (ti)dt^y ^4(^4 )j (t4 )dMMt3 I <
ji=P+1 t ts /
0 / T ts
< E / 03 (O j (*)/ 02 (t2) j (t2)x ^ ? T ^ 2
x E / WO j (t1 WO^ji (OdtAdtn =
ji=P+1 t ts /
= J l{t2<ts}03 (t3)02(t2)x
[t,T ]2
/ 00 t2 T \ 2
x E / WO j (t1 WO j (0^4 dt2dt3 <
\ji=p+11 tS )
K
< -j 0 (144)
p2
if p ^ oo, where constant K does not depend on p;
P / o \ 2 P / ^
E ( E Cj4j2j2jJ = E E J ^4 (0 j (0 / 03 (Ofe (t3)x ji ,j4=0 \j2=P+1 / ji ,j4=0 \j2=P+1 t t
ts t2 \ 2
WOj (t2^ WOj (O^MMM^ I = tt
P / o T ^
= E E I WO j (O / WO j (t1)x
jij4=0 \vj2=P+1 t t
t4 t4 x 2
x^ WOfe (^2^ WOfe (O^MMM^ ) =
ti t2
P / T t4
= E / (t4^ WOfe (t1)x
¿1,j4=^ \ t t
o r r \2
x 53 / ^2(¿2)^2 (¿2) / ^3(^3)^¿a (t3)dt3dt2dM^ I <
¿2=P+1ti I )
< £ ( /V (¿4)^4 (¿4)/ (¿1)^1 (¿1)x
¿1 ¿4=^ i t
OO t4 t4 X 2
o rt rt
x 53 / ^2(^2)^2 (¿2 ) / ^3(^3)0J2 (¿3)dt3dt2dt1 dt4
¿2=P+1 t1 t2
= J I{t1<t4>^4 (¿4M (¿1)x
[t,T ]2
/ o t4 t4 \ 2
x 53 / ^(¿2)<fe (¿2) / ^(¿3)<fe (¿3)di3diJ dhdi4. (145)
\¿2=P+1tl /2 /
Note that, by virtue of the additive property of the integral, we have
OO t4 t4
o rt rt
/ W^fe(¿2)/ ^(¿3)^2(i3)di3di2 = (146)
¿2=P+1 t1 t2
= 53 J ^3)^2(¿3^ ^2)^2(i2)di2di3-
¿2=P+1 t t
t1 t3 o i 5
- 53 J ^3)^2(¿3^ ^2)^2(i2)di2di3-
¿2 =P+1 t t
o ^ 11
- / ^3(i3)0j2 (¿3)diW «¿2)^2 (¿2)di2. (147)
¿2=P+1t1 t
However, all three series on the right-hand side of (147) have already been evaluated in (140) and (143). From (145) and (147) we finally obtain
P / o \2 K
E E C^nn) <-r-£ 0 (148)
j1,j4=0 \j2 =P+i '
if p ^ o, where £ is an arbitrary small positive real number for the polynomial case and £ = 0 for the trigonometric case, constant K does not depend on p.
In complete analogy with (143), we have for
P / o \ 2 P / ^
E E
= E E / 04 (O0j2 (0 / 03 (0 j (t3)x ji ,js=0 \j2=P+1 ' ji ,js=0 y'2=P+1 t t
ts t2 \ 2
xj WOj (t2^ WOj (O^MMM^ I = tt
P / o T ts
= E E J WO j (O j WO j (t2)x ji,js=0 \j2=P+1 t t
t2 T \ 2
xj WOj WOj (O^tA I =
tt3
P / o T r
= E E J ^(O j (O j WO j (t1)x
ji,js=0 \j2=P+1 t t
ts T \ 2
x J WO0j2 (t2)dt2dt^ y WOj (O^^A I = ti t3
P / T ts
= E y WOj (*)/ WO^ji (t1)x
ji,js=0 \ t t
2
x E J WOj (t2)dt2dt^y WOj (O^^A I <
j2=P+1 ti ts
T ts
oo
< E J ^(O j (O j WO j (Ox
ji,js=0 \ t t
tS T
to 3 „
X ^ J ^(¿2) j (¿2 ^4(^4) j (¿4)^4^1 ^3
j2=p+1 ti is
: J l{ti<ts}^3 (t3 M (t1)x
[t,T ]2
/ o t3 T \2
x / ^(¿2)^2 (¿2)diW ^4(¿4)^2 (¿4^4 dMi3 <
\¿2=p+lt; t3 /
K
< - 0 (149)
if p ^ o, where £ is an arbitrary small positive real number for the polynomial case and £ = 0 for the trigonometric case, constant K does not depend on p.
We have for
p /to \ 2 p / r
E E = E E / ^(¿4) j (t4M ^3 (t3)0js (t3)x
ji ,j2=0 \js=P+1 ' ji ,j2=0 yjs=P+1 t t
ts t2 \ 2
fyfäj (¿2^ ^(¿Oj (¿OdMMM^ I =
tt
/ T T
p /to ^
= E E I WO j (¿0 / ^2) j (¿2)x
ji,j2=0 \j's=P+1 t ti
T T \ 2
t2 ts
p / T T
= E / &)/ «^j (¿2)x
jij2=0 \ t ti
T t \ 2
to T T
x E / ^(tj) j(¿3) / ^(¿4) j(t^MMM^ I < js=p+1i, tS
2
oo ( T T
< E / ^ (t1)0ji 02 (Oj (t2)x
ji,j2 =0 \ t ti
T t \ 2
oo T t
x E / WOj (t3M WOj (t^MMM^
js=P+112 ts
= J !{ti<t2}0i (t1 M (t2) x
[t,T ]2
/ o T T
x E / ^(O j(*)/ WO j(¿4)dt4dt^ dt2dt1. (150)
\js=P+1 t2 ts
It is easy to see that the integral (see (150))
J WOj (Oy WOj (t4)dt4dt3
t2 t3
is similar to the integral from the formula (146) if in the last integral we sub-
stitute t4 = T. Therefore, by analogy with (148), we obtain
P / o \2 k
E E Cnnnn) 0 (151)
j1,j2 =0 \j3 =P+i '
if p ^ o, where £ is an arbitrary small positive real number for the polynomial case and £ = 0 for the trigonometric case, constant K does not depend on p.
Now consider (E30HE32]). We have for (130) (see Step 2 in the proof of Theorem 7)
oo \ 2 / P o \ 2
Ey Cj2j1j2j1 I = 1 Y] Cj2j1j2 j1 I <
j2=P+i j1 =P+i / \j1 =0 j2=P+i /
P / o \ 2
< (p + 1)E( E Cj2j1j2j1 . (152)
j1=0 \j2=P+i /
Consider (128) and (149). We have
P / o \ 2 P / o x 2
53 ( 53 ^'¿^¿^¿^¿4 1 ^^ I 53
¿1=0 N¿2 =P+1 / ¿1j3=0 \?2=P+1
/ \ 2 P / o N 2 K
<
ji =js
v ( V C]mnn < -5-, (153)
¿1j3=0 N¿2 =P+1 '
where £ is an arbitrary small positive real number for the polynomial case and £ = 0 for the trigonometric case, constant K does not depend on p. Combining (152) and (153), we obtain
V V C V<(P+1)^< ^ 0
/ . / , ^ ./:./ ./:./ I ^ 2-e - ^1-e ^ U ¿2=P+1 ¿1 =P+1 /
if p ^ o, where constant K1 does not depend on p. Similarly for (131) we have (see (127), (148))
oo \ 2 / P o \ 2
53 53 ^¿¿¿l I = [ 53 ^¿^¿'l ] <
¿2=P+1 ¿1 =P+1 / Vl =0 ¿2=P+1 /
P / o \ 2
< (p + 1)E E ^¿1 , (154)
¿1=0 N¿2 =P+1 /
P / o \ 2 P / o x 2
53 ( 53 ^¿¿¿l ] = ( 53 ^¿^^l ¿1=0 \72=P+1 / ¿1 ,¿4=0 N¿2 =P+1
P / o \2 K
< E E ^,2,1 (155)
¿1,¿4=0 \72=P+1 /
where £ is an arbitrary small positive real number for the polynomial case and £ = 0 for the trigonometric case, constant K does not depend on p. Combining (154) and (155), we obtain
2
(p + 1)K K1
<
ji=j4
oo oo
E E ( '.' •'• •/• ./ J < 2-e - pl-e > 0 j2=p+1 ji =p+1 ' 1 1
if p ^ o, where constant Ki does not depend on p. Consider (132). Using (60), we get
oo oo
oo oo
oP
53 53 Cjsjsjiji = ^3 53 Cjsjsjiji ^3 53 Cjsjsjiji
js=P+1 ji=P+1 js =P+1 ji=0 js=P+1 ji=0
1o
2 53 ^ 'j-J-J J-
js =P+1
oP
EE Cjsjsjiji,
(jiji WO js=P+1 ji=0
(156)
where (see (21))
C
jsjsjiji
ü'ij'iwo
ts
T t4
= J ^(O j(t4^ WO j(O/ WOWO^MMA ttt
From the estimate (82) for the polynomial and trigonometric cases we get
(157)
OO
53 Cj3j3j1j1 j3=P+i
where constant C is independent of p.
Further, we have (see (151))
\2
P o x 2
Cj3j3 j1j1 I —
(j'mWO
C
< —,
p
Po
Cj
ji=0 js=P+1
<(p+1)53 53 Cjsjsjiji
ji=0 \j's=P+1 2
oo
(p+1) E E
Cjsjsj2ji
ji,j2=0 \j's=P+1
2
<
ji=j2
< (p+i) E E c
........ (_P I I) A' ........ A'i
jsjs j2ji I < O- £ < 1_
ji,j2=0 \j's=P+1
p2
p1
(158)
where constant Ki does not depend on p. Combining (1156)-^IT58), we obtain
oo oo
C
js=P+1 ji =P+1
jsjsjiji I < 1_ £ ' p1 £
K2
< --► 0
2
P
£
2
if p ^ o, where constant K2 does not depend on p.
Let us prove (133l)-(fT351). It is not difficult to see that the estimate (1571) proves (133).
Using the integration order replacement, we have
53 ^¿^¿'¿'l ¿1=P+1 ¿¿J WO
o « 4 2
53 / ^4 (¿4)^3(¿4 W ^2 (¿2)^¿'1 (¿2) / ^(¿Ofe (¿1)di1di2di4 =
¿'l=P+1 t t t o T / T \ t2
= 53 / ^2(¿2) J Ik(¿4)«i4)di4 ^¿1 (¿2) J ^(¿1 (¿1 )di1di2, (159)
¿-1 =P+1 t V t2 / t
00
53 Cjij2j2ji ji=p+1
(j2 J2WO
T t4 ts
TO ft rt rt
= 53 / ^(¿4)j(¿4) / ^(¿3)^2(t3M (ti)^ji(¿1)dMt3dt4 =
ji=p+11 t t
T t4 t4
TO ft rt rt
= 53 /^(¿4)j (¿4 )/ ^(¿1 )j (¿1)/ ^(¿3)^2(i3)di3di1 di4 = ji=p+11 t ti TO T t4 X t4 tA
53 J ^4(¿4) j (¿4^ (¿1) j (¿1) /- / I W^fe)^«4 = ji=p+11 t \t t / TO T / t4 \ t4
53 J ^(¿4^ ^3(i3)^2(i3)di3 j(¿4) ^ ^(¿1) j(ii)di1di4- (160)
ji=p+11 V t / t
TO t tW ti \
53 J ^4(¿4) j(¿4) y W^fe)^ I j(ii)di1di4. (161)
ji=p+11 t V t /
Applying the estimate (82) (polynomial and trigonometric cases) to the right-hand sides of (159)-(161), we get
œ
<
j3=P+1
œ
<
jl=p+1 (j2j2)^(-)
c p:
c p:
where constant C is independent of p. The estimates (I! 351). The relations (I! 24I)-(IT31) are proved. Theorem 12 is proved.
(162)
(163) >3) prove (134),
7 Expansion of Iterated Stratonovich Stochastic Integrals of Multiplicity 5
Theorem 13 [48], [49]. Suppose that (x)}°=0 is a complete orthonormal system of Legendre polynomials or trigonometric functions in the space L2([t,T]). Furthermore, let ^i(t),...,(t) are continuously differentiable nonrandom functions on [t,T]. Then, for the iterated Stratonovich stochastic integral of fifth multiplicity
9fT
= / (t5 ) ... ^l(tl)dwt(il) ... dwt(:s)
(164)
the following expansion J *
T,t
(i1) Z (:5) j5
j1,-j5=0
that converges in the mean-square sense is valid, where ¿1,..., i5 = 0,1,..., m,
T t2
Cj5...ji = WOj (t5) ... (t1 )0ji (t1)dt1 . . . dt5
and
T
Cj° = J j(t)dw
t
(:) T
are independent standard Gaussian random variables for various i or j (in the case when i = 0); another notations are the same as in Theorem 1.
Proof. Note that in this proof we write k instead of 5 when this is true for an arbitrary k (k e N). As follows from Sect. 4 (see (EIHE3), (EE), (SB), (89), (91), (92)), Conditions 1 and 2 of Theorem 7 are satisfied for complete orthonormal systems of Legendre polynomials and trigonometric functions in the space L2([t,T]) (a = 1/2, ^ = 1). Let us verify Condition 3 of Theorem 7 for the iterated Stratonovich stochastic integral (1E4). Thus, we have to check the following conditions
lim
. E . £
jqi Jq2 >j93 =0 VA 1 =P+1
= 0,
jgi =jg2
oo
oo
im E IE E j
jgi =P+1 jg3 =P+1
jl
j91 =
= 0,
(165)
(166)
jgi jg2 >■% jg4
Pim E E Cj , ..
jqi =0 \jg3 =P+1
j1
= 0, (167)
(jg2 jgiW •) >jg i = jg2 >jg3 = jg4 >02==0i ■+1
where ({g1,g2}, {03,04}, {q1}) and ({g1, g2}, {q1, ^3}) are partitions of the set {1, 2,..., 5} that is {01,02,03,04,21} = {01,02,^1,^2,^3} = {1, 2,..., 5}; braces mean an unordered set, and parentheses mean an ordered set.
Let us find a representation for Cj, ^ I. , 1 that will be convenient
K ...j1 ljg1 =jg2 , 02>01+1
for further consideration. Using the integration order replacement in the Riemann integrals, we obtain
T tl+2 tl + 1 t t2
J hk (tk).. .J h/ (t/) J hi-1(ti-1) ...J h1(ix)dt1...
t t t t t
... dt/-1dt/dti+1... dtk =
T tl+2 t;+i tl+1 t; + i t; + i
= J hk (tk) ...J h/+1(t/+1^ h1(t1^ h2(t2) ...J h/-1(t/-1^ h/ (t/)dt/ x
t t t ti t;_2 ti — 1
2
2
P
2
P
xdt/-1... dt2d^dt/+1... dtk =
T tl+2 / t,+ 1 \ tl + 1 tl + 1 tl+1
= J hk (tk) ...J h/+1(t/+1H J hi (t/)dt/ J y ...J h/-1(t/-1)x t t \t / t t1 tl-2
xdt/-1... dt2d^dt/+1... dtk—
T tl+2 t; + 1 t; + 1 t;+1 / t,-1 \
— J hk (tk) ...J h/+1(t/+1^ h1(t1^ h2(t2) ...J h/—1(t/—1) J h/ (t/)dt/ x t t t t1 ti-2 \ t /
xdt/—1... dt2dt1dt/+1... dtk =
T t;+2 / t;+1 \ t;+1
= J hk (tk) ...y h/+1(t/+1n J h/ (t/)dt/ J h/—1(t/—1)...
t t t t t2
.. y h1 (t1)dt1... dt/—1dt/+1... dtk—
t
T tl+2 t;+1 / t;-1 \ t;-1
—Jhk <tk)... j hl+1(tl+1^ hl-1(tl-1) (/h/ (t/)dt/ W hl-2(tl-2)...
t t t t t
t2
...J h1 (t1 )dt1... dt/—2dt/—1dt/+1 ...dtk, (168)
t
where 1 < l < k and h1 (t),..., hk (t) are continuous functions on the interval [t,T]. By analogy with (168) we have for l = k T t, t2
J h/(t/) J h/—1 (t/—1).. y h (t1)dt1... dt/—1dt/ =
t t t T T T T
= / M^) / h (t2) ...j hU^j h ft ^^ . . . ^ =
t t1 t, — 2 t,-1
T \ T T T
/W^J Jh1(i,)jMfc)...j ^ft-1)^...¿Mil-
,t / t h ti — 2
T T T / ti—1 \
- /hi(ti) / h2 te) ...j h-i(i,-i) I / Idt1-1... =
t ti ti — 2 \ t / T \ T t2
^ hi (ti)iti I J hi-1(ti-1)... J h1 (t1)it1... ii/-1-
t t t T / ti —i \ ti —i t2
-J hi-1(ti-1) J hi (ti )iii J hi-2(ti-2) ...J h1(t1)it1. ..iti-1. (1E9)
t \ t / t t
The formulas (1E8), (1E9) will be used further. Our further proof will not fundamentally depend on the weight functions ^1(r),..., V(t)• Therefore, sometimes in subsequent consideration we write V1(t),..., V(t) = 1 for simplicity.
Let us continue the proof. Applying (I1E8I) to Cjk...ji+1 jiji—1 ...js+1jijs—1...j1 (more precisely to hs(t.) = Vs(¿^j (is)), we obtain for l + 1 < k, s — 1 > 1, l - 1 > s + 1
oo
•••ji+1jiji —1 -jS + 1ji js—1...j1 = (170)
ji=P+1
T ti+2 ti+1 ti
53 / j(tk) . J j(¿i+oy^ji(ti^ j 1 (ti-1)... =p+11 t t t ts+2 ts+1 ts
... J fe + 1 (iS+1^ ^ 0js —1 (iS-1) ... t t t
t2
(i1)it1 ...¿i.- A^...^1iti iii+1 =
t
QQ T ti+2 ^+1 ti
53 / j (tk) ...y 0ji+1 (ti+1^ ^ji (ti) J j 1 (ti-1)... ji=p+11 t t t
oo
ts+2 / ts+1 \ ts+1
.J 0js+1 (ts+0 I J (ts)dts J 0js-1 (ts —1) ... t \ t / t t2
J j (t1)dt1... dts—1dts+1... dt/—1dt/dt/+1... dtk—
t
T t,+2 t, + 1 t,
E /^jk (tk ) ...y^+1 (t/+1^ (t/y (t/ —1) ... j, =P+1 t t t t ts+2 ts+1 / ts — 1 \ ts-1
...J 0js+1 (ts+0 / 0js-1 (ts — 0 I J j (ts)dts I J 0js-2 (ts—2) ... t t t t t2
. . J j (t1)dt1... dts—2dts—1dts+1... dt/—xdt/dt/+1... dtk =
t
to to
= 53 Aj'fc-"j'i+1j'ij'i-1-js + 1j'ijs-1-"j'1 Bjk ■■■jl + 1jljl-1---js + 1jl js-1...j1 .
j, =P+1 j, =P+1
Now we apply the formula (168) to
AjV"ji+1jdi-1."js + 1jijs-1".j1 and Bjfe ■■■j,+1 j, j,-1 ...js+1 j, js-1 ■■■j1
(more precisely to h/(t/) = (t/)0j,(t/)). Then we have for l + 1 < k, s — 1 > 1, l — 1 > s + 1
to
53 Cjk ■■■jl + ljljl — 1■■■js+ljljs — 1■■■jl =
j,=P+1
f 4
= 53 Fd)(t1,..., ts—1, ts+1, ..., t/—1, t/+1,..., tk) x
[t,T]k-2 d=1
k
x n (tg) j(tg)dti.. . dts_idts+i... dt/-idt/+i... dtk =
g=i g=l,s
4 4
EC *(d) = v^ c
jV-j+lj-l-js+ljs-l-jl / v j'fc •••jq •••j1
d=i d=i
(171)
q=/,s
where
b . . . , ts—1, ts+1, . . . , t/-1, t/+1, . . . , tk)
to ts+1 tl+1
ji =P+1 t t
(172)
Fp(2)(t
b . . . , ts—1, ts+1, . . . , t/—1, t/+1, . . . , tk) =
to ts_1 tl—1
1 {11 < ...<ts — 1 <ts +1 < ...<t i — 1 <ti +1 <... <t k }
ji =P+1 t t
(173)
F(3)(t
1, . . . , ts — 1, ts+1, . . . , t/—1, t/+1, . . . , tk) =
TO ts—1 ti+1
j =P+1 t t
(174)
F(4)(t
1, . . . , ts— 1, ts + 1, . . . , t/ —1, t/+1, . . . , tk) =
TO ts+1 ^1
j =P+1 t t
(175)
By analogy with (171) we can consider the expressions
jk_ 1 ...j2ji , (176)
ji =P+1
TO
53 Cjk ...ji+ijiji_ 1...j2ji (1 +1 — k), (177) ji =P+1
TO
E j_ 1 ...j.+j ...ji (s — 1 > 1). (178)
ji =p+1
Then we have for (176)-(178) (see (168), (169))
to „ 2 k—1
E j_ 1 j = J EG(d)(t2,...,tk—(tg)j(tg)dt2...dtk—1,
ji=P+1 [tT]k_2 d=1 g=2
(179)
to „ 2
j =P+1 [t Tifc-2 d=l
/ / j^y V</25 • • • , i/-1) il+l, • • • , tk) x
[t,T
k
x n (tg) j (tg)dt2 • • • dt/_idt/+i • • •dtk, (180)
g=2 3='
to „ 4
Cj'iifc-i--is+iiiis-i-ji = 53 Dyd) (t1' • • • ' ts-1' ts+1' • • • , tk-1) x
=y+1 [t T]k-2 d=1
k-1
x n ^g (tg) j (tg)dt1 • • ^-1^+1 • • •dtk-1, (181)
3=i 3=s
where
TO T
GP1)(t2, • • • , tk-1) = l{t2<...<tfe-i> E / ^k (t (t )dT / WT (t )dT,
-ij
j=y+1 i i
TO tk-i H
GP2)(t2, • • • , tk-1) = -l{i2<...<tfc-i} 53 J ^k(t(t)dT J WT(t)d
j =y+1 t t
Ey1)(t2, • • • ,t/-1,t/+1, • • • , tk) =
to y ^
e / ^(t)&<(t)d
t (t (t )dT,
j =y+1 t t
Ey2)(t2, • • • ,t/-1,t/+1, • • • , tk) =
y
to ti-i ^
i{t2<...<t'-i<t'+i<...<tfc} 53 / (t)^(T)dT / (T)dT'
j =y+1 t t
DP1) (t1, • • • , ts-1, ts+1, • • • , tk-1) =
TO T tS/i
j =y+1 t t
DP2) (ib . . . , is—1, is+1, . . . , tk-1) =
to T ^
l{t1<...<ts—1<ts + 1<...<tk—1} E / V W^ji (T)i^ Vs(T)0ji (т)iт,
ji =P+1 {
DP3) . . . , is—1, is+1, . . . , tk-1) =
to tk— ts+1
1{t1<...<ts — 1<ts+1<...<tk— 1} / Vk(t)<j(t)iT /
ji =p+1 i i
D4 (t1,..., t.-1, t.+1,..., tk-1) =
to tk— ts—1
1 {11 < ...<t s — 1 <ts +1 < ...<t k — 1}
e / Vk(t)0ji(t)iT /
ji=p+1 t t
(T )^ji (t )iT,
Now let us consider the value Cj, 1. ,.. To do this, we will make
jk ...j1 ljS1 =jg2 ,02=01 + 1 '
the following transformations
T ti+2 ti+1 ti ti —1 t2
J hk (tk)... J h/+1(t/+1^y hi (ti) J hi (ti-1^ hi-2 (ti-2) ...y h1(t1)it1...
t t t t t t
. . . iti—2iti—1 iti iti+1 . . . itk =
T ti+2 ti + 1 ti + 1 ti+1
= J hk (tk)... J hi+1(ti+1^ h1(t1^ h2 (t2)... J hi-2(ti-2)x
t t t t1 ti—3
/ ti+1 ti — 2\ /ti + 1 ti —1 \
x - hi(¿i—1) - hi(ti)itiiti-1iti-2 . . . it2it1iti+1 ...itk =
T ti+2 / ti+1 ti + 1 \ ti+1
= J hk (tk)... J hi+1(ii+1H y hi (ti )iti y hi (ti—1)iti—1 y h1(t1)x
t t \ t t / t
ti+1 ti+1
xj h2(i2).. ./ ^c^-2 ...¿w^ -
t1 ti-3
T t'+2 / t'+i \ t' + i t' + i
-J hk (tk) • •• J h/+1(t/+1H J hi (t/)it/ J h1(t1^ h2 (t2) •••
t t \ t / t ti
t'+i / t'-2 \
•■•/hl-2(tl-2) (Jh/ ^H dгl-2■■•dt2dtldгl+l •■■dtk -
t'—3 \ t /
T t'+2 / t'+i t'-i \ t' + i
-J hk (tk)••• J h/+1(t/+1n J h/(t/-1^ h/ (t/)dt/dt/-J J h1(t1)x
t t \ t t / t
t'+i t'+i
x/ №)••,^^•dt2dt1 am--dtk+
ti t'-3
T t'+2 t' + i t'+i t' + i
+ y hk (tk) •••J h/+1(t/+1^ h1 (t^ h2(t2) •••J h/—2 (t/—2) x
t t t ti t'-3 / t'-2 t'-i \
x ( f (tl—0 f h/J • „w^ • •,, =
T t'+2 / t' + i t'+i \ t' + i
= J hk (tk) • • • y h/+1(t/+1) I y h/ (t/h/(t/-1)it/-1 I y h/-2(t/-2)x
t t \ t t / t t'-2 t2
x y h/-3(t/-3) • ••J h1 (t1)it1 • • • it/-3it/-2it/+1 • • • itk-
tt
T t'+2 / t' + i \ t'+i
-J hk (tk) • • • J h/+1(t/+1M y h/ (t/)it/1 y h/-2(t/-2)x t t t t / t'-2 \ t'-2 t2 x (/h/ ^ 1W^3) ••J h1(t1)it1 • • •itl—зitl—2itl+l • • •itk -
\ t / t t
T t'+2 / t' + i t'-i
-J hk (tk )••• y h/+1(t/+1M y h/ (t/-1^ h/ (t/)it/it/-1 | x
t t \ t t
ti+1 ti —2 t2
xj / fc^) ... / M^ ... M^1...itk+
t t t
T ti+2 ti + 1 i ti —2 ti —1 \
+ j hk (tk)... J hm(tm) / h/_2(t/_2) i / h/ ^ / h/ (t/X^ ) x
t t t \ t t / tl — 2 t2
x J h/—3 (t/—3) ...J h1 (t1 )it1... it/-3iti-2iti+1... itk, (182)
tt
where l + 1 < k,l — 2 > 1, and h1(T),..., hk(t) are continuous functions at the interval [t,T].
Applying (182) to Cjk...j+jjji—2......j1, we obtain for l + 1 < k, l — 2 > 1
to
Cjk •••ji+1 ji ji ji —2......j1 =
ji=P+1 4
' 7pd)(t1,... ,t/-2,ti+1,... ,tk)x
r 4
/ 53H(d)(t1,...,t/-2,ti+1,...,tk):
[t,T]k—2 d=1
k
x n Vg(tg) j (tg)it1 . . . it/-2it/+1 ... itk
g=1 g=i—1,i
44
( d) _ \ A ( d)
j1
d=1 d=1
jk...ji+1ji —2...j1 = / v jk ...jq....
where
, (183)
q=/-1,1
Hp1)(t1, . . . ,t/-2,t/+1, . . . ,tk) =
to ti+1 ti+1
i{t1<...<ti—2<ti+1<...<tk} 53 IV(t)^'(T)iW v/-1(t)^ji(т)iт, (184)
jl= +1 t t
H^2)(t1, . . . ,t/-2,t/+1, . . . ,tk) =
to ti+1 ti—2
i{t1<...<ti—2<t'+1<...<tk} 53 / V(T)^j' (T)iT / v/-1(t)^jiw^ (185)
jl= +1 t t
Hp3)(ti, . . . ,i/-2,t/+i, . . . =
to w T
i{i1<...<ii-2<ii+1<...<ifc} J (t) / ^(186)
ji =p+1 t
1, . . . , ti-2, tl+1, . . . , tk)
ti-2 oo 1-2
= 1{ti<...<ti-2<ti+i<...<tfc^ J ^/-i(r(t)J ^(0)<j(0)d0dr. (187)
ji =p+1 t t
By analogy with (183) we can consider the expressions
to
53 ...ji + ljiji ' (188)
ji =p+1
TO
53 Cjijijfe-2...ji. (189) ji=p+1
Then we have for (188), (189) (see (182) and its analogue for tm = T)
to „ k
53 Cjk...ji+ijiji = J LP(t3' . . . , ik) fj ^g(tg) j(tg)dt3 . . . dtk, (190) [t,T
to „ 4 k-2
ji ji jk-2 ...ji = 53MPid)(t1' . . . '¿kW n 'WgJ j61 • • • U6k-2,
[t,T ]k
ji=p+1 [t Tifc-2 g=3
TOTO n "± A/—^
53 CCjijijk-2...ji = 53 Mpd) (t1,..., tk—2) n ^g(tg ) j(tg)dt1 . . .dtk—2'
(191)
ji =p+1 [tTik-2 d=1 g=1
where
t3 T
TO ^ „
Lp(t3, . . . 'tk) = 1{t3<...<tk} E / ^ (t )0ji (t ) / wwji (0)d0dr,
TO
1{ti<...<tk-2} E I ^k(T)0ji (T)dT / ^k—1(t)0ji (T)dT'
ji =P+1 t t
ji=P+1 t
Mp1)(t1'...,tk—2) =
T T
oo
T
tfc-2
1{il<...<ik-2} E / ^ Wj (t)^ / ^-1(t)^ (t)^t,
j=P+l t
M(3)(ti,...,tk-2)
oo
T
1{ti<...<tk-2} E / ^k-l(r)0ji (t W ^k (Wj (0)d0dr,
j =p+1 t
M(4)(ti,...,tk-2)
tfc-2
1{ti<...<tfc-2^ / ^k-l(t)0ji (t) / ^k (0)<j (0)d0dt.
=p+1 t
It is important to note that Cjj ^ , (d = !,•••, 4) are
Fourier coefficients (see (171), (183)), that is, we can use Parseval's equality in the further proof.
Combining the equalities (I171I)-(ITT5) (the case > gi + 1), using Parseval's equality and applying the estimates for integrals from basis functions that we used in the proof of Theorems 11, 12, we obtain for (171)
2
oo
E I E
jqi jq2 'j93 =0 v'si =i'+1
jg1 =jg2 ,g2>gi + 1
OO
E E Cj5...ji
q=si,32
jgi =js2 ,g2>gi+1
4 x 2
(d)
EV^ C*(d)
1 Z-/ Cj5...jq...ji
ji,...Jqv..j5=° \d=1 q=si,32
<
q=gi,g2/
— ^^ ( Cj5.. jq ...j
ji,...,jq,...,j5=° \d=1 q=si,32
q=gi,g2/
P
2
P
p
2
4
l
TO „ 4
E / • • • '^gi-l'^gi+l' • • • '^52-1 'Wl' • • • '¿5)x
J1,...,jq,...,j5=0 UTH5 d=1
- \[t,T]3
5
X n (tq) j (tq)dtl . . . dtg1-ldtg1 + l . . . d^-ld^ + l . . . dts
q=1
q=si,32
J
(tl' • • • '¿gl-l'Wl' • • . 'tg2-l'tg2 + l' . . . '¿5)
X
3 x d=l
xdtl... dtg1-ldtg1+l... dtg2-ldtg2+l... dts ^
<
C 2
/ (Fp(d)(tl ' . . . ,tg1-l'tg1 + l' . . . ,tg2-l'tg2 + l' . . . 'O) X
d=l[t,T ]3
xdtl... dtg1-ldtg1+l... dtg2-ldtg2+l... dts ^
K
<— 0
p2-e
(192)
if p ^ 00, where £ is an arbitrary small positive real number for the polynomial case and £ = 0 for the trigonometric case, constant K does not depend on p. The cases (176)-(178) are considered analogously.
Absolutely similarly (see (1192))) combining the equalities (1183I)-(ITH7) (the case = g1 + 1), using Parseval's equality and applying the estimates for integrals from basis functions that we used in the proof of Theorems 11, 12, we get for
oo
E I E Cj5...j1
j91 j92 'j93 =0 \jS1 =p+1
p /to
= £ E C»..,
j1,...,jq,...,j5=0 \jS1 =P+l
q=S1,S2
jg1 =jg2 ,g2=g1 + 1
jg1 =jg2 ,g2=g1 + 1
2
2
2
p
2
El V C
j1,...,Jq,...,j5=0 \d=1 9=31^2
<
oo
< E (Ecr'f
j1,...,jq,...,j5=0 \d=1 q=si,32
q=gi,g2,
E / EH^.-
„•__n J 1
j1,...,jq,...,j5=0 [, T13 d=1 9=31,32 \ [t ' T 1
x n (tq) j(tq)dt1 • • • dtg1-1dtg1+2• • • dt5
9=1 9=31,32
/
: / (E H<d»(i1,...,t91-1,i
[t,T ]3 ^ d=1
b • • • , tg1-1, tg1+2, • • • , t5) I dt1 • • • dt51-1 dt51+2......dt5 <
<
4 P 2
1, • • • , tg1-1, tg1+2, • • • , t5^ dt1 • • • dtg1-1dtg1+2 •
d=1[t,T ]3
• • dt5 <
K
<— 0
p2-e
(193)
if p ^ oo, where £ is an arbitrary small positive real number for the polynomial case and £ = 0 for the trigonometric case, constant K does not depend on p.
The cases
From we obtain
are considered analogously. 3) and their analogues for the cases (T761)-(T78), (188), (189)
E I E Cj5 .j1
j91 j92 j93 =0 \j31 =p+1
j31 =j32
(194)
where constant K is independent of p. Thus the equality (T65) is proved. Let us prove the equality (T66). Consider the following cases
1- g2 > g1 + 1, g4 = g3 + 1 2. g2 = g1 + 1, g4 > g3 + 1,
2
4
p
2
4
1
2
2
3. g2 > gi + 1, g4 > g3 + 1, 4. g2 = gi + 1, g4 = g3 + 1.
The proof for Cases 1-3 will be similar. Consider, for example, Case 2. Using (59), we obtain
oo oo
E E E
jq1 =0 \jS1 =P+1 jS3 =P+1
jg1 =jg2 Jg3 =jg4 ,g4>g3+l,g2=g1 + 1
TO p
E E E Cj5...j1
j91 =0 \jS1 =P+1 jS3 =0
jg1 =jg2 jg3 =jg4 ,g4>g3+l,g2=g1 + 1
p / p TO
E E E
j91 =0 \ =0 jS1 =p+1
<
(195)
jg1 =jg2 jg3 =jg4 ,g4>g3+l,g2=g1 + 1
pp
TO
< (p+^E E E C*-j1
j91 =0 jS3 =0 \ jS1 =p+1
jg1 = jg2 jg3 = jg4 ,g4 >g3+1 ,g2=g1 + 1
2
pp
(p+^E E E Cj5..j1
j91 =0 jg3 jg4 =0 \ jg1 =p+1
jg1 =jg2 ,g4>g3+l,g2=g1 + 1
<
jg3 jg4
pp
oo
< (p+ ^E E E Cj-j
j91 =0 jg3 jg4 =0 vjg1 =p+ 1
(196)
jg1 =jg2 ,g4>g3+l,g2=g1 + 1
It is easy to see that the expression (196) (without the multiplier p + 1) is a particular case (g4 > g3 + 1,g2 = g1 + 1) of the left-hand side of (194). Combining (194) and (196), we have
2
TO TO
E E E
jq1 =0 V jg1 =p+1 jg3 =p+1
<
jg 1 = jg2 ;jg3 = jg4 ,g4 >g3+1 ,g2 =g1 + 1
<(E±IW<Kl^0
p2-e pl-e
(197)
if p ^ o, where constant K1 does not depend on p.
2
p
2
p
2
2
p
Consider Case 4 (g2 = g1 + 1, g4 = g3 + 1). We have (see (60))
oo oo
E E E
Jqi =0 Vjgi =P+ 1 jag =P+ 1
jgi jS2 'jS3 jS4
oo
TO p
E E E-E I j
jqi =0 \j'si =P+1 Vías =0 JAS =0
J i
p i 1 TO
E I 9 E (
Jqi =0 \ Jgi =p+1
1
jgi jA2 'jss jA4
p TO
EE CJ5...ji
jgi =jS2 '(jss jas jss =0 jsi =p+1
2
<
- 2 E I E f
Jqi =0 \jgi =p+1 pip TO
+2 E E E Cj5..ji
Jqi =0 VJas =0 Jai =p+1
+
j3i =jS2 ' (jSS j33 •)
(198)
(199)
jgi jA2 'jss jA4
An expression similar to . We have
was estimated (see
(197)). Let us
estimate
TO
E I E CJ5...ji
Jqi =0 \ Jgi =p+1
jsi = jS2 '(jss j33 •)
TO
(T - ()E E Cj5..j
Jqi =0 V Jgi =p+1
<
Jgi =Jg2, (j'ss JasW
pp
^ (T -1) EE E C,.*
Jqi =0 Jas =0 \Jai =p+1
(200)
Jai =js2, Üas Jas )^jgs
where the notations are the same as in the proof of Theorem 7.
The expression (200) without the multiplier T — t is an expression of type (I124l)-(IT291) before passing to the limit lim (the only difference is the replace-
p^TO
ment of one of the weight functions ^1(r),..., ^4(t) in (T24)-(T29) by the product ^/+1(r(t) (l = 1,..., 4). Therefore, for Case 4 (g2 = g1 + 1, g4 = g3 + 1),
2
p
2
p
2
p
2
2
p
2
p
2
we obtain the estimate
p
oo oo
E E E Cj5...j1
jq1 =0 \jg1 =p+1 jg3 =p+1
<
jg1 =jg2 jg3 =jg4 ,g4=g3+l,g2=g1 + 1
K
(201)
where constant K is independent of p.
The estimates (197), (201) prove (166). Let us prove (167). By analogy with (200) we have
2
pTO
E I E Cj5...j1
j91 =0 \jg3 =p+1
(jg2 jg1 W-),jg1 =jg2 =jg4 ,g2=g1+1
pTO
E I E Cj5...j1
j91 =0 \jg3 =p+1
(jg 1 jg 1WO ,jg3 = jg4 >g2=g1+1
pTO
(t -1) £ £ C,5...,1
j91 =0 \ jg3 =p+1
<
(jg1 jg1 W,jg3 =jg4 ,g2=g1+1
pp
oo
< (T -1)£ E E
jqi =0 jgi=0 \jg3 =p+1
Thus, we obtain the estimate (see
(jg1 jg1 Wg1 ,jg3 =jg4 ,g2=g1+1
(202)
pTO
E I E Cj5...j1
j91 =0 \jg3 =p+1
and the proof of Theorem 12)
2
<
(jg2 jg 1W •) ,jg 1 = jg2 = jg4 >g2=g1+1
K
(203)
where £ is an arbitrary small positive real number for the polynomial case and £ = 0 for the trigonometric case, constant K does not depend on p.
The estimate (203) proves (167). Theorem 13 is proved.
2
2
2
8 Estimates for the Mean-Square Approximation Error of Iterated Stratonovich Stochastic Integrals of Multiplicity k (k e N)
In this section, we estimate the mean-square approximation error in Theorems 7, 10 for iterated Stratonovich stochastic integrals of multiplicity k (k e N).
Theorem 14 [13], [48], [49]. Suppose that every ^ (t) (l = 1,..., k) is a con-
x
tinuously differentiate nonrandom function at the interval [t,T]. Furthermore, let (x)}°=0 is a complete orthonormal system of Legendre polynomials or trigonometric functions in the space L2([t,T]). Then the following estimates
MIJ-^)]^ - E Ckj n Zf)} < kix
[ V ji,...jfc =0 l=1 J )
[k/2] , , 2 ^
J- + ^ ^ M < 2'"''5,2r_1'5r2r •••i<?fe-2r ^ ^
P r=1 ({{gi,S2},...,{S2r-1,S2r }},{91,...,9fc-2r}) \ {S1,S2,...,S2r-1,S2r,91,...,9fc-2r } = {1,2,.",k} /
(204)
M \( J-^]^""} - E Cjk ...j1 (s) n j0) [ < K2(s)x
[ V =0 l=1 / J
[k/2] r 2 ^
P r=1 ({{S1,S2},...,{S2r-1,S2r }},{91,...,9fc-2r}) \ {S1,S2,...,S2r-1,S2r,91,...,9fc-2r } = {1,2,.",fc} /
X
(205)
hold, where s £ (t, T] (s is fixed), p £ N, i1,..., = 1,..., m,
R(p)0i>02v,02r-i,02r- (% •••iqfc-2r) = R(p)gi,g2,---,g2r-i,g2r •••«qfc-2r)
T=s
R!гptgl'g2'•••'g2r-1'g2r(iqi•••iqk-2r) is defined by (72), J*[^(k)]Tilt•••ik) and J^]^"*) iterated Stratonovich stochastic integrals (I2EI) and (80), Cjkj and Cjkj (s) are
are
Fourier coefficients (20) and (78), constants K1; K2(s) are independent of p; another notations are the same as in Theorems 1, 7, 10.
Proof. Note that Conditions 1, 2 of Theorems 7, 10 are satisfied under the conditions of Theorem 14 (see (81)-(84), (86), (88)-(93)). Then from the proof of Theorem 7 we have that the expression (77) before passing to limit l.i.m. has the form
p k
E cv.j,nzf = J№(k)]£r')p+
j'lv-j'fc =0 1=1
[k/2] / 1
2r
r=W (sr ,...,Sl)GÂfc,r
+ ^ R(p,)gi,g2,...,g2r-i,g2r^ 2r) | (206)
({{si ,32},...,{S2r-1,S2r}},(91,...,9fc-2r }) {si ,S2,S2r—i ,S2r ,qi ,...,qk-2r }={1,2,...,k}
where J[^(k)]y1'"ifc)p is the approximation for the iterated Ito stochastic integral (2), which is obtained using Theorems 1, 2, i.e.
p / k [k/2]
J№(k)]rifik)p = E c,Jnj)+ E(—i) ji,".jfc=0 \/=1 r=l
r k—2r
X
E n 1{ig2s-1 = ig2s =°}1{jg2s-1 = jg2s ^ C ) ' (207)
({{31,52},...,{S2r-1,S2r »,{91,...,9fc-2r}) S=1 1 = 1 /
{si,32,...,S2r-1,S2r,91,...,9fc-2r } = {1,2,.",k}
I[^(k)]Ti1t-is1-1is1+2-isr"1V+2 'ifc)p is the approximation obtained using (207) for the iterated Ito stochastic integral J[^T^*)[sr'"''s1] (see (36)).
Using (206) and Theorem 9, we have
pk
)
j'iv-jfc=0 /=1
E Cjk... n
[k/2]
r=1 (sr ,...,si)eAfc,r
+ 1 J[^tV'^ - J[W^r^ ) +
[k/2] 1 . 2r
+ E E ^ JLV JT.Î
r=l (sr,...,si)GÄ,
r
— I [^(k)](i1-is1-1is1 + 2...isr-1V + 2---ifc ) | +
[k/2]
+ y^ y^ R(p)g1,g2,-,g2r-1,g2r (i91 •••«gfc-2r )
r=1 ({{S1,S2},...,{S2r-1,S2r }},{91,...,9fc-2r }) {S1,S2,...,S2r-1,S2r,91,...,9fe-2r } = {1,2,.",fc}
= j + J ty^fö"^ )p — J [^fe^M +
[k/2] , r=1 (sr,-,S1)GÂfc,r 2
— I [^(k)j(i1...is1-1iS1 + 2..-iSr-1V + 2-.-ik ) j +
[k/2]
+ y^ y^ R(i5)g1,g2,-,g2r-1,g2r •••iqfc-2r) (208)
r=1 ({{S1,32},...,{S2r-1,S2r }},{91,...,9fc-2r }) {S1 ,S2,...,S2r-1,S2r,91,...,9fc-2r }={1,2,...,k}
w. p. 1, where we denote J[^fe"^)[sr""'s1] as I[^(k)]
(¿1".is1-1is1+2-"isr — 1 i Sr +2-"ife)
T,) •
In [13] (Sect. 1.7.2, Remark 1.7) it is shown that under the conditions of Theorem 14 the following estimate
\k
M | (j[^k)%"ik) - Jrh \ ;■];-'" '')''} < klPk{Tp t] (209)
holds, where J[^^V^} is defined by (2), J[^W]^"^ has the form (207), p e N, i1,..., = 1,..., m, constant Pk depends only on k.
Applying (209), we obtain the following estimates
(n...ifc)p iL/ikhiii-.ifc) » 1 - C
M { ( - k) ) } < (210)
(211)
where p £ N, constant C does not depend on p.
From (208), (210), (211) and the elementary inequality
, 2 /2 2 0\
(ai + a2 + ... + ) < n (ax + a2 + ... + an) , n £ N
we obtain (204).
The estimate (205) is obtained similarly to the estimate (204) using Theorem 1.11 in [13], Theorem 10 and the estimate [13] (Sect. 1.8.1, Remark 1.12)
M \ (- < klPh{s ~t)f
p
where
t2
j[^(k)](;|...ik) = (tk)... ^1(t1)dwt(;i)...dwi:k),
[k/2] E
ji,-..,jfc =0 M=1 r=1
j[^(k)]i:r:')p = e Cj-j.(s) nzf +E(-1)'x
k- 2r
X
E II 1{:S2s-i = Ss =0}1{jS2s-i = jS2s } II Z.C
({{fli,S2 },...,{S2r-i,S2r }},{?i,...,9fc-2r}) S=1 1 = 1
{fli,S2 ,...,S2r-i,S2r,9i,...,9fc-2r }={i,2,.",fc}
where s £ (t,T] (s is fixed), Cjk...j.(s) is the Fourier coefficient (78), p £ N, i1,... ,ik = 1,..., m, constant Pk depends only on k; another notations are the same as in Theorems 2, 10. Theorem 14 is proved.
s
r
9 Rate of the Mean-Square Convergence for Expansions of Iterated Stratonovich Stochastic Integrals of Multiplicities 1 to 5
In this section, we consider the rate of convergence for approximations of iterated Stratonovich stochastic integrals. It is easy to see that in Theorems 11-13 the second term in parentheses on the right-hand side of (204) is estimated for k = 3,4, 5. Combining these results with Theorem 14, we obtain the following theorems.
Theorem 15 [13], [48], [49]. Suppose that [^(x)j 0 is a complete ortho-
normal system of Legendre polynomials or trigonometric functions in the space L2([t,T]). Furthermore, let ^i(t),^2(t),^3(t) are continuously differentiable nonrandom functions on [t,T]. Then, for the iterated Stratonovich stochastic integral of third multiplicity J*[^(3)]y1/2i3') defined by (131) the following estimate
m J (j-i^p™ - £ cjm&WA ) < §
[ \ j1j2j3 =0 /J P
is fulfilled, where p £ N, ii,i2,i3 = 1,... ,m, constant C is independent of p; another notations are the same as in Theorem 1.
Theorem 16 [13], [48], [49]. Let [^ (x) j 0 be a complete orthonor-
mal system of Legendre polynomials or trigonometric functions in the space L2([t,T]). Furthermore, let ^i(t),... , ^4(t) be continuously differentiable non-random functions on [t,T]. Then, for the iterated Stratonovich stochastic integral of fourth multiplicity J*^4']^'"*4' defined by (3) the following estimate
MI (j-i^Tr'- £ Cm...*j'..4
holds, where p £ N, ¿1,..., i4 = 1,..., m, 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 s = 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 17 [13], [48], [49]. Assume that (x)j 0 is a complete orthonormal system of Legendre polynomials or trigonometric functions in the space L2([t,T]) and ^i(t),... , ^5(t) are continuously differentiable nonrandom functions on [t, T]. Then, for the iterated Stratonovich stochastic integral of fifth multiplicity J*[^(5)]£V",5) defined by (3) the following estimate
MHrr'hv-^ t
is valid, where p G N, ii,...,i5 = 1,... ,m, constant C is independent of p, s is an arbitrary small positive real number for the case of complete orthonormal system of Legendre polynomials in the space L2([t,T]) and s = 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.
We should also note the following theorem for the case k = 2.
Theorem 18 [13] (Sect. 2.8.1). Suppose that (x)}°=0 is a complete orthonormal system of Legendre polynomials or trigonometric functions in the space L2([t,T]). Furthermore, let ^1(t),^2(t) are continuously differentiable nonrandom functions on [t,T]. Then, for the iterated Stratonovich stochastic integral of second multiplicity J*[^(2)]Tr1ti2) defined by (3) the following estimate
m I (rm^ - £ ^W)'} ^ f
is fulfilled, where p G N, i1 ,i2 = 1,...,m, constant C is independent of p; another notations are the same as in Theorem 1.
Note that the analogue of Theorem 18 for the case k = 1 follows from The expansion (29) for the narrow particular case ¿i = ... = = 0 can
be obtained under the condition of convergence of limiting traces [58] (Theorem 5.1), [59] (Theorem 4.1), [56] (Remark 1.5.7, Proposition 4.1.2) (the definition of limiting traces can be found in [59]).
10 Theorems 4—6, 11—13, 15—18 from Point of View of the Wong—Zakai Approximation
The iterated Ito stochastic integrals and solutions of Ito SDEs are complex and important functionals from the independent components wT' (i = 1,..., m) of the multidimensional Wiener process wT, t £ [0,T]. Let wT'p (p £ N) be some approximation of wT' (i = 1,... ,m). Suppose that wT'p converges to wT' if p —y oo in some sense and has differentiable sample trajectories.
A natural question arises: if we replace wT' by wT'p in the functionals mentioned above, will the resulting functionals converge to the original functionals from the components wT' (i = 1,..., m) of the multidimensional Wiener process wT? The answere to this question is negative in the general case. However, in the pioneering works of Wong E. and Zakai M. [60], [61], it was shown
that under the special conditions and for some types of approximations of the Wiener process the answere is affirmative with one peculiarity: the convergence takes place to the iterated Stratonovich stochastic integrals and solutions of Stratonovich SDEs and not to the iterated Ito stochastic integrals and solutions of Ito SDEs. The piecewise linear approximation as well as the regularization by convolution [60]-[62] relate to the mentioned types of approximations of the Wiener process. The above approximation of stochastic integrals and solutions of SDEs is often called the Wong-Zakai approximation.
It is well known that the following representation takes place
T
wW - wf = £ I h (0)d0 j, j = / hj (^)dw« (212)
t "t - sj , sj / vn^^e
j=0 t t
where t > 0, t G [t, T], (x)}°=0 is an arbitrary complete orthonormal system of functions in the space L2([t,T]), j are independent standard Gaussian random variables for various i or j. Moreover, the series (212) converges for any t G [t,T] in the mean-square sense.
Let Wr)p — wt(i)p (p G N) be the mean-square approximation of the process Wr) — w(i), which has the following form
T
p
j=0 t
From (213) we obtain
wTi)p - wp = £ J j(0)d»cji). (213)
dwTi)p = £ h (t)Cj°dr. (214)
j=0
Denote
dwT0)p = dr, p G N. (215)
Consider the following iterated Riemann-Stieltjes integral
T t2
^k (tk)... ^i(ii)dwt( ;i)pi... dwt ifc)pfc, (216)
where ^1(t),...,ik(t) are nonrandom functions on [t,T], p1,...,pk G N, i1,...,ik = 0,1,..., m.
Let us substitute (214) and (215) into (216)
T ^ pi Pk k
I ik(tk)... I i(ii)dw' ;i)pi...dw<;k)Pk = £...£nj>, (217)
t t ji=0 j'fc=0 1=1
where
T
j = | & (t )dw«
t
p
are independent standard Gaussian random variables for various i or j (in the case when i = 0),
t t2
cjfc ...ji = / (tk ) j (tk)... fa (ti)0ji (ti)dti
/1
t t
is the Fourier coefficient corresponding to the function
For the particular case pi = ... = pk = p, the formula (217) has the form
T t2
k
iij- v- ^^ t1
/i,"-,/fc=0 1=1
^(tk)... (ii)dw<:i)p...dw«:k= Ê nz/r
To best of our knowledge [60]-[62] the approximations of the Wiener process
in the Wong-Zakai approximation must satisfy fairly strong restrictions [62] (see Definition 7.1, pp. 480-481). Moreover, approximations of the Wiener process that are similar to (213) do not satisfy the conditions of Definition 7.1 and Theorem 7.1 [62]. Also, these approximations of the Wiener process were not
considered in [60], [61]. Therefore, the proof of analogue of Theorem 7.1
for approximations of the Wiener process based on its series expansion should be carried out separately.
From the other hand, Theorems 4-6, 11-13, 15-18 and Theorems 1, 2 (k = 1) from this article can be considered as the proof of the Wong-Zakai approximation based on the iterated Riemann-Stieltjes integrals (216) of multiplicities 1 to 5 and the Wiener process approximation (213) on the base of its series expansion. At that, the mentioned Riemann-Stieltjes integrals converge (according to Theorems 4-6, 11-13, 15-18 and Theorems 1, 2 (k = 1)) to the appropriate Stratonovich stochastic integrals (3). Recall that (x)}°=0 (see (1212), (213), and Theorems 4-6, 11-13, 15-18) is a complete orthonormal system of Legendre polynomials or trigonometric functions in the space L2([t,T]).
The Wong-Zakai approximation is widely used to approximate stochastic integrals and SDEs. In particular, the Wong-Zakai approximation can be used
to approximate the iterated Stratonovich stochastic integrals in the context of numerical integration of Ito SDEs in the framework of the approach based on the Taylor-Stratonovich expansion [2]-[13 .
For example, the authors of the works [2](Sect. 5.8, pp. 202-204), [5] (pp. 8284), [22] (pp. 438-439), [31] (pp. 263-264) use the Wong-Zakai approximation within the frames of approximation of iterated Stratonovich stochastic integrals based on the Karhunen-Loeve expansion of the Brownian bridge process. However, in these works there is no rigorous proof of convergence for approximations of the mentioned stochastic integrals.
From the other hand, the theory constructed in this article (also see Chapters 1 and 2 of the monograph [13]) can be considered as the proof of the Wong-Zakai approximation for the iterated Stratonovich stochastic integrals (3) of multiplicities 1 to 5 based on the Wiener process series expansion using Legendre polynomials and trigonometric functions.
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:2003.14184v28 [math.PR], 2022, 869 pp. DOI: https://doi.org/10.48550/arXiv.2003.14184
[14] Allen, E. Approximation of triple stochastic integrals through region subdivision. Commun. Appl. Anal. (Special Tribute Issue to Professor V. Lakshmikantham), 17 (2013), 355-366.
[15] 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.
[16] 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.
[17] Gaines, J.G., Lyons, T.J. Random generation of stochastic area integrals. SIAM J. Appl. Math., 54 (1994), 1132-1146.
[18] 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.
[19] Ryden, T., Wiktorsson, M. On the simulation of iterated Ito integrals. Stoch. Proc. and their Appl., 91, 1 (2001), 151-168.
[20] 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.
[21] 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.
[22] Kloeden, P.E., Platen, E., Wright, I.W. The approximation of multiple stochastic integrals. Stoch. Anal. Appl., 10, 4 (1992), 431-441.
[23] 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
[24] 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
[25] 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
[26] 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
[27] Foster, J., Habermann, K. Brownian bridge expansions for Levy area approximations and particular values of the Riemann zeta function. arXiv:2102.10095v1 [math.PR]. 2021, 26 pp.
DOI: https://doi.org/10.48550/arXiv.2102.10095
[28] Kastner, F., Rößler, 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: https://doi.org/10.48550/arXiv.2201.08424
[29] Malham, S.J.A., Wiese A. Efficient almost-exact Levy area sampling. Stat. Prob. Letters, 88 (2014), 50-55.
[30] 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.
[31] Platen, E., Bruti-Liberati, N. Numerical Solution of Stochastic Differential Equations with Jumps in Finance. Springer, Berlin, Heidelberg. 2010,
pp.
[32] 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
[33] 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
[34] 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
[35] 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
[36] 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
[37] 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
[38] 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
[39] 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: https://doi.org/10.48550/arXiv.1905.03724
[40] 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: https://doi.org/10.48550/arXiv.1912.02612
[41] 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
[42] 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
[43] 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
[44] Kuznetsov, D.F. Expansion of iterated Ito stochastic integrals of arbitrary multiplicity based on generalized multiple Fourier series converging in the mean. arXiv:1712.09746v22 [math.PR]. 2022, 111 pp.
DOI: https://doi.org/10.48550/arXiv.1712.09746
[45] 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:1801.01079v13 [math.PR]. 2022, 67 pp. DOI: https://doi.org/10.48550/arXiv.1801.01079
[46] 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:1801.00231v18 [math.PR]. 2022, 105 pp.
DOI: https://doi.org/10.48550/arXiv.1801.00231
[47] Kuznetsov, D.F. The hypotheses on expansions of iterated Stratonovich stochastic integrals of arbitrary multiplicity and their partial proof.
arXiv:1801.03195v13 [math.PR]. 2022, 104 pp.
DOI: https://doi.org/10.48550/arXiv.1801.03195
[48] Kuznetsov, D.F. Expansions of iterated Stratonovich stochastic integrals based on generalized multiple Fourier series: multiplicities 1 to 5 and beyond. arXiv:1712.09516v16 [math.PR]. 2022, 174 pp. DOI: https://doi.org/10.48550/arXiv.1712.09516
[49] Kuznetsov, D.F. Expansion of iterated Stratonovich stochastic integrals of fifth multiplicity based on generalized multiple Fourier series. arXiv:1802.00643v9 [math.PR]. 2022, 94 pp.
DOI: https://doi.org/10.48550/arXiv.1802.00643
[50] 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 Up-ravlenia, 2 (2020), 89-117. Available at:
http://diffjournal.spbu.ru/RU/numbers/2020.2/article.1.6.html
[51] 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
[52] 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
[53] 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
[54] Stratonovich, R.L. Conditional Markov Processes and Their Application to the Theory of Optimal Control. Elsevier, N. Y., 1968, 350 pp.
[55] Ito, K. Multiple Wiener integral. J. Math. Soc. Japan, 3, 1 (1951), 157169.
[56] Budhiraja, A. Multiple stochastic integrals and Hilbert space valued traces with applications to asymptotic statistics and non-linear filtering. Ph. D. Thesis, The University of North Caroline at Chapel Hill, 1994, VII+132 pp.
[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
[58] Johnson, G.W., Kallianpur, G. Homogeneous chaos, p-forms, scaling and the Feynman integral. Trans. Amer. Math. Soc., 340 (1993), 503-548.
[59] Budhiraja, A., Kallianpur, G. Two results on multiple Stratonovich integrals. Statistica Sinica, 7 (1997), 907-922.
[60] Wong, E., Zakai, M. On the convergence of ordinary integrals to stochastic integrals. Ann. Math. Stat. 36, 5 (1965), 1560-1564.
[61] Wong, E., Zakai, M. On the relation between ordinary and stochastic differential equations. Int. J. Eng. Sci., 3 (1965), 213-229.
[62] Ikeda, N., Watanabe, S. Stochastic Differential Equations and Diffusion Processes. 2nd Edition. North-Holland Publishing Company, Amsterdam, Oxford, New-York. 1989, 555 pp.
[63] Liptser, R.Sh., Shirjaev, A.N. Statistics of Stochastic Processes: Nonlinear Filtering and Related Problems. Nauka, Moscow. 1974, 696 pp.
[64] Luo, W. Wiener chaos expansion and numerical solutions of stochastic partial differential equations. Ph. D. Thesis, California Institute of Technology, 2006, 225 pp.
