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DIFFERENTIAL EQUATIONS AND
CONTROL PROCESSES N. 4, 2023 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 proof of the expansion of iterated Ito stochastic integrals
with respect to the components of a multidimensional Wiener process based on generalized multiple Fourier series and Hermite
polynomials
Abstract. The article is devoted to a new proof of the expansion for iterated Ito stochastic integrals with respect to the components of a multidimensional Wiener process. The above expansion is based on Hermite polynomials and generalized multiple Fourier series in arbitrary complete orthonormal systems of functions in a Hilbert space. In 2006, the author obtained a similar expansion, but with a lesser degree of generality, namely, for the case of continuous or piece-wise continuous complete orthonornal systems of functions in a Hilbert space. In this article, the author generalizes the expansion of iterated Ito stochastic integrals obtained by him in 2006 to the case of an arbitrary complete orthonormal systems of functions in a Hilbert space using a new approach based on the Ito formula. The obtained expansion of iterated Ito stochastic integrals is useful for constructing of high-order strong numerical methods for systems of Ito stochastic differential equations with multidimensional non-commutative noise.
Dmitriy F. Kuznetsov
Peter the Great Saint-Petersburg Polytechnic University e-mail: sde_kuznetsov@inbox.ru
Key words: iterated Itô stochastic integral, multiple Wiener stochastic integral, Ito stochastic differential equation, generalized multiple Fourier series, multidimensional Wiener process, Hermite polynomial, mean-square convergence, expansion.
Contents
1 Introduction 68
2 Preliminary Results 75
2.1 Expansion of Iterated Ito Stochastic Integrals based on Generalized Multiple Fourier Series..................................... 75
2.2 Modification and Generalization of Ito's Theorem. Proof on the Base of the Ito Formula and Without Explicit Use of the Multiple Wiener Stochastic Integral . 78
3 Main Results 114
3.1 Generalizations of Theorem 2 to the Case of an Arbitrary Complete Orthonormal Systems of Functions in the Space L2([t,T]) and ), ... (t) G L2([t,T]) . . 114
3.2 Modifications of Theorems 6, 7 for the Case of an Arbitrary Complete Orthonormal Systems of Functions in the Space L2([t,T]) and $(t1;..., tk) G L2([t, T]). 115
4 Comparison with Other Results and Conclusions 117 References 119
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 = 1,..., m) of this process are independent. Consider an Ito stochastic differential equation (SDE) in the
integral form
s s
/m „
a(xT,r)dr + ^ / Bj(xT,T)dwj), x0 = x(0,u), u e (1) 0 j=1 0 Here xs is n-dimensional stochastic process satisfying the equation (1). The nonrandom functions a(x, t), Bj (x, t) : Rn x [0, T] ^ Rn (j = 1,..., m) guarantee the existence and uniqueness up to stochastic equivalence of the strong solution of equation (1) [1]. The second integral on the right-hand side of (1) is the Ito stochastic integral. Let x0 be an n-dimensional random variable, which
, ^ -v
is F0-measurable and M{|x01 ) < to (M denotes a mathematical expectation). We assume that x0 and wT — w0 are independent when t > 0. In addition to the above conditions, we will assume that the functions a(x,T), Bj (x, t) (j = 1,..., m) are sufficiently smooth functions in both arguments.
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]-[10]. 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
t t
J} = / ^(tk). . i WOdw^ ... dw^, (2)
]T>V"= J ^k (tk) ..J (i1)dwj;° ... dw£>, (3)
t t
where ^1(t),... (t) are nonrandom functions on [t,T], wT^ (i = 1,... ,m) are independent standard Wiener processes and wT0) = t, i1,..., ik = 0, 1, ...,
m,
J and
denote Ito and Stratonovich stochastic integrals, respectively.
Generalization of the method of expansion of iterated Ito stochastic integrals (2) based on generalized multiple Fourier series (see Theorem 5.1 ([6], p. 236) and Sect. 5.1 ([6], pp. 235-245)) composes the subject of the article.
Note that another approaches to the expansion and mean-square approximation of iterated Ito and Stratonovich stochastic integrals (2) and (3) can be found in [2]-[5], [21]-[38].
Suppose that ^i(t),...,(t) £ L2([t,T]). Define the following function (the so-called factorized Volterra-type kernel) on the hypercube [t,T]k
I^i(ii) ...^k (tk), t1 <...<tk
, (4)
0, otherwise
where t1,... ,tk £ [t,T] (k > 2) and K(¿1) = (¿1) for ¿1 £ [t,T].
Suppose that (x)}°=0 is an arbitrary complete orthonormal system of functions in the space L2([t,T]).
It is well known that the generalized multiple Fourier series of K (t1;..., tk) £ L2([t, T]k) is converging to K(t1;..., tk) in the hypercube [t, T]k in the mean-square sense, i.e.
lim
PlvvPk ^^
K-K
pi—pk
= 0,
L2([t,T ]k)
where
pl pk
KP1 ...pk
ji=0 jk=0 1=1
C =
Cjk ...jl =
[t,T ]k
is the Fourier coefficient, and
/k
K (t1,...,tk )ü (ti )dt1... dtk
.m,k 1=1
l/l
L2([t,T ]k)
( \1/2 J /2(t1,...,tk)dt1 ...dtk
\[t,T ]k
/
Consider the partition {rj}N=0 of [t,T] such that
(5)
(6)
(7)
t = t0 < ... < tn = T, AN = max Arj ^ 0 if N ^ œ, ATj = Tj+1 — Tj.
0<j<N—1
(8)
k
Theorem 1 [6] (2006) (also see [7]-[20]). Suppose that ^(т),... (т) are continuous nonrandom functions on [t,T] and {ф-(x)}°=0 is an arbitrary complete orthonormal system of continuous or piecewise continuous functions in the space L2([t,T]). Then
Pi Pk / k
j [^rk)=pi1;.-, £ ■•• ю M n j»
ji=0 jk=0 \/=i
1;1;Ш; ^ фj fo ^W^ ■ ■ ■ ф^ fo ^W^ , ^
N^œ ,, ~~ „ 1 k '
(/i,...,/kJGGk
where
Gk = Hk\Lk, Hk = {(/i,...,/k) : li,... ,1k = 0, 1,..., N - 1},
Lk = {(/i,...,/k) : 1i,...,1k = 0, 1,..., N -1; lg = (g = r); g,r = 1,...,k},
l.i.m. is a limit in the mean-square sense, ii5..., ik = 0,1,... ,m,
T
j = J j (t)dw« (10)
t
are independent standard Gaussian random variables for various i or j (in the case when i = 0), Cjfc...j1 is the Fourier coefficient (7), AwTj = wTj.++1 — wTj (i = 0,1,...,m), {Tj} =0 is a partition of [t,T], which satisfies the condition (8).
A number of generalizations and modifications of Theorem 1 can be found in [10], Chapter 1 (see also bibliography therein).
Let us consider corollaries from Theorem 1 (see (9)) for k = 1,..., 5 [6]
Pi
J[V/%' = l;i.m; £Cji j1», (11)
ji=0
Pi P2
J^ ^ = P1^ E E j cj;i»zj22» - 1{;i=;2=0}1{ji=j2} ), (12)
ji=0 j2=0
Pi P2 P3
j[^r» = i.i.m ££ Ej jji-
jl=0 i2=0 j3 =0
-1{:i=:2=0}l{^=j2}Cj33) - 1{:2=:3=0}1{i2=i3}C]:i) - 1{:i=::=0}iij^MCj2' ) , (13)
J
( :i...:4)
Pi
P4
4
l.i.m.
ii=0 i4=0
1=1
1 1 Z (:3)Z (:4» 1 1 Z (:2»Z (i4'
-1{:i=:2=0}1{ii=i2}Si3 Zi4 - 1{:i=:3=0}1{ii=i3}Zi2 Zj4 -
1{:i=:4=0}1{ii=i4}Zi2 Zj3 - 1{:2=:3=0}1{i2=i3}Zii j -
1 I A:i)A:3) 1 1 A:i)A:2) _L_
1{:2=:4=0}1{i2=i4}Zii j - 1{:3=:4=0}1{i3=i4}Zii j +
+ 1{:i=:2=0}1{ii=i2}1{:3=:4=0}1{i3=i4} + 1{:i=:3=0}1{ji=j3}1{:2=:4=0}1{j2=j4} +
+ 1{:i=:4=0}1{ii=i4}1{:2=:3=0}1{i2=i3}
(14)
J
{:i=:2=0} {:i=:4=0} {:2=:3=0} {:2=:5=0} {:3=:5=0}
(:i...:5)
Pi
P5
= l.i.m.
ii=0
EiMn zi
¿5=0
(:i) i
+ 1{:i= =i2=0} {ii =i2} {:3 :4=0}
+ 1{:i= =i2=0} {ii =i2} {:4 :5=0}
+ 1{:i= =i3=0} {ii =i3} {:2 :5=0}
+ 1{:i= =i4=0} {ii =i4} {:2 :3=0}
+ 1{:i= =i4=0} {ii =i4} {:3= :5=0}
Z (:3)Z (:4) Z (i5
{ii=i2}Zi3 Zi4 Zj5
Z (:2) Z (:3)Z (i5
{ii=i4}Zi2 Zi3 Zi5
Z (:i)Z (i4) Z (i5
{i2=i3}Zii Zi4 Zi5
Z (:i)Z (:3)Z (:4
{i2=i5}Zii Zi3 Zi4
Z (:i)Z (:2 ) Z (:4
{i3=i5}Zii Zi2 Zi4
Z (:5 {i3=i4}Zi5
Z (i3
{i4=i5}Zi3
Z (:4 {i2=i5}Zi4
Z (:5 {i2=i3}Zi5
Z(:2 {i3=i5}Zi2
1
1
{:i
{:i
1{i2 = 1{i3 = 1{:4 + 1{:i= + 1{:i= + 1{:i= + 1{:i=
+ 1{:
i=
:3=0} i5=0} :4=0} :4=0} :5=0} :2=0} :3=0} :3=0} :4=0} :5=0}
,1=1
Z (:2 {ii=i3}Zi2
Z (:2 {ii=i5}Zi2
Z(:i {i2=i4}Zii
Z(:i {i3=i4}Zii
Z(i4)Z(i5 Zi4 Z
i4 i5
i3 Zi4
i3 Zi5
i2 Zi5
Z (:i)Z (i2 ) Z (:3) +
{i4=i5}Zii Zi2 Zi3 +
{i'i=i2} {ii=i3} {ii=i3} {i'i=i4} {i'i=i5}
1 { :3 = i5=0} { i3 Z (:4) =i5}Zi4
1 { :2 = i4=0} { i2 . ,Z(i5)
1 { :4 = i5=0} {i4 ■ iZ(i2)
1 { :2 = i5=0} { i2 Z (:3) =i5}Zi3
1 { :2 = i3=0} { i2 . ,Z(i4)
5
+1 {i 1 = i5 =0} 1 {ji =j5 } 1 { i2 =i4=0} 1 {j2 =j4 } Cjs'3 ) + 1{i1=i5=0}l{ji=j5}1{i3=i4=0}l{j3=j4}C]2i2) + + 1{i2=i3=0}l{j2=j3}1{i4 = i5=0}l{j4=j5}C] ^ + 1{i2 = i4=0} 1{j2=j4} 1{i3 = i5=0} 1{j3=j5}Zj ^ +
+ 1{i2=i5=0}1{j2=j5}1{i3 = i4=0}1{j3=j4}Cj1il) ), (15)
where 1A is the indicator of the set A.
Consider a generalization of the formulas (11)-(15) for the case of arbitrary multiplicity k (k G N) of the iterated Ito stochastic integral (2).
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
({j9h92}, • • -,{92r-l,92r}}, {<?!, • • -Ak-2rJ), (16)
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 (16) is a partition and consider the sum with respect to all possible partitions
y ^ agig2,---,g2r-ig2r,qi...qk-2r, (17)
({{fl1,S2 },..., {fl2r-1,32r >},(91,...,9k-2r }) {si,S2,...,S2r-1,S2r,91,...,9k-2r }={1,2,-.,k}
where agig2,...,g2r-ig2r,qi...qk-2r G R
Below there are several examples of sums in the form (17)
^ agig2 = a12,
({S1,S2}) {S1,S2}={1,2}
y^ ag1g2,g3g4 = a12,34 + a13,24 +
({{S1,S2},{S3,S4}}) {S1,S2,S3,S4}={1,2,3,4}
y^ ag1g2,q1q2 = a12,34 + a13,24 + a14,23 + a23,14 + a24,13 +
({S1,S2},{91,92}) {S1,S2,91,92}={1,2,3,4}
E
agig2,qiq2q3
({31,32},{91,92,93}) {31,32,91,92 ,93} = {1,2,3,4,5}
— «12,345 + «13,245 + «14,235 + «15,234 + «23,145 + «24,135 + +«25,134 + «34,125 + «35,124 + «45,123,
E
«gig2,g3g4,qi
({{31,32},{33,34}}, {?l}) {31,32,33,34,91 } = { 1,2,-3,4,5}
— «12,34,5 + «13,24,5 + «14,23,5 + «12,35,4 + «13,25,4 + «15,23,4 + «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-
Now we can formulate Theorem 1 (see (9)) in another form.
Theorem 2 [8] (2009) (also see [9]-[15]). Suppose that ^1(r), (t)
are continuous nonrandom functions on [t,T] and (x)}°=0 is an arbitrary complete orthonormal system of continuous or piecewise continuous functions in the space L2([t,T]). Then the following expansion
P1 Pk / k [k/2]
J[^ (k)]Trk 1 — ,,, Li.m. E ■ ■. £ C..J nj1+E<-1)
p1,...,pfc ...
j1 =0 jfc =0 \Z=1 r=1
k-2r
x
S n 1{i32s-1 = i32s =0}1{j32s-1 = j32s } H ^ I (18)
({{31,32},...,{32r-1,32r }},{91 ,...,9k-2r}) S=1 1=1 /
{31,32,32r — 1,32r ,91 ,...,9k-2r }={1,2,.",k}
that converges in the mean-square sense is valid, where i1,..., ik — 0,1,..., m,
[x] is an integer part of a real number x, H —f 1, = 0; another notations are
0 0
the same as in Theorem 1.
Further in this article, we will consider a generalization of the expansion (18) to the case of an arbitrary complete orthonormal systems of functions in the space L2([t, T]) and ^1(t),..., (t) G L2([t, T]). Moreover, we will consider a modification of (18) based on the Hermite polynomials.
It should be noted that there is a work [39] in which an expansion similar to (89) was obtained (see Sect. 4 for details). A comparison of our results with the results from [39] and with other publications will be given in Sect. 4.
r
2 Preliminary Results
2.1 Expansion of Iterated Ito Stochastic Integrals based on Generalized Multiple Fourier Series
Suppose that $(ti,...,tk) e L2([t,T|k), ii ,...,ik = 0, 1,...,m, dw(0) = dr.
Let us introduce the following notation for the sum of iterated Ito stochastic integrals
t t2
JwTr*) d=f E J---J *(ti. • • ■ ■ tk )dw!:i)... dwi-), (19)
where all permutations (t1,... ,tk) when summing are performed only in the values dwt::)... dw^). At the
same time the indices near upper limits of integration in the iterated stochastic integrals are changed correspondently and if tr swapped with tq in the permutation (t1,..., tk), then ir swapped with in the permutation (i1,..., ik). In addition,
T t2
J...J ^(ti—tk )dw(::) ...dwt(:k)
is the iterated Ito stochastic integral.
Let us give an exumple of the sum (19) for k = 3
T is t2
(t1,t2,tS) tit
T ts t2
r^iT«« = ^ y y|$(ti,t2,t3)dw<:i'dwi:2>dw<:s>=
T (s ¡2 T t2 ts
J J y>$(ti,t2,t3)dwi;i)dwi:2)dwiss)+JJ y>$(ti,t2,t3)dwi;i)dwiss)dwi:2)+
t t t t t t T ts ti T ti ts
+// /^(ti,t2,t3)dwi:2)dwi ;i)dwiss)+JJ /^(ti,t2,t3)dwi:2)dwiss)dwi t t t t t t T t2 ti T ti t2
+JJ |$(ti,t2,t3)dwiss)dwi :i)dwC2)+jj/$(ti,t2,t3)dwiss)dwi:2)dwi :i). t t t t t t
Theorem 3 [10], [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 pk
J[#>]&• ■ :k> = l . i-m. £ ... £ Cjk . . j J''[ j... j]& • ■ H)
ji=0 jk=0
converging in the mean-square sense is valid, where J[^(k)]y1t"ifc) is the iterated Ito stochastic integral (2), J''[ j ... j ]T,t":k) is defined by (19) or has the form,
T t2
j"[0ji...j]Trk) = E Jj(tk)..Jj(ti)dwt(;i)...dwi:k),
(ji,...,jfc) t t
where
E
means the sum with respect to all possible permutations j..., jk). At the same time if jr swapped with in the permutation (j,...,jk), then ir swapped with in the permutation (i1,..., ik). Another notations are the same as in Theorems 1 and 2.
Proof. Using (19), we have
T t2
j[^(k)]Tr:fc)=/^&)..J^i(ti)dw( : i)...dwt:k) = j''[K]T:,v:k) w.p. 1, tt
(20)
where K = K(t1,... ,tk) is defined by (4).
Applying the linearity property of the Ito stochastic integral and (20), we obtain w. p. 1
J [^Tp > = j'wTr"' = j''[Kpi...pk ]Trk) + J''[K - Kpi...pk ]Tif:'' =
p i Pk
= E . . . E Cjk■■■j i J"[0j i . . . j+ J''[RP i...Pk, (21)
ji=0 jk =0
where
Rpi...pfc (t1, ) = K(t1, — Kpi...pfc (t1; • • • 5 ^k);
K(t1; • • • ,tk) and Kpi...pk(t1; • • • ,tk) are defined by (4) and (6), respectively; the Fourier coefficient Cjk...j has the form (7).
Note that (see (19))
J [RPi...Pfc ]t.
(«i..-ifc)
T t2
^ J
^iv-^k) t
2 / Pi Pk k \
• K(t1-----tk)-E • ••ECk-jin^(ti)
{ V ji=0 jk=0 i=1 /
x
xdwtn) • • • dw
(«fc) tk ;
where notations are the same as in (19).
According to the standard moment properties of the Ito stochastic integral [1| and the properties of the Lebesgue integral, we get the following estimate
m{ (/ '[Rpi...
] (ii ...ik) Pk ]T,t
<
<
T t2
Pi Pk
(tiI."Itk) t t
k (t1, • • •, tk) -e • ji=0
xdt1 • • • dtk =
ECjk ..^n ^ (ti )
x
jk=0
1=1
(22)
Pi Pk
Ck I IK(t1,•
[t,T ]k
■ tk) - E • • • E Cjk-ji n ^ (ti) ) dt1 • • • dtk
ji=0 jk =0 1=1
= Ck
KK
Pi...Pk
L2([t,T ]k)
(23)
where constant Ck depends only on the multiplicity k of the iterated Ito stochastic integral J [^iT^^), and permutations (ti, ...,tk) when summing in (22)
2
2
k
2
k
2
are performed in the expression dti... dtk. At the same time the indices near upper limits of integration in the iterated integrals from (22) are changed cor-respondently.
Combining (5) and (23), we get
lim m{ (j"[R,i . . . ^Vl =0. (24)
From (21) and (24) we obtain the following expansion for the iterated Ito stochastic integral (2)
pi pk
J [V;(k>iTrk) = l-i-m; E -EC,,..» J"[<j ...j iTTk), (25)
ji=0 jk=0
where J''[ j ... jiT^"^ is defined by (19).
It is easy to see that J''[ j ... j i T- k) can be written in the form
T t2
jU-i...&jTrk) = E /j(tk)(ti)dwi:i)...dw(:k), (26)
(j i,..-;j'fc) t t
where
E
(j i,.--,jfe)
means the sum with respect to all possible permutations (ji,...,jk). At the same time if jr swapped with in the permutation ji ..., jk), then ir swapped with in the permutation (ii,..., ik).
The relations (25) and (26) complete the proof of Theorem 3. Theorem 3 is proved.
2.2 Modification and Generalization of Ito's Theorem. Proof on the Base of the Ito Formula and Without Explicit Use of the Multiple Wiener Stochastic Integral
In this section, we generalize Theorem 3.1 from [40i (1951) which gives the relaionship between the multiple Wiener stochastic integral and the Hermite polynomials. Recall that in [40i the case ii = ... = ik = 0 (the case of a
scalar standard Wiener process) has been considered. In the main result of this section, we will consider the case ¿1,..., ik = 0,1,..., m (the case of a multidimensional Wiener process). Moreover, our proof diffes from that given in [40] and is based on the Ito formula. Also, we do not explicitly use the multiple Wiener stochastic integral in the proof of Theorem 4. Although it should be noted that the sum (19), which plays a central role in the proof of Theorem 4, is equal w. p. 1 to the multiple Wiener stochastic integral with respect to the components of a multidimensional Wiener process (see the proof in [10], Sect. 1.11 for details).
Let us introduce some notations.
We will say that the condition (*) is fulfilled for the multi-index (i1... ik) (i1,...,ik = 0,1,..., m) if m1,..., mk are multiplicities of the elements i1,...,ik, respectively, i.e.
mi m2 mr
{¿1, . . . , ik} = {¿1, • • • , ¿1, ¿2, • • • , ¿2, • • • , V, • • • , V},
where r = 1,..., k, braces mean an unordered set, and parentheses mean an ordered set. At that, m1 + ... + mk = k, m1,... , mk = 0,1,..., k, and all elements with nonzero multiplicities are pairwise different.
It is not difficult to see that
J" [j•.. j ) = J
1-"«fc ) jll
mi m2
mk
mi m2 m^
(fi^n" ••• f^îk)
w. p. 1, where we suppose that the condition (*) is fulfilled for the multi-index
(i1...¿k) and {jgi,...,jgmi+m2+...+mfc} = hgi,...,jgfc} = {j1,... jk}.
Suppose that
Jhw, • • • , Jhij , jh2,i, • • • , jh2j , • • • , , • • • , , (27)
s-v-' s-v-' >—:—v-^J
n1,i n2,1 n^,;
where ni,/ + n2,/ +... + ndl,/ = m,; ni,/,n2,/,... ,ndj)z = 1,... ,m,; d, = 1,... ,m,; l = 1,..., k. Note that the numbers mi5..., mk, gi,..., depend on (ii,..., ik) and the numbers ni;/,..., ndl;/, hi;/,..., hdl;/, d, depend on {ji..., }. Moreover, {jg1 ,...,j'gfc} = {ji,...,jk}.
Let Hn(x) be the Hermite polynomial of degree n
2/o dn
or
[n/2]
Hn(x) = n! E
(—1)mx
m,„n—2m
m=0
m!(n — 2m)!2r
(n e N).
(28)
For example,
Ho(x) = 1, Hi(x) = x, H2(x) = x2 — 1, H3(x) = x3 — 3x, H4(x) = x4 — 6x2 + 3, H5(x) = x5 — 10x3 + 15x.
Let us formulate the following modification and generalization of Theorem 3.1 from [40] for the case ii,..., ik = 0,1,..., m.
Theorem 4 [10], [15]. Suppose that the condition (*) is fulfilled for the multi-index (ii... ik) and the condition (27) is also fulfilled. Furthermore, let (x)}°=0 is an arbitrary complete orthonormal system of functions in the space L2([t,T]). Then
J'
j1
... j]
(il ...ik) = jk ]T,t =
n
/=i
Mcj;;),)...^(cj:M, if i,=0
1{ml=0} + 1{ml>0} <
V
, VSjhl,J V3hdltlJ
ndl,l
if i/ = 0
/
w. p. 1, where Hn(x) is the Hermite polynomial (28), 1A is the indicator of the set A, ii,... = 0,1,... ,m; ni,/ + n2,/ + ... + ndl,/ = m/; ni,/,n2,/,... ,ndj)/ = 1,..., m/; d/ = 1,..., m/; l = 1,..., k; mi + ... + mk = k; the numbers mi,..., mk, gi,..., depend on (ii,..., ik) and the numbers ni;/,..., ndl;/, hi,/,..., hdi,/, d/ depend on {ji,..., jk}; moreover, j,... j} = {ji,..., jk};
T
j = J 0j(r)dw[i) (i = 0,1,..., m; j = 0,1, 2,...) t
are independent standard Gaussian random variables for various i or j (in the case when i = 0) and dw[0) = dr.
Proof. First, consider the case ii = ... = ik = 1,..., m and ji ..., jk £ {0} U N. This case has been considered in [40], but we give a different proof here. By induction, we prove the following equality
T t2
p! y 0/ (tp) ..J 0/ (ti)dwiii) ...dw^x t t
T t2
x E / 0jq (tq) ...j 0ji (ti)dwt i)... dw^ =
(j1,-,j9) t t
t t2 ti t2
= E / 0jq (tq) ...J 0ji (ti)/ 0/ (tp) ...J 0/ (ti)x , /,...,/ ) t t t t
P
xdwt(i)... dw^dw^ ... dw^ (29)
w. p. 1, where p £ N, l = j,..., jq, and
E
(qi,...,qn)
means the sum with respect to all possible permutations (qi,..., qn).
Consider the case p =1. Using the Ito formula, we get w. p. 1 for s £ [t, T]
t2
t )dw[iw 0jq (tq)... i 0ji (ti)dwt(ii)... dwti =
s
T ¿2
T )0jq (t) J 0jq-l (tq—i) ...J j (ti)dw(!) . . . dwV 1dT+ t t t S T ¿2
+ J 0/ (T) J j (tq) ..J j (ti)dwt(ii) . . . dw^^ t t t
T T ¿2 \
+ / 0j (t)( i 0/ (^dw^l 0jq-i (tq—i) ..J j (ti)dwt(i) ...dw^M dwTi).
tt
(30)
Hereinafter in this section always s e [t,T]. Differentiating by the Ito formula the expression in parentheses on the right-hand side of equality (30) and combining the result of differentiation with (30), we obtain w. p. 1
J(/)s,t J(jq ...jl)s,t = S T ¿2
= / 0/ (T )0jq (T )J 0jq-1 (tq—i) ..J 0ji (t^dw^ ...dwii^ 1 dT + t t t
+ J(/jq ...j1)s,t +
s t 9 ¿2
+ / 0jq (t )/ 0/ (0)0jq-1 (0)J 0jq-2 (tq—2) ..J j (Odw^... dw^- 2 dfldw^ t t t t
+ J(jq /jq-l"..^^ +
+ J 0jq (T 0jq-1 (A)* t t
/9 9 ¿2 \
* I J 0/ (u) dwii^ 0jq-2 (tq—2) ..J 0j1 (Odw^... dwt(i- J dw9i)dwTi), \t t t /
where
¿2
0jq (tq) . I 0j1 (ti)dwi(i) . . . dw(i) = J(jq ...jl^.
s
s
Continuing the process of iterative application of the Ito formula, we have w. p. 1
J(jq -.71^ =
= J(/jq -.71^ + J(jq/jq-1...j1 )s,t + ... + J(jq ...j"^^
S T ¿2
+ J 0/ (t )0jq (t ) J 0jq-l (tq—i) ..J 0jl (tOdw^... dwt(i-1 dT + ... t t t S ¿3 ¿2
... + / 0jq (tq) ...J 0j2 M / 0/ (T )0jl (t )dTdwt(i)... dwi(i). (31) t t t
Summing the equality (31) over permutations j..., jq), we get
E J(/)s,t J(jq...jl)s,t = E J(/jq"¿1^ + S(s) (32)
(j1,...,jq ) (j1,-..,jq,/)
w. p. 1, where
S (s) =
/ s T ¿2
= E / 0/ (T )0jq (T) / 0jq-l (tq—i) -.J 0jl (ti)dwt(!) . . . dw(i^l dT + ...
(j'l,".jq) \ t t t
s ¿3 ¿2 \
... + / 0jq(tq) . . .j 0j2 fe) / 0/(T)0jl(t)dTdwt(i)... dwif . (33) t t t /
Consider
s s ¿2
J 0/ (T )0jq (T )dT J 0jq-1 (tq—i) . J 0jl (ti)dwi!) ...dwii^ 1 . t t t
Applying the Ito formula, we get w. p. 1
s s ¿2
(T )0jq (T )dW 0jq-1 (tq—i) ... 0jl (ti)dwi!) ...dwii^ 1 =
T t2
T )0jq (T) 0jq-i (tq-1 ) - 0ji (ti)dwt(ii) ..
. i dT +
t
s
+ / 0jq_i (tq-i ) X
jq_iv ^q-
t
tq-1 tq-1 t2
X I j 0/ (t )0jq (t )d^ 0jq-2 (tq-2) ..J 0ji (ti)dwi(i) ...dwt(i-2 ) dwt(i_) i. t t t
By iterative application of the Ito formula (as above), we obtain w. p. 1
s s t2
T )0jq (T )dT / 0jq-1 (tq-i ) ... 0ji (ti)dwt(i) ...dwt(ll__ 1 =
T t2
T )0jq (T) / 0jq-1 (tq-1 ) - 0ji (ti)dwt(i) ..
. dT + . . .
s t2 ti
... + / 0jq_i (tq-i) ...J 0ji (ti) / 0/ (T )0jq (t )dTdw(ii)... dw(i_i. (34) t t t
Summing the equality (34) over permutations (j,..., jq), we get
s s t2
E / 0/ (t )0jq (t )dT J 0jq_i (tq-i).../ 0ji (ti)dwt(ii) ...dwt(i_i = Si(s),
(j1,...,jq) t t t
(35)
w. p. 1, where
Si(s) =
t2
E ( j 0/ (T )0jq (T ) / 0jq_1 (tq-i) - f 0ji (ti)dwt(il} . . . dw^i dT +
(j1v"Jq ) V t
s
s
s t2 ti
. . . + [ 0jq_1 (tq-i) ... i 0ji (ti^ 0/ (T )0jq (t )dTdw(i) . . . dwt(i_ J . (36)
It is not difficult to see that
S(s) = Si(s) w. p. 1. (37)
Moreover, due to the orthogonality of {0j(x)}°=0 and (35), (37), we have
S(T) = Si(T) = 0 w. p. 1. (38)
Thus (see (32), (38)), the equality (29) is proved for the case p =1. Let us assume that the equality (29) is true for p = 2, 3,..., k - 1, and prove its validity for p = k.
From (32) for the case q = k - 1, j = ... = jk-i = l we obtain
(Ji)s,t (k - 1)! (Jk-i)s,t = k! (Jk)s,t + S2(s) (39)
w. p. 1, where
S2(s) = S (s)
(k > 2) and S2(s) = 0 (q = k - 1, k = 1),
ji=...=jq=/, q=k-i t2
J 0/ (tr) -..J 0/ (ti)dw(ii)... dwti =f (Jr )s,t (r £ N) and (= 1. tt
Taking into account (33), (35)-(37) and the orthonormality of {0j(x)}°=0, we have
S2(T) = (k - 1)!(Jk-2)T,t. (40)
Combining (39) and (40), we obtain the following recurrence relation
k! (Jk)t,( = (Ji)T,( (k - 1)! (Jk-i)T,t - (k - 1)! №-2)^ (41)
w. p. 1.
s
Using (41) and the induction hypothesis, we get w. p. 1
T ¿2
k! J 0/ (tk) ..J 0/(ti)dw(!) ...dw^x ¿¿
T ¿2
x £ / 0jq(tq) . J 0jl (ti)dwt(!)... dwt(i) =
T ¿2
tq) . . . w 1 w q
(j1,jq) i t
T / T ¿2
( (k 1)! [ 0 (t. ^ [ 0 (t. )d„ r(i) r(i)
0/ (t ) dwt!^ (k — 1)! / 0/ (tk—i) - - - / 0/ (ti)dw{;)... dw{fck-l *
¿
T ¿2
T ¿2 \
x E / 0jq (tq) ...J 0jl (ti)dwi!)... dwiiM —
Cilv-Jq) t t '
T ¿2
— (k — 1)! y 0/ (tk—2) ..J 0/ (ti)dwi!)... dwg- 2 * ¿¿
T ¿2
x E / 0jq(tq) . J 0ji (ti)dwt(!)... dwt(i) =
(j1,jq) t t
t t ¿2 ¿1 ¿2
J 0/ (t) dwT!) E / 0jq (tq) 0ji (ti)/ 0/ (tk—i) - J 0/ (ti) *
t (j1,---,jq, /,...,/ ) t t t t
k-1
xdwC/1)... dw.(i) dwi!)... dwC1) —
¿1 ¿k-1 ¿1 ¿q
T ¿2 ¿1 t
2
(k — 1) £ / 0jq (tq) 0jl (ti)/ 0/ (tk—2)0/ (ti)x
(j1,...,jq, /,...,/ ) t
k-2
* dw(!)... dw,, dwj!)... dwt(i). (42)
¿1 ¿k-2 ¿1 ¿q
Let [j| be the symbol I which does not participate in the following sum with respect to permutations
£
(jiv--Jq, /.---,/ ) k_1
Using (32), we have w. p. 1
t2
ti
t) dw« £ J 0jq (tq)... / 0ji (ti) / 0/(tk-i)
0/(ti)x
(j1,...,jq, /,...,/ ) t
k_1
xdwii)... dw)dw}i;... dw)i; =
(i) (i)
(i)
't'fc_r' (I
s
£
(j1v--Jq, /,...,/ ) t
t2 ti
0jq(tq) . . . /0ji (ti^ 0/(tk-i) .
t
t i rjA q
tt
0/ (ti)x
k_1
xdw(i)... dw,(i) dwt(i)... dwt(i) =
(i tfc_i (1 tq
£
(j1,...,jq, /.---,/ ) k_1
k_1
k_1
... + J.
+ J
+ ... + J
k_1
k_2
k_1
3(s) =
(ji,-..,jq, /,...,/ ) k
(43)
where
3(s) =
t2
£
I tl(Vl) - I (t)Jlitl)X
(j1,---,jq,/,---,/_) V t k_1
t
s
s
2
q
1
t
s
2
¿1
t'n
* j 0/(ti—i).. .j 0/(t;)dw(!)... dw(i-1dwt(!)... dw£ldT +... ¿¿
s ¿3 ¿2
+ ...J 0jq (tq) ...J 0j2 (t2)
t t t t ¿2
* 10/ (tk—i) ..J 0/ (ti)dw(!)... dw(i-1 dтdwC21)... dw^ ¿¿
s ¿2 ¿1
+ J <f>jq{tq) • • • J fe(il) J <f^{r)(f>l{T)x
t t t T ¿2
* I 0/ (ti—2) ... / 0/ (tii)dw(!)... dwJJ-2 dTdw^ ... dw^ + ...
(i) i-"7/-J— /-] TTT(i)
k-2
¿2
... + / 0jq (tq) ... 0jl (ti)*
jqV ^q; t t
¿1
¿3 ¿2
x / / rM^rdw^
.. dw,(i) dwC!)... dwfi)
¿k-1 ¿1 ¿q
¿¿
s
Using (33), (35)-(37), we get w. p. 1
S3(s) =
s s ¿2
£ / <f\J\(T)<f>i(T)dr J (pJq(tq)... J (f)n(ti)x
(jl,■■■¿q, /,...,/ ) f t t
k-1
¿1 ¿2
* / 0/ (tk—2) ... I 0/ (tii)dwt(i)... dw(!) dwt(!)... dwj
¿1 ¿k-2
tq
t2
(k- 1) E j ^¡\(r)Mr)dr j cf>jq(tq)... J (ii>->i?) /,...,/ ) t t t
k_2
ti
x / 0/ (tk-2) ... j 0/ (ti)dw|i)... dw(i_ 2 dwt(ii)... dwt(i) +
s s t2
+ E / ^W^W^7" J ii(Vi )■■■ J /,...,/ ) t t t k_1
ti t2
x y 0/ (tk-i) ..J 0/ (ti)dw(i)... dw(i_ i dw^... dwt(i_ i + tt
tq_1
s s tq t2
+ E / J J ^(tq-z) ■ ■ ■ J
(jl,--,jq-2,jq /,...,/ ) t t t t k_1
ti
t'
x / 0/ (tk-i) ... / 0/ (ti)dw(i)... dw(i_ i dwti... dw^ dw^
s s t3
+ E / ^WfeW^ J hqitq) ■ ■ ■ J <f>h(h)x
(h ,--,jq /,...,/ ) t t t
fc_1
t2 t2 x y 0/ (tk-i) ...J 0/ (ti)dw(i) . . . dw(i_ i dwt(ii) . . . dw^ tt
(44)
Applying (44) and the orthonormality of {0j(x)}°=0, we finally have
T
S3(T) =
(k - 1) E J 0jq (tq)
t2
0jq (tq) ... 0ji (ti)x
(j1,---,jq, /.---,/ ) t
k_2
s
s
t
2
¿2
* y 0/ (tk—2) ...J 0/ (tii)dw(!)... dw(i- 2 dwt(!)... dwt(i). (45)
¿¿
Combining (42), (43), (45), we obtain w. p. 1
T ¿2
k! j 0/ (tk) ..J ^(tOdw^ ...dw^i^ ¿¿
T ¿2
* £ 10jq(tq) - J 0jl (ti)dwi!)... dwt(i) =
(j1,...,jq) t t
T ¿2
= £ j0/(tk)..J0/(ti)dwt(!)...dw^* ( /,...,/ ) ¿ ¿
T ¿2
* £ 10jq(tq) - J 0jl (ti)dwi!)... dwi(i) =
T ¿2
tq) . . . w 1 w q
(j1,...,jq) i t
T ¿2 ¿1 ¿2
£ J 0jq (tq) 0jl (ti)/ 0/ (tk ) - J 0/ (ti)*
(j1,...,jq, /,...,/ ) t t t t
*dw^... dw|i)dwi!)... dwi(i), (46)
where l = ji,..., jq.
The equality (29) is proved. From the other hand, (46) means that
q+n n q
J"[<j>h ... ^ 6l.. = J"\4>i ■ ■ ■ ' m-i • • • (47)
nn
w. p. 1, where n, q = 0,1, 2 ...; l = j2, ..., jq and
q
J' '[0jl --.0jq ]T2t--- !) = 1
for q = 0.
¿
1
Consider polynomials Hn(x,y), n = 0,1,... defined by [41]
Hn(x,y)=[—eax-a y'2
(Ho(x,y) =f 1).
(48)
a=0
It is well known that polynomials Hn(x,y) are connected with the Hermite polynomials (28) by the formula [41]
[n/2]
Hn(x,y) = ynl2Hn[^-)=n\Y,
VV,
¿=0
(-1 )lxn~2lyl i\(n-2i)\2r
(49)
For example,
Hi(x,y) = x, H2(x,y) = x2 - y, H3(x,y) = x3 - 3xy,
/\/10 O / Q O
H4(x, y) = x - 6x y + 3y , H5(x, y) = x - 10x y + 15xy . From (28) and (49) we get
Hn(x, 1) = Hn(x). Note that [41] (also see [10] (Chapter 6, Sect. 6.6) for details)
T t2
(50)
I M^dw^ ...dw^J = —H,
1
T T
(1) i
n!
T)dwT1), 02(r)dT =
t
t
T
T
n!
!"n
I (M^wW,! = J
(1)
(51)
w. p. 1, where n £ N, Hn(x,y) is defined by (48) (also see (49)), and Hn(x) is the Hermite polynomial (28).
From (51) we have w. p. 1
t t2
J'% ... =n\ I Utn) ...J Mt^dw^ ... dw£> =
tt T \ / T
n!
)dw<1M = Hn i &(t)dw<1>
(52)
where n € N.
Combining (47) and (52), we obtain
q+n / T
J"[<t>h ...h <!>№;■ A) = Hn\ / ■ J"[<t>h ... ^ • •1) (53)
yJq • • • VlJT,t — I I Yl\T)aWT I " LYjl'-'YjqJ
n \t
w. p. 1, where n, q = 0,1, 2 ...; l = ji,..., .
The iterated application of the formula (53) completes the proof of Theorem 4 for the case ii = ... = = 1,..., m and j,..., € {0} U N.
To prove Theorem 4 for the case ii = ... = = 0,1,..., m and j,..., € {0} U N, we need to prove the following formula in addition to the previous proof
T t2 T t2
P! j fa (tP) "J & (ti)dti . . . £ J j (tq) . . .J j (ti)dti . . . =
t t O'lv-Jq) t t
t t2 ti t2
= £ / j (tq).. J j (ti) / (tp) ...J & (ti)dti... dtpdti... dtq,
(j1,---Jqt t t t
P
(54)
where p € N,
E
means the sum with respect to all possible permutations (j,..., ).
First, consider the case p =1. We have
/88 t2 \
d J ^ (0)^/ j (tq) . . . / j (ti)dti ...dtq =
t t t
8 t2
= (s) / j (tq) . . . J j (ti)dti . . . dtqds+
tt /8 t2 8 \
+ 0jq (SH J 0jq-1 (tq-i) . . . J j (ti)dti . . . dtq-i • / (0)d0 I ds.
t t t
q
Then
s s t2
J 0/ (0)d^ 0jq (tq) ...J 0ji (ti)dti . . . dtq =
t t t
= I(/jq ---ji)s,t +
s / T t2 T
+ J 0jq (T ) J 0jq_1 (tq-i) . J 0ji (ti)dti . . . dtq-i -J 0/ (0)d0 | dT,
t t t t
where
s t2
J 0j (tr) . . .J 0ji (ti)dti . . . dtr = /(jr ---ji)s,t . (55)
tt
Continuing this process, we get
s
j 0/ E I(jq---ji)s,t = E 7(/j'q---ji)s,t. (56)
t (j1,---,jq) (j1,---,jq,/)
The equality (54) is proved for the case p =1. Let us assume that the equality (54) is true for p = 2,3,..., k - 1, and prove its validity for p = k.
From (56) for j = ... = jq = l, q = k - 1 we have
(/i)s,t (k - 1)!(/k-i)s,t = k! (Ik)s,t, (57)
where k £ N and
s t2
i 0/(tk) . . i 0/(ti)dti . . . dtk = (Ik)s,t , (/0)s,t = 1.
Using (57) and the induction hypothesis, we obtain
k!(/k)s,t E /(jq---ji)s,t = (/i)s,t (k - 1)!(/k-i)s, E /(jq---ji)s,t =
(j1,---,jq ) (j1,---,jq )
= E ^-J'l /,...,/ W = E ^g^t^q-Jl /,...,/ )S,U (58)
(j1,---,jq, /,...,/ ) fc_1 (j1,---,jq, /,...,/ ) fc_1
fc_1 fc_1
where /(^...j^s^ is defined by (55) and [7] is the symbol I which does not participate in the following sum with respect to permutations
£ .
(j1 v--Jq, /,---,/ )
k-1
By analogy with (56) we obtain
£ Wt
(j1,---jq, /,---,/ ) k-1
k-1
(iii-j?) /,...,/ ) \ fc-i fc-i
k-1
k-1 k-2 k-1
£ Z./,..../>■/• (59)
(j1,---,jq, /,.-.,/ ) k
k
Substituting s = T into (58), (59) and combining (58), (59), we conlude that the equality (54) is proved for p = k. The equality (54) is proved.
Note that
T t2 / T \n/T n
n! J Utn) .J Mti)dti ...dtn = n\^ij (f)i(r )dr J = IJ (f)i(r )dr
t t t t
(60)
where n € N.
After substituting (60) into (54), we have for p = n
(T V
I J (f>i{r)dr I E J(jq...h)T,t = £ (61)
\t / (jlr-Jq) (jl,-,jq,l,...,l)
The equality (61) means that
q+n
I /«r)drl ■J'yn...lpjq]g--(i\ (62)
Jq ^rt ' ' ' Tly\T,t I I ' V 7 | Lr Jl ' ' ' rjqi
n \t
where n, q = 0,1, 2 ... and J'^j... j]tî"0) = 1 for q = 0.
The relations (53) and (62) prove Theorem 4 for the case ¿i = ... = ik = 0,1,... ,m and ji, ... G {0} U N.
Remark 1. Note that the equality (54) can be obtained in another way. Let = {(t1;... ,tq) G [t,T]q : 3 i = j such that t = tj} be the "diagonal set" of [t, T]q (q = 2, 3,...) [42]. Since the Lebesgue meashure of the set is equal to zero [42], then (see (19))
q
/ (63)
[t,T ]q
From (63) we have
p q
= J 0ji (ti)... 0jq (tq)dti... dtq y 0/(ti)... 0/(tp)dti... dtp =
[t,T ]q [t,T ]p
= J 0ji (ti) ... 0jq (tq) 0/ (ti) . . . 0/ (tp)dti . . . dtpdti . . . dtq =
[t,T ]p+q
p+q
= (64)
It is not difficult to see that the equality (64) is nothing but the equality (54) written in another form.
To complete the proof of Theorem 4, we need to consider the case ii,... ,ik = 0,1,... ,m and ji,..., jk £ {0} U N.
Obviously, the proof of Theorem 4 will be completed if we prove the following equalities
T t2
E / j (tq) .../ j (ti)dwt(i1) ...dwt(iq )x
(j1v--Jq) t
T
x E / j (tn) (t/i)dw(i)... dwt(n) =
(jl ;---;jn) t
T
t t2
t1
E J j(tq)..7 ^ (ti)J j(tn) - J jM)x
(j1,---Jq j'i ,-j'J t t t t
xdw,(,i)... dwt(,i)dwt(;1)... dwt(iq),
t1 n
(65)
t
2
t
2
T t2
E / j (tq) - J j (ti)dw(;1) ...dwt(;q )x
(j1v--Jq) t t
t t2
x £ / j (in) -¡j (iDdw«:" ...dw<?> =
(j1,---Jn) t t
t t2 t1 t2
= E / j (tq) ...J j / j (tn) ...J j (ti)x
(j1,---Jq,j1,---,jn) t t t t
xdwt(0)... dwt(0)dwt(;1) ... dwt(;q) (66)
w. p. 1, where n, q € N, dw[0) = dr, ii,..., iq = 1 in (65) and ii,..., iq = 0 in (66),
£
(j1>---Jg)
means the sum with respect to all possible permutations j ..., ). At the same time if jr swapped with j in the permutation (j,..., ), then ir swapped with in the permutation (ii,..., ).
The equalities (65) and (66) mean that
J"[ j . . . j j . . . fe ]ii,1t---;q i---i) = J"[ j . . . j iTi--^) ^ j//[0j1 . . . j ]T/t--l), (67)
J"[j ... j0i ... jj^^--'-0--^ = J"[0, ... 0jq) - J"[0,1... jfi-00 (68)
w. p. 1, where ii,..., iq = 1 in (67) and ii,..., iq = 0 in (68).
First, we prove the equality (65). Consider the case n = 1. Using the Ito formula, we get w. p. 1
t2
0ji (fl^ / 0jq (tq) ... i 0ji (ti)dwt(;i) . . . dwt('q) =
t t t
= J(i ;q---;i) +
(j1 jq---j1)s,t t2 T
+ / 0jq(T) ( I 0jq_i (tq-i) . . . / 0ji (ti)dwt(;i) . . . dw^/ 0ji (^)dw^ ) dw(;q)
tt
= j (i 'q---;i) , j (;qi ;q_1---;i) + + J ('q---'i1 ) (69)
(j'1 jq---ji)s,t (jq j'i jq_ 1 ---j 1 )s,t . . . (jq---ji j'i )s,t , ( )
where
t2
r(;i) j,T,.(;r) def T(;r---;i)
j (tr)... 0ji (ii)dwJi" ... dw^' = J^^, (70)
ii,...,ir = 0,1,..., m. From (69) we obtain
t2
0ji Wdw^ £ J 0jq (tq)... [ 0ji (ti)dwt(;i)... dw£q)
(j'i,---,j'q) t
s s t2
E / 0ji 0jq (tq) ..J 0ji (ti)dwt(;i) ...dwt(
(ji,---,jq) t t t
E/ j (i ;q---;i) . j (;qi ;q_1---;i) + + j (;q---;ii )
V (j1jq---ji)s,t (jqj 1 jq— 1 ---j1)s,t . . . (jq---j1j1)s,t
(j1,---,j'q )
s
s
s
= £., Jjq-j 1i)>8,t (71)
(j 1 ?--- ,jq' )
w. p. 1, where jj- - -j 1 )8t is defined by (70). The equality (65) is proved for the
case n = 1.
Let us assume that the equality (65) is true for n = 2,3,..., k — 1, and prove its validity for n = k.
Applying (32), (33), (35)-(37), we obtain w. p. 1
8 t2 E / j (tk) •../ j (ti )dw(i' ...dw«i> =
(j '>---j'k) t t
8 8 t2 = J j (#)dw^ J] J j 1 (tk—i) ..J j ' (tOdw^ ... dwt(i-1 —
t (j ',---,jk- 1) t t
8 8 t2 — I] J j Wjk- 1 / (tk—2) ...Jj (ti )dw(i>... dw(i-2. (72)
(j i>---j'k- 1) t t t
Substituting s = T in (72) and applying the orthonormality of (x)}°=0, we get w. p. 1
t t2
E / j (tk) .J ^ii (t i )dw( i> ...dwji' =
(j ''---'jfe) t t
T T t2
= J j(0)dw^i) E / j 1 (tk—i).. J hi (ti)dwt(ii)... dw(i- 1 — t (j i>---Jk- 1) t t
T t2
(i) dw(i)
k k- k-2 k—2 j 1 wt . . . dwtk-2
(j iv-Jk- 1)
E 1W=*-1 } / ^k-2 (tk—2) ... ^j!(ii)dw(l1i' . . . 2 , (73)
where 1A is the indicator of the set A.
Using (73) and the induction hypothesis, we obtain w. p. 1
T t2
E / 0j (tk) - J 0ji (ti)dwt(ii)... dwt(ii) x (j1'---'jfc) t t
T t2
x E / 0jq(tq) . J 0ji(ti)dwt(;i)... dw£q) =
(j1v--,j'q) t t
T T t2
/ 0jk (^dw^ E / 0jk_i (tk-i) ...J 0ji (ti)dwt(ii)... dwt(i_ 1 x t (ji,---,jfc_i) t t
T t2
x E / 0jq (tq) -.J 0ji (ti)dwt(;i)... dw£q) -
(j'i,---,j'q) t
T t2
E 1 j =jk_1^ 0jk_2 (tk-2) ...J 0ji (ti)dwt(i) . . . dwi(i_ 2 x
(j1v--jfc_1) t t
T t2
x E / 0jq(tq).. J 0ji(ti)dwt(;i)... dw£q) =
(j1>---,j'q) t t
T
= 0jk (tf)dw(i)x
t t2 ti t2
E / 0jq (tq)... i 0ji fa) / 0jk_i (tk-i)... i 0ji (^i)x
/ ... J . J \ J J J J
(ji,---,jq,j1,---,jfc_1) t t t t
xdwt(i)... dwt( ,i) dwt(;i)... dwt(;q)-
(i tfc_i (1 tq
T t2
E 1 j =jfc_iM 0jk_2 (tk-2) ... 0ji (ti)dwt(i) . . . dwt(i_ 2 x
'fc_ -'fc_1
(j1;---;jfc_1)
T t2
x E / j (tq) .J j 1 (ti)dwt(; 1)... dwt(;q). (74)
(j 1v--Jq) t
Further, applying the induction hypothesis, we have w. p. 1
T t2
E 1{jk=jk- 1} 0jk-2(tk—2) .. &i(ti)dwt(i)... dwt(i-2x
k k-
(j iv-j'k- 1)
T t2
x E / j(tq) .. J k 1 (ti)dwt(; 1) . .. dw£q) =
(j 1v--Jq) t
T t2
E 1{jk =jk- 1 }J 0jk-2 (tk—2) ...J h ' (ti)dwJ i} . . . dwtkk-2 +
(j i ,---,jk—2) t t
T tk —2 t2
+ E 1{jk =jk—2 } / j 1 (tk—2) f j 3 (tk—3) ... / '(ti)x
(j 1 r-j'k—3 — 1)
xdwt(i) •••dwt(1—3 dwt(i— 2 + ...
T t3 t2
+ E 1{jk =H}J h—2 (tk—2) .../ j M/ j1 (ti)x
C^v-Jk—1) t t t
X^W^ •••dwt(1— 2 I x
T t2
x E / j(tq) . . J k 1 (ti)dwt(; 1) .. . dwt(;q) =
(j 1>---j'q) t
T t2
1{jk =jk — 1} E / 0jk—2 (tk —2) ...j h ' (ti)dwt(i) . . . dwt(i— 2 + (j ^---.j'k —2) t t
T tfc_2 t2
+ 1{j =jk_2} E / 0jk_i (tk-2^ 0jk_3 (tk-3) ...J 0ji (ti)x (j1 r--Jfc_3 Jfc_1) t t t
xdwt(i) ...dwt(i_ 3 dwt(i_ 2 + ...
T t3 t2
... + 1 j =ji} E / 0jk_2 (tk-2) ...J 0j2 (t2^ 0jk_i (ti)x (j2,---jfc_i) t t t
X^W^ . . . dwt(i_2 I x
T t2
x E / 0jq (tq) ...j 0ji (ti)dwt(;i)... dw£q) =
(j1 >--- jq) t
t t2 ti t2
1 j =jk_i} E / 0jq (tq)... i 0ji fa) / 0jk_2 (tk-2)... i 0ji fa)x
/ ... 1 .1 \ J J J J
(j1 ,---,jq,j1,--- ,jfc_2) t t t t
xdw^... dw^ dwt(;i) ... dwt(;q) +
(i tfc_2 (1 tq
T t2 ti
+ 1{j =jk_2} E / 0jq(tq) - 0ji(t0 / 0jk_i(tk-2)x (j ,---,j ,j ,---,j ,j )
(j1;---;jq ,j1 >--->jfc_3 'jfc_ 1 ) t t t
tk_2 t2 x f 0jк—3 (tk-3) ■■■[ j (ti ^... d-w^ 3 dw'^ 2 dw<;i)... dwf') +
T t2
+1{j'=ji} E / 0jq (tq) .../^i (ti)x (ji,--
ti t3 t2
xf j _2 (tk-2) ■■■[ j (4) / 0j-i _1 (tDdw^dw^ ... dw;i—2 dw<;'>... dwf') =f
d=f S4(T). (75)
By analogy with (34) we obtain w. p. 1
T T t2
/ fo/ (t^ (t )dTj j 1 (tr—i) .../ j (ti)dw(i1) ...dwi;:;1) = t t t
T T t2
= / fo (t (t )/ j1 (tr—i) .../ h 1 (ti)dw(; 1) ...dwi;— 1 )dr +...
t t t
T t2 t 1
... + / j1 (tr—i) ...J fo 1 (ti) / 0/ (t (t )dTdwt(; 1)... dw(;:— 1), (76) t t t
where ii,..., ir—i = 0,1,..., m.
Using iteratively the Ito formula, as well as (76) and combinatorial reasoning, we get w. p. 1 (see Remark 2 below for details)
T
J j (0)dw^i)x
t
t t2 t1 t2
x E / j (tq)... i j (ti) / j1 (tk—i)... i ' (ti )x
/ ... I . I \ J J J J
(j 1,---,jq1,---,jk— 1) t t t t
x dwt i)... dwt( i) dwt(; 1)... dwt(iq) =
t tk— t tq
t t2 t1 t2
= E / j (tq) ...J j (ti)/j (tk) ...J fo '(ti)x
(j 1,---JqJ i,---,jk) t t t t
x dwt i)... dwt( i)dwt(; 1)... dwt(iq) +
t tk t tq
, T t2 ti (
+ E (/ 0jq (tq)... / 0ji (ti) / 0j '№0j'_i (0) / 0j_2 (tk-2) (j1,---,jq,j1,---,j '_1 A t t t t
t2
t2
J 0ji (O^... dwíl—2 dwfdwi;^... dw«;q)+
T t2 ti t'fc_1 (
+ i 0jq (tq) ... / 0ji (ti) / 0j_i (ti-i^ 0j(0)0j_2 (0) / 0j_3 (tk-3)
t2
...J 0j1 (ti) dw(i)... dw^ 3 dw(0)dwt(i) i dw( ;i) ^ - ( ;q)
,t' . ,.„wt' uw( „wt' u„(i . . . dwf +
(1 t k_3 ( tfc_1 (1 tq
t t2 ti t3 t2
+ / 0jq (tq)... i 0ji (ti) / 0j_i (tk-i)... i 0j2 (t2 ) / 0j(0)0ji (0)dw(0)x
xdwt(i)... dwt( ,i) dwt(;i)... dwt(;q)
t2 tfc_i (1 tq
t t2 ti t2
E / 0jq (tq) ...J 0ji (ti )/ 0j'(tk) ■■■/ 0ji (ti)x
(j1 ,---,jq,j1,---,j ^ t t t t
xdwt(i)... dwt(i)dwt(;i)... dwt(;q) + (i tk (1 tq
✓ T ( t2 ti
+ E / 0j'(0)0j'_i (°)J 0jq (tq) . J 0ji (ti ) / 0j_2 (tk-2) . . .
(j1 ,---,j'q ,j1,---,j fc_2 A t t t t
t2
0ji (ti)dw|i)... dw|i_ 2 dwt(;i)... dwt(;q )dw(0) +...
t t2 t 1 t2 t 1
+ foq (tq) .. j (tiW fojk—2 (tk —2) .. jM) / j (Wjk—1 (^)dw^0)x
x dwt i)... dwt i) dwt(; 1)... dwt(;q) > +
t tk—2 t tq
✓ T 0 t2 t1
+ E I / j (^)fojk—2 (0) / j (tq) --J j (ti) / fojk—1 (tk —i)x
-jq3,jk—t t t t
tk—1 t2 x / fojk—3 (tk—3)-~J j (ti)dw<i» ...¿wii—^¡^dw';1' •••dw«q')dw<0) + ...
tt
t t2 t1 t;k—1 t2
+ i fojq (tq) ... i j (ti ) / fojk—1 (tk —i^ fojk—3 (tk —3) ... i j (ti)x
ti
x i j (0)fojk—2 (0)dw(0)dw(i) ...dwt(i—3 dw(i—1 dwt(i1) ...dwt(;q^ +
✓ T 0 t2 t1
+ E / j (°) I j (tq) ...J j (ti) / fojk—1 (tk—i)
(j1 ?--- jq j'2 v-Jk — t t t t
t3
j (t^dw^... dw^ dwt(i1) ... dwt(;q)dw00) +
t2 tk—1
t t2 t1 t3 t2
+ / j (tq)... i j (ti) / fojk—1 (tk—i)... f j (t2 ) / j (0)j (0)dw00)x
x dwt i)... dwt( i) dwt(i1)... dwt(iq)
t2 tk—1 t1 tq
t t2 ti t2
E / 0jq (tq) .j 0ji (ti)/0j(tk) ./ 0ji (ti)x
(j1,---,jq,j1,---,j^ t t t t
xdw( 1 )... dwt(i)dwt(;i)... dwt(;q) +
(i tk (1 tq
T T t2 ti
+ [ 0j(0)0j_i(0)d0 E / 0jq(tq)... i 0ji(ti) / 0j'_2(tk-2)
t/ ^ J J J
t (j1,---,jq ,j1,---,j fc_2) t t t
t2
0ji(ti)dw|1) ■■■dwí(1^dwt(;i)...dwt;q) +
(1 t k_2
T T t2 ti
+ i 0j(0)0j_2(0)d0 E / 0jq(tq)... f 0ji(ti) / 0j_i(tk-i)x
,... j . j . j j J J
t (j1,---,jq,j1 ,---,jk_3,jk_1) t t t
t k_1 t2 xj j_3(tk-3) . J j(Odw«... dw;i—3dw<1__dw™... dwf') + ...
tt
T T t2 ti
... + / 0j(0)0ji (0)d0 E / 0jq(tq)... i 0ji (ti) / 0j_i (tk-i)...
t/ x t/ t/ t/
t (j1,---,jq ,j2 ,---,j k_1) t t t
t3
J 0j,(t/2)dw|1) ...dwl 1_ 1 dwt(ii)... dwi;q) t
t t2 ti t2
= E / 0jq (tq) ...J 0ji (ti )/ 0j'(tk) ...J 0ji (ti)x
(j1 ,---,jq,ji,---,jfc) t t t t
xdwl")... dwt(1)dwt(1i)... dwt(qq) + S4(T). (77)
From (74), (75), and (77) we conclude that the equality (65) is proved for n = k. The equality (65) is proved.
q
Remark 2. It should be noted that the sums with respect to permutations
( j ,---,jq ,j ,---,jk — )
in (77), containing the expressions j(0)j—1 (0),..., j(0)foj '(0), should be understood in a special way. Let us explain this rule on the following sum
T t2 t1 0
£ J j (tq ) -J j (tl) / j (0)foik- 1 (0) / j—, (tk—2) . . .
(j 1,---JqJ 'j-Jfe — 1) t t t t
t2
..J fo '(ti)dwt(i)... dwt(i—2 dw00)dwt(; 1)... dwt(;q). (78)
t
More precisely, permutations j/ ..., jq, j/, ..., Jk_ J when summing in (78) are performed in such a way that if j* swapped with j* in the permutation
(j+k—^..., Ji) = (j,..., jl, jk—/, jk—2,..., J'l) , then i* swapped with id in the permutation
k-2
Moreover, j swapped with j in the permutation
(^Vk—1 ,...,40 = (j , j ^ 1, 2 ^..j.
A similar rule should be applied to all other sums with respect to permutations
( j ,---,jq ,j ,---,jk — )
in (77) that contain the expressions j (0)j—2 (0),..., j (0)foj '(0).
Let us prove the equality (66). Consider the case n = 1. By analogy with (69) and (71) we obtain
8 8 t2
/ j(t,)...J^(ii)dw'; 1 >...w';«> = £ jgj
t (j 1,---,jq) t t (j 1'---'jq0
w. p. 1, where J((j;r---j1)st is defined by (70). The equality (66) is proved for the
case n = 1.
Let us assume that the equality (66) is true for n = 2,3,... ,k - 1, and prove its validity for n = k.
In complete analogy with (56) we get
s s t2
/ 0j'(0)d0 / 0j'_1 (tk-1)■ ■■ 0j(t1 )dt1 ■■■dtk-1 =
= J(0...0) + J(0...0) + . . . + J(0...0) . (79)
= J(jj_i---j')s,t + J(j '_i j j_2---j ')s,t + ■ ■ ■ + J(j'_ 1 ---j j)s,t. (79)
Applying (79), we have
t t2
E j (tk)...j(ti^■ ..dw^
(j i'---,j D t t
V^ / 7(0---0) + j(0---0) + + j(0---0)
Z^ I J(j i---j ')s,t + J(j'_ j j_2---ji)s,t + ■ ■ ■ + J(j'_ i---j j)s,t
(j i,---,j i)
t t t2
0j'(0)d0 E / 0ji(tk-i) ■■ 0j'(ti)dwt(0) ■ ■ ■ dwt(0_ i. (80)
t (j i,---,j i) t
t
Using (80) and the induction hypothesis, we obtain w. p. 1
T t2
E / 0j '(tk) ■■■ / 0j i(ti)dwt(0) ■■■dwt(0)x
,(0) ^ ,(0).
(j i'-'j' D 1 t
T t2
x E / 0jq(tq) ■ ■ J 0j 1 (ti)dwt(; 1) ■ ■ ■ dw£q) =
(j i'---'jq) t t
T T t
2
0j'(0)d0 E I 0j'_i(tk-i) ■ ■ ■ I 0ji(ti)dwt(0) ■ ■ ■ dwt(0_i x
(ji,---,jfc_1 ) t
T t2
x E / 0jq(tq) ■ ■ ■ / 0ji(ti)dwt(;i) ■ ■ ■ dwt(;q) =
(j1'---,jq) t t
T T t2 = / 0j(0)d0 £ Jj (tq ) ■■^0ji (ti )x
t (ji'-- -,jq,ji,---,jk_i) t t ti t2
xjj_1 (tk-i) ■ ■ ■ /0ji(ti^...dw|0— idw«;^■ ■ ■ dw<;q)
tt
T T t2
= E /0j'(0)d^0jq (tq ) ■■■|0ji (ti )x
(ji,-- -,jq,j1,---,j '_1 ) t t t
t1 t.
2
x 0j;_i(tk-i)■■■ j(tl)dw( 0).■■dw<0_idw«;i)...dwi;«(81)
An iterative application of the Ito formula leads to the following equality
T T t2
j 0^(0)^/ 0jq (tq) ■ ■■/ 0ji (tl)x
t t t ti t2
x j j_1 (tk-i) ■ ■ ■ / 01 (ti^... dw|0— idw«^... dwiq») =
tt
= J (0;q---;i0-- -0) + J (;q 0;q_i---;i0-- -0) + j (;q-„¿10---0) +
= (j 'jq---j1jfc_1---j1)T,t (jqjfcjq_1---j1jfc_1 ---j1)T,t - - - (jq---j1j j_1 ---j1)T,t
+ ^ (;q---;i0---0) + + ^ (;q---;i0---0) (82)
+ J(jq---jij'_ij j_2---ji)T,t + ■ ■ ■ + J(jq---jij'_i---jj)T,t (82)
w. p. 1.
Combining (81) and (82) we finally obtain w. p. 1
T t2
E J 0jq (tq) ■■■/0ji (ti)dwt(;i) ■■■dwt(;q):
(j1'---,jq) t t
T t
T
x £ J j (tk) .../j (il)dw<?' ...dw«0> =
(j1>".,jfc) t t
t t2 ti t2
= E / j (tq) •••/j (ti )/ j (tk) •••/ j (ti)x
(j1 j'ivj') t t t t
xdwt(0) ...dwt(0)dwt(i1)... dwfq).
t i tk t 1 q
The equality (66) is proved for n = k. The equality (66) is proved. Theorem 4 is proved.
Let us consider the following theorem.
Theorem 5. Suppose that {<(x)}°=0 is an arbitrary complete orthonormal system of functions in the space L2([t,T]). Then the following representation
k [k/2] J"[<ji... jiTt""' = 11j> + £(—1)
1=1 r=1
r k—2r
X E n 1{ig2s- 1 = ig2s =0}1 {jg2s- 1 = j»2. }H j (83)
({{Si,S2},...,{S2r- 1 ,92T }},(q1,...,qfc-2r}) S = 1 1=1
{S1,S2,...,S2r-1,S2r,q1,...,qfc-2r } = {1,2,...,k}
is valid w. p. 1, where i1,...,ik = 0,1,...,m, J'' [j ... j ) is defined
by (19), [x] is an integer part of a real number x, = 1, Y1 = 0; another
0 0
notations are the same as in Theorems 1,2.
Remark 3. It should be noted that the formulas (29), (64), (67), (68) follow from (83). It is only necessary to set the values of the corresponding indicators of the form 1A from the formula (83) equal to 0 or 1.
Proof. The proof of Theorem 5 is carried out by induction using the following recurrence relation
j'' [j... ]Tr' = j'' [ j ]«• j'' [j ■.. <j'-1 fe"-1'
k-i
E1{;i=;k=0}1{ji=jkr J''[0ji ■ ■ ■ 0ji_i0ji+i ■ ■ ■ 0jk_i]T;1---;i_i;i+1---;k_i) w.p. 1. (84)
1=1
Let us prove the recurrence relation (84). Using iteratively the Ito formula, the orthonormality of {0j(x)}°=0, as well as (76) and combinatorial reasoning, we obtain w. p. 1 (see Remark 4 below for details)
j' '[j ^ • j' '[j ---0« _i iTr ^ =
T T t2
=j j E / j- (tk-i) ■■■/0ji «i^wi;0 ..■dw<:—ii) =
t (ji,---,jk_i) t t
T T t2
= E j^jkw^y*0jk_.(tk-i)■ /0ji(t^■■■dw((:—ii)=
(j1,---,jk_1) t t t
T t2
= E J0jk(tk)■ ■ ■ /0ji(ti)dwi;i)...dwir)+ (j1'---,jk) t t
/ T ( t2
+1
{; k=; k_i =0} / 0j k (0)0jk_1 0jk_2 (tk-2) ■ ■■ 0ji (t1)x
k_1 =
(j1,---,j k_1 )
xdwt(;i) ■ ■■dwt_2 )dw(0)+
T tk_1 ( t2
+1{; k=;k_2=0} / 0jk_i(tk-iW 0jk(0)0jk_2(0) / 0jk_3(tk-3) ■ ■■ 0ji(ti)x
xdwi;i)... dwt(:——з)dw(0)dwt(:—_l)+.■■
T t3 t2
+ 1{;k=;i=0} / 0jk_i(tk-i) ■ ■■/ 0j2(t2^ 0jk(0)0ji(0)x t t t
xdw^dw^ ...dwfc1)
T t2
= E /<* (tk) ••7<j1 (t1)dwt(;1) ...dwg') + (j1,---,jfc) t t
f T 0 t2
+ E 1{;'=;'-1=0^y №<j*-1(0) J <j'-2 (tk—2) ...J <j1 (t1)x (j1,--,jfc-2) ^ t t t
xdw«;1» ...dw«:--2)dw<0) +...
T t2 t1
... + /<j'-2(tk—2)... f <1 (to /j(0)<jfc_1 (0)dw00)dwi;1)...dwt(;--2)
f T 0 tfc_1
+ E 1{;'=;fc-2=0^ ^ y j (^)<jfc-2(e) J <j'-1 (tk—1) J <j'-3(tk—3)
(j1v,jfc-3,jfc-1) v t t t
t2
..J <j1 (t1)dwt(;1)... dw(;--3)dwt(;--1)dw00) +... t
T tfc-1 t2 t1
••• + j <j'-1 (tk —1) j <j'-3 (tk —3) <j1 (t0 / j (^)<jfc-2 (^)X
t t t
xdw00)dwi;1)... dwt(;--3)dwt(;--1^ +...
f T 0 t3
• • • + E 1{;'=;1=0^y Wj(0) J <j'-1 (tk—1) . J <j2 (t2) x (j2,---,j'fe-1) ^ t t t
xdwt;2) •••dwt(;-^l)dw0o) +...
T t3 t2
+ / 0jk_i (tk-i) ... i 0j2 M / 0jk (0)0j (^w^ ■■■dwí:—_i)
T t2
= E J0jk(tk)■ ■ ■ /0ji(ti)dwi;i)■■■dwt(:k)+ (ji'---,jk) t t
T T t2
+ / 0jk (0)0jk_i (0)d0 E 1{;k=;k_i=0} / 0jk_2 (tk-2) ■■■J 0ji (tl)x t (j1,---,jk_2) t t
xdwt(;i) ■■■dw|:—_2)+
T T tk_i
+ J 0jk (0)0jk_2(0)d0 E 1{;k=;k_2=0^ 0jk_i (tk-0 / 0jk_3(tk-3)
t (j1,---,jk_3,jk_1) t t
t2
■■■[ j (to^ ■■■dw(:——з_dwí:__l'_+■..
T T t3
+ / 0jk (0)0ji (0)d0 E 1{;k=;i=0^ 0jk_i (tk-l) ■■■J 0j2 (t2) x t (j2,---,jk_1) t t
xdwC2) ■■■dwi(k—-1l) =
J''[0ji ■ ■ ■ 0jkiT,1"-^ + 1{;k=;k_i=0}1{jk =jk_i} • J''[0ji ■ ■ ■ 0jk_2iTi--^ 2) +
+ 1{;k=;k_2=0}1{jk=jk_2} • J''[0ji ■ ■ ■ 0jk_30jk_i]t,1('' k 3 k 1 +
■ ■ ■ + 1{;k=;i=0}1{jk=ji} • J''[0j2 ■ ■ ■ 0jk_i]T;j2t---;k_l) =
= J' '[0ji ■■■0jk ]T;it---;k)+
k—1
+ E 1{;,=;'=0}1{j,=j'} • J''[<j1... <j,-1 <j,+1... j-1 ]T;;t--;,-1;,+1---;fc-1). (85) 1=1
The equality (84) is proved. Theorem 5 is proved.
Remark 4. It should be noted that the sums with respect to permutations
L
(j1,---,jfc-1)
in (85), containing the expressions
1{;k=;k-1 =0} j (Wjfc-1 . . . , 1{;fc=;1=0}<jfc Wj
should be understood in a special way. Let us explain this rule on following sum
T 0 t2
E 1{;fc=;fc-1=0}/ j№j-1(e) J <j'-2(tk—2) .../j(t1)x (j1,---,jfc-1) t t t
xdwt;1) •••dwt(;-^2)dw0o)• (86)
More precisely, permutations (j1;..., jk—1) when summing in (86) are performed in such a way that if jr swapped with in the permutation (j1;..., jk—1), then ir swapped with id in the permutation (i1;... ,ik—2,ik—1) (note that ik—1 = 0). Moreover, j swapped with <jd in the permutation
(j , . . . , j-j = (j , . . . , <j'-2 , 1{;fc = ;fc-1=0} • j • ,
where j-1 (t) = 1{;fc=;fc-1=0} j (t)<j'-1 (t).
A similar rule should be applied to all other sums with respect to permutations
L
(j1,---,jfc-1)
in (85) that contain the expressions
1{;fc=;fc-2=0}<jfc(^)<jfc-2№> . . . » 1{;fc=;1=0}<jfcWj №.
https://doi.org/10.21638/11701/spbu35.2023.405 Electronic Journal: http://diffjournal.spbu.ru/ 113
3 Main Results
3.1 Generalizations of Theorem 2 to the Case of an Arbitrary Complete Orthonormal Systems of Functions in the Space L2([t,T]) and ^(t), (t) G L2([t,T])
Theorems 3-5 imply the following two theorems on expansion of iterated Ito stochastic integrals (2).
Theorem 6 [10], [15]. Suppose that the condition (*) is fulfilled for the multi-index (¿i... ) (see Sect. 2.2) and the condition (27) is also fulfilled. Furthermore, let 01(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 pk
J№<k>]ft"ik> = l.i.m V ... £ Cjk...ji x
' p.....pi„—^-no ' * ' *
¿1=0 jk=0
/
k
X
1=1
\
n
l{m;=0} + l{m;>0} <
z(0) \ni,' (r(0) yv if , _n
(87)
converging in the mean-square sense is valid, where Hn(x) is the Hermite polynomial (28), 1A is the indicator of the set A, i1,...,ik = 0,1,...,m; ni,/ + n2,/ + ... + ndl,/ = m/; ni)/,n2,/,... ,ndj)/ = 1,... ,m/; d/ = 1,... ,m/; l = 1,..., k; m1 + ... + mk = k; the numbers m1,..., mk, g1,..., depend on (i1,..., ik) and the numbers n1;/,..., ndl;/, h1;/,..., hdl;/, d/ depend on {j'i, ..., }; moreover\ (igi, . . . , j'g* } = {ji..., };
T
cf _J ^Wdw^ (i _n, 1,..., m; j _ 0,1, 2,...)
t
are independent standard Gaussian random variables for various i or j (in the case when i _ 0) and dw[0) _ dr; another notations as in Theorems 1, 2.
Theorem 7 [10], [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 Pk ( k [k/2]
jh^-» = pi i.i.m.œ £... £ Ck..ji nj1 + B—Drx
ji =0 jk=0 \/=1 r=1
r k—2r
x E n 1{ig2s-i = ig2s =0}1{jg2s-i = jg2s } II » ) (88)
2s— i a2s ' ^2s— i "»2s-
({{Si ,92},...,{92r — i,92r}},{9i v-^k — 2r }) S=1 /=1
{si ,s2,...,s2r—i,s2r,qi,...,qk—2r }={i,2,...,k}
converging 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 Theorems 1, 2, 5.
3.2 Modifications of Theorems 6, 7 for the Case of an Arbitrary Complete Orthonormal Systems of Functions in the Space
L2([t,T]) and ) e L2([t,T]).
Replacing the function K(t1,... , tk) of the form (4) in Theorems 6, 7 by the function $(t1,... ,tk) e L2([t,T]), we get the following two theorems.
Theorem 8 [10], [15]. Suppose that the condition (*) is fulfilled for the multi-index (i1... ik) (see Sect. 2.2) and the condition (27) is also fulfilled. Furthermore, let $(t1,... ,tk) e L2([t,T]) and {0j(x)}°=0 is an arbitrary complete orthonormal system of functions in the space L2([t, T]). Then the following expansion
pi pk = l.i.m. £ x
Pi, ... ,Pk ^^ Z-' Z-'
ji=0 jk =0
/
k
x l{m; =0} + l{m;>0}
1=1
V
j)jj, if i/ =
0
(89)
Z
(0»
j:
i,i
nil
. Zj
(0»
j:
dl,l
if i/ = 0
/
ndl,l
converging in the mean-square sense is valid, where the sum of iterated Ito stochastic integrals J"^]^"^^ is defined by (19),
C —
/k
^ (ti)dti ...dtk (90)
[t,T ]k 1=1
is the Fourier coefficient, Hn(x) is the Hermite polynomial (28), 1A is the indicator of the set A, i1,..., ik = 0,1,..., m; n1;/ + n2;/ + ... + ndl/ = m/; ni,i ,n2,/,.. .,ndi,/ = 1,...,m/ ; d/ = 1,...,m/ ; l = 1,...,k; mi + ... + mk = k ; the numbers m1,...,mk, g1 ,...,gk depend on (i1,...,ik) and the numbers ni,/,... ,ndi)/, hi,/,..., hdl,/, d/ depend on {j1,..., jk}; moreover, {j^!,... j} = {ji,..., jk};
T
j = / 0j(r)dw[i) (i = 0,1,..., m; j = 0,1, 2,...) t
are independent standard Gaussian random variables for various i or j (in the case when i = 0) and dw(0) = dr.
Theorem 9 [10], [15]. Suppose that $(t1,...,tk) G L2([t,T]k) and (x)}°=0 is an arbitrary complete orthonormal system of functions in the space L2([t,T]). Then the following expansion
Pi Pk ( k [k/2]
J*"ik) — p ^ £■ ■ ■ £ C...11 n+
Pi,---,Pk ^^
11=0 1k=0 \/=1 r=1
r k—2r
x E n 1{ig2s-1 = ig2s=o} 1{jg2s-1 = jg2s } II C
({{31,32},...,{s2r-1,32r »,{91,...,9fc-2r}) S=1 1=1
{31,32,...,32r-1,32r,91,...,9fc-2r }={1,2,.",k}
converging in the mean-square sense is valid, where the sum of iterated Ito
stochastic integrals J''[$]T:t"ik' is defined by (19), the Fourier coefficient Cjk... has the form (90); another notations are the same as in Theorems 5,8.
1i
4 Comparison with Other Results and Conclusions
Before starting this section, we recall that the sum of iterated Ito stochastic integrals (19), which plays a central role in the proofs of Theorems 6-9, is equal w. p. 1 to the multiple Wiener stochastic integral with respect to the components of a multidimensional Wiener process (see the proof in [10], Sect. 1.11).
It should be noted that an analogue of Theorem 8 (more precisely, the expansion like (89) for the case i1,..., ik = 1,..., m) was obtained in [39]. The mentioned expansion is formulated in [39] using the multiple Wiener stochastic integral and the Wick product. Also note that the proof in [39] is different from the proof given in this article. Let us describe these differences.
In [39], the author interprets the multiple Wiener stochastic integral from a finite-dimensional kernel Kp, .,p(t1,..., tk) of the form (6) as a linear operator and proves that this operator is bounded. We note that the proof from [39] is essentially based on Theorem 3.1 from [40].
In our proof of Theorems 6-9 we use the sum of iterated Ito stochastic integrals (19) several times and do not explicitly use the multiple Wiener stochastic integral. Moreover, our proof of Theorems 6-9 is based on the Ito formula and does not use Theorem 3.1 from [40]. The methodology of our proof is a direct development of the approach we used to prove Theorem 5.1 in [6] (2006).
Note that the results of [39], as well as the results of this article, are based on our idea [1] (2006) on the expansion of the kernel (4) (or $(t1,...,tk) e L2([t,T]k)) into a generalized multiple Fourier series (see [1], Chapter 5, Theorem 5.1, pp. 235-245 or [10], Chapter 1 for details).
We also note a number of works [40], [42]-[46] in which the properties of multiple Wiener stochastic integrals were studied using measure theory, in particular, the formulas for the product of such integrals were obtained.
First of all, let us compare Theorem 5 with Proposition 5.1 from [43]. An analogue of the right-hand side of (83) for nonrandom x1,..., xk is constructed in [43] using diagrams (see the formula (5.1) in [43]). This means that the application of the formula (5.1) from [43], unlike the formula (83), is difficult when performing algebraic transformations.
Further, we note that the formula (5.1) from [43] was applied to the representation of the multiple Wiener stochastic integral somewhat differently than the formula (83). Namely, using Proposition 5.1 [43]. Let us expain this difference in more detail.
Proposition 5.1 from [43] in our degree of generality and in our notations can be written as
J'' [ j ... j ]£-ik) =
= J"
3 • • • 3 3 ■ ■ ■ 3 ••• 3 • • •
J3\ • • • Y3ijY32 ■ ■ ■ Y32. mi m2
(¿1 ...ir,
...ik)
T,t
mp
77/ [" / / l(n---«m1) T// r I I 1 («m1 + l---«m2 ) r// f / / ] ^mi + ...+mJ)_1 + l-«fc)
= J [3 — jlr,i 1 'J [ t j2 ■■'fe]T,i1 2 ■ ''■■ J №p — TOplT,( p
(91)
w. p. 1, where
m
m
i
2
J'' [ j ■ ■ ■ 3]
( .. .i'TTT,^ )
( ii T,t
, J"
m2
, -i ( i ^ j2 ■ ■ ■ r j2 ]T,t ,
mp
J'' [j • ■ ■ ^jp]
('■--
^ imi + ...+mp_i + i '.'ik '
jpj T,t
m
i
are defined by the right-hand side of the formula (5.1) from [43], mi + ■ ■ ■+mp = k, mi, ■ ■ ■ ,mp > 0, jq = jd (q = d, q,d = 1, . . . ,p), ii, ■ ■ ■, ik = 1, . ■ ■ ,m.
This actually means that in [43] an analogue of the formula (83) is constructed for the special case j = ■ ■ ■ = jk. Moreover, the specified analogue is based on the formula (5.1) [43] obtained using diagrams.
Comparing the formulas (83) and (91) (or (5.1) from [43]), it is easy to understand that the transition from (83) and (91) is obvious. It is only necessary to set the values of the corresponding indicators of the form 1A from the formula (83) equal to 0 or 1. The reverse transition from the formula (91) to the formula (83) is not obvious. Note that the formula (83) (not the formula (91)) is convenient for the numerical integration of Ito stochastic differential equations (see [10], Chapter 5 for details).
Let us turn to the comparison of Theorem 5 with another interesting work [46] (2019). As it turned out, a version of Theorem 5 was obtained in terms of Wick polynomials and for the case of vector valued random measures in [46] (see Theorem 7.2, p. 69). However, much earlier the formula (83) (Theorem 5) is obtained in our monograph [8] (2009) as part of the formula (5.30) (see [8],
p. 220). Moreover, particular cases of the formula (83) were obtained even earlier in our works [6] (2006) and [7] (2007). More precisely, particular cases k = 1,..., 5 of the formula (83) were obtained in [6] (2006) as parts of the formulas on the pages 243-244 and partiular cases k = 1,..., 7 of the formula (83) were obtained in [7] (2007) as parts of the formulas on the pages 208-218.
We also note that we have found an explicit expression for the Wick polynomial of degree k of the arguments (j1',..., j'1 (see the formula (83)), which is very convenient for the numerical simulation of iterated Ito stochastic integrals (2) [13]. Note that the representation of the Wick polynomial of the arguments Zj1',..., in terms of the product of Hermite polynomials is less convenient for the numerical simulation of iterated Ito stochastic integrals (2). For example, the expression for J'^j j j j ]Ti1ti2i3i4' in terms of the product of Hermite polynomials, even under the condition i1 = i2 = i3 = i4, already contains 15 different expressions (see [10], Sect. 1.10). At the same time, all these 15 expressions are contained in one formula (83) provided that k = 4 and ii = i2 = i3 = i4. It is very convenient, since in computer simulation using the formula (83), in addition to modeling of random variables Zj i1',..., j', it remains only to set the values of the corresponding indicators of the form 1A from the formula (83) equal to 0 or 1.
It should be noted that in [44] (Theorem 6.1) a diagram formula was obtained for the product of two multiple Wiener stochastic integrals with respect to vector valued random measures. The formula (65) can be derived from the diagram formula [44]. Although the proof of the diagram formula [44] is much more complicated than our proof of the formula (65).
To conclude this article, we say a few words about expansions (87) and (88). The transition from the expansion (88) to the expansion (87) is obvious. It is only necessary to set the values of the corresponding indicators of the form 1a from the formula (88) equal to 0 or 1. The reverse transition from the formula (87) to the formula (88) is also possible but not obvious. However, Theorems 4 and 5 provide a transition from (87) to (88) and vice versa. Note that the expansion (87) is interesting from the point of view of studying the structure of the expansion of iterated Ito stochastic integrals. On the other hand, the expansion (88) is exceptionally convenient for applications [13], [14]. For example, in [13], [14], approximations of iterated Ito stochastic integrals of multiplicities 1 to 6 in the Python programming language were successfully implemented using (88) (k = 1,..., 6) and Legendre polynomials.
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