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DIFFERENTIAL EQUATIONS AND
CONTROL PROCESSES N. 2, 2024 Electronic Journal, reg. N $C77-39410 at 15.04.2010 ISSN 1817-2172
r
http://diffjournal. spbu. ru / e-mail: [email protected]
Stochastic differential equations Numerical methods Computer modeling in dynamical and control systems
A new approach to the series expansion of iterated Stratonovich
stochastic integrals with respect to components of a multidimensional Wiener process. The case of arbitrary complete orthonormal systems in Hilbert space
Abstract. The article is devoted to the development of a new approach to the series expansion of iterated Stratonovich stochastic integrals with respect to components of a multidimensional Wiener process. This approach was proposed by the author in 2022 and is based on generalized multiple Fourier series in complete orthonormal systems of functions in Hilbert space. In the previous parts of this work, expansions of iterated Stratonovich stochastic integrals of multiplicities 1 to 6 were obtained. At that, the expansions were constructed using two specific bases in Hilbert space. More precisely, Legendre polynomials and the trigonometric Fourier basis were used. In this paper, expansions of iterated Stratonovich stochastic integrals of multiplicities 1 to 4 are obtained on the base of arbitrary complete orthonormal systems of functions in Hilbert space. Sufficient conditions for the expansion of iterated Stratonovich stochastic integrals of arbitrary multiplicity are formulated in terms of trace series. The results of the article will be useful for construction of strong numerical methods
Dmitriy F. Kuznetsov
Peter the Great Saint-Petersburg Polytechnic University e-mail: [email protected]
with orders 1.0, 1.5 and 2.0 (based on the Taylor-Stratonovich expansion) for Ito stochastic differential equations with non-commutative noise. Key words: iterated Stratonovich stochastic integral, iterated Ito stochastic integral, Ito stochastic differential equation, multidimensional Wiener process, generalized multiple Fourier series, mean-square convergence, expansion.
Contents
1 Introduction 75
2 Preliminary Results 76
2.1 Expansion of Iterated Ito Stochastic Integrals of Arbitrary Multiplicity k (k G N) Based on Generalized Multiple Fourier Series Converging in the Mean...... 76
2.2 Stratonovich Stochastic Integral........................... 80
2.3 Expansion of Iterated Stratonovich Stochastic Integrals of Arbitrary Multiplicity
k (k g N) Under the Condition on Trace Series................... 81
2.4 Expansions of Iterated Stratonovich Stochastic Integrals of Multiplicities 1 to 6.
The Case of Legendre Polynomials and Trigonometric Fourier Basis....... 87
2.5 Connection Between Iterated Stratonovich and Ito Stochastic Integrals of Arbitrary Multiplicity k (k g N)............................. 89
2.6 Multiple Wiener Stochastic Integral With Respect to Components of a Multidimensional Wiener Process.............................. 90
3 Main Results 93
3.1 Generalizations of Expansion of Iterated Stratonovich Stochastic Integrals of Arbitrary Multiplicity k (k g N) Under the Condition on Trace Series...... 93
3.2 Generalization of Theorems 11-13 to the Case When the Conditions 0o(x) = l/^/T^t and ripi(T)ripi-i(r) g L2([t,T]) (I = 2, 3,..., k) are Omitted.......114
3.3 Another Definition of the Stratonovich Stochastic Integral ............120
3.4 Expansion of Iterated Stratonovich Stochastic Integrals of Multiplicity 2. The Case of an Arbitrary CONS in the Space L2([t)T]) and ^(t),^(t) G L2([t)T]) 122
3.5 Expansion of Iterated Stratonovich Stochastic Integrals of Multiplicity 3. The Case of an Arbitrary CONS in the Space L2([t, T]) and (t),^2(t),^3(t) = 1 . 124
3.6 Expansion of Iterated Stratonovich Stochastic Integrals of Multiplicity 4. The Case of an Arbitrary CONS in the Space L2([t, T]) and (t),... ,^4(t) = 1 . . 129
3.7 Generalization of the Results from Section 3.5 to the Case ),^2(t),^3(t) g L2([t,T])........................................142
3.8 Generalization of the Results from Section 3.6 to the Case (t),... ) g L2([t,T])........................................148
3.9 On the Calculation of Matrix Traces of Volterra-Type Integral Operators .... 151 4 Conclusion 161 References 161
1 Introduction
Let (П, F, P) be a complete probability space, let {Ft,t £ [0, T]} be a nonde-
creasing 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 £ [0, T]
and has independent components wt(;) (i = 1,..., m). Consider an Ito stochastic
differential equation (SDE) in the integral form
t t P m „.
xt = x0 + a(xT,т)dr + ^ / Bj(xT, т)dwj\ x0 = x(0,w), w £ (1) 0 j=1 0
Here xt is the n-dimensional stochastic process satisfying (1). The functions a, Bj : Rn x [0, T] ^ Rn guarantee the existence and uniqueness up to stochastic equivalence of a solution of (1) [1]. The second integral in (1) is the Ito
г 21
stochastic integral. Further, x0 is F0-measurable and M{|x01 ) < то (M denotes a mathematical expectation). We also assume that x0 and wt — w0 are independent when t > 0.
Consider the following families of iterated Ito and Stratonovich stochastic integrals:
t t2
J ^(k)]Trfc) = / Фк (tk) ..J ^1(t1)dwi;i)... dwi:k), (2)
tt * T * t2
ЛФ(к)]T^-;k) = J Фк(tk)..J Ф1 (t1)dwt;i)...dwt:k), (3)
tt
where ф1(т),... ,фк(т) : [t,T] ^ R, i1,... ,ik = 0, 1, ..., m, w[0) = т,
J and
denote Ito and Stratonovich stochastic integrals, respectively.
It is well known that the stochastic integrals (2) and (3) play an important role when solving Ito SDEs numerically using Taylor-Ito and Taylor-Stratonovich expansions [2]-[14]. From the other hand, Ito SDEs have many applications, which explains the relevance of the problem of their numerical solution [2]-[13].
Note that ^i(t),..., ) = 1, ii,..., ik = 0,1,..., m (the case of classical Taylor-Ito and Taylor-Stratonovich expansions) [2]-[8] and ^(t) = (t — t)qi, ql = 0,1,... (/ = 1,..., k), i1,...,ik = 1,..., m (the case of unified Taylor-Ito and Taylor-Stratonovich expansions) [9]-[14].
This article is Part III of the work devoted to a new approach to the series expansion and mean-square approximation of iterated Stratonovich stochastic integrals (3) ([15] and [16] are Part I and Part II of the mentioned work, respectively).
We also note other approaches to the mean-square approximation of iterated Ito and Stratonovich stochastic integrals (2) and (3) [2]-[5], [17]-[36].
2 Preliminary Results
2.1 Expansion of Iterated Ito Stochastic Integrals of Arbitrary Multiplicity k (k G N) Based on Generalized Multiple Fourier Series Converging in the Mean
Suppose that ^1(t),...,(t) G L2([t,T]). Define the following function (Volterra-type kernel) on the hypercube [t,T]k :
I^i(ti) ...^k (tk), ti <...<tk
, (4)
0, otherwise
where t1,... ,tk G [t,T] (k > 2) and K(t1) = (t1) for t1 G [t,T].
Assume that (x)}°=0 is a complete orthonormal system (CONS) of functions in the space L2([t,T]). It is well known that the generalized multiple
/
Fourier series of K(ti,... , tk) £ L2([t,T]k) is converging to K(ti,... , tk) in the hypercube [t,T]k in the mean-square sense, i.e.
lim
KK
= 0,
L2([t,T ]k)
where
/ .
i/2
f
li2([i,T ]k)
J f2(ti,...,tk )dti ...dtk
V[t,T ]k y
Pi Pk k
KPi ...Pk (ti,...,tk ) = £ ...EcwiII ^ (ti),
ji=0 jk=0 /=i
/k
K (ti,...,tk )H (t/ )dti ...dtk (5)
7=1
[t,T ]k 7=i
is the Fourier coefficient.
Consider the partition {rj}N=0 of [t,T] such that t = t0 < ... < tn = T, AN = max Arj ^ 0 if N ^ to, ATj = Tj+i — Tj.
0<j<N—i
(6)
The following theorem marked the beginning of a systematic study of the problem of strong approximation of iterated Ito and Stratonovich stochastic integrals (2) and (3) that have been most fully studied to date in [14].
Theorem 1 [11] (2006), [12]-[16], [37]-[52]. Suppose that ^(t),...,^(t) are continuous nonrandom functions on [t,T] and {0j(x)}°=0 is a CONS of continuous functions in the space L2([t,T]). Then
Pi Pk / k
j ^r) = pU-to e... E ^ n cf —
ji =0 jk =0 \/=i
— l.i.m. £ j (t/1 ^w^... j (T/k)AwTik) ) , (7)
(/i...../k)£Gk i k J
where Jis defined by (2), ii,... ,ik = 0,1,... ,m, l.i.m. is a limit in
]?r*k) is
the mean-square sense,
Gk = Hk\Lk, Hk = {(1i,...,1k): 1i,...,1k = 0, 1,..., N — 1},
Lk = ): Il = 0, 1,..., N -1; lg = lr (g = r); g,r = 1,...,k},
eg T r T
if = / <Aj(T)dw<->
t
are independent standard Gaussian random variables for various i or j (in the case when i = 0), Cjk...j is the Fourier coefficient (5), AwTj = wj+1 — wTj (i = 0, 1, ...,m), {Tj}=0 is a partition of the interval [t,T] satisfying the condition (6).
A number of generalizations and modifications of Theorem 1 can be found in [14], Chapter 1 (see also bibliography therein).
Let us consider corollaries from Theorem 1 (see (7)) for k = 1,..., 5 [11]
pi
J [^'f' = l.i.m. £ Cji j1', (8)
' pi^œ ' é J1 ji =0
Pi P2
J[^<S>]^Ïi2) = pbSÎfeo £ E Cj2j4 Zj:',Zj22) — 1{!i=.2=0}1{ji=j2} ) , (9)
ji=0 j2=0
Pi P2 P3
<:f3) = i.i.m Ejjjj-
ji=0 j2=0 j3 =0
-1{:i=:2=0}1{ji=j2}Cj(33) - 1{:2=:3=0}1{j2=j3}j' - 1{:i=:3=0} 1{ji=j3 j' ) , (10)
Pi P4 /4
J"iTr1=pi'i-œ £•■■£c^II j'
ji=0 j4 =0 \/ = 1
1 1 /■ fe'z (:4' 1 1 Z fe'z (i4 '
-1{:i=:2=0}1{ji=j2}Zj3 z j4 - i{:i=:3=0}i{ji=j3}Zj2 zj4 -
1 1 z (:2 ' Z (:3' 1 1 z (:i'Z (:4 '
-1{:i=:4=0}1{ji=j4}Zj2 j - i{:2=:3=0}1{j2=j3}Zji j -
1 -, z (:i 'z (:3' 1 1 Z (:i'z (:2'_L
-1{:2=:4=0}1{j2 =j4}zji zj3 - 1{:3=:4=0}1{j3=j4} j j + + 1{:i=:2=0}1{ji=j2} 1{:3=:4=0}1{j3=j4} + 1{:i=:3=0} 1{ji=j3} 1{:2=:4=0} 1{j2=j4} +
+ 1{H = M=0}l{ji=j4}1{i2 = i3=0}l{j2=M
(11)
P1
P5
5
J [#>fr!5) = l.i.m £ j'
\ l-1
j1=0 j5-0
{¿1- ¿2-0}
— 1 {¿1- ¿4-0}
— 1 {¿2- ¿3-0}
— 1 {¿2- ¿5-0}
— 1 {¿3- ¿5-0}
+^¿1= -¿2-0} {j1 =j2} 1
+^¿1= -¿2-0} {j1 =j2} 1
+^¿1= -¿3-0} 1{j1 -j3} 1
+^¿1= -¿4-0} {j1 -j'4}1
+^¿1= -¿4-0} {j1 -j'4}1
+^¿1= -¿5-0} {j1 - j5 } 1
+ ^¿2 = -¿3-0} { j2 -j3} 1
Z(i3)Z(i4) Z(i5 {j1-j2} zj3 zj4 z j5
Z (i2 ) Z (i3)Z (i5 {j1-j4} Zj2 Zj3 Z j5
Z (i1)Z (i4) Z (i5
{j2-j3}Zj1 Zj4 Z j5
Z (i1)Z (i3)Z (i4
{j2-j5} Zj1 Zj3 Zj4
Z (i1)Z (i2 )Z (i4
{j3-j5}Zj1 Zj2 Z j4 •1{i3-i4-0} •1{«4-«5-0} •1{i2-i5-0} •1{«2-«3-0} •1{i3-i5-0} •1{i2-i4-0} • 1{«4-¿5-0}
Z (i5 { j3-j4} Z j5
Z (i3 {j4-j5} Zj3
Z (i4
{j2-j5}Zj4
Z (i5
{j2-j3}Zj5
Z (i2 {j3-j5} Zj2
Z (i3
{j2-j4}Zj3 ^¿1
1
1
1
1
1
{¿1-¿3-0}
{¿1^5-0}
{¿2-¿4-0}
{¿3-¿4-0}
{¿4-¿5-0}
+ ^¿1^2-0} + ^¿1^3-0} + 1{¿l—¿3-0} + 1{¿1 —¿4-0} + ^¿1^5-0} + 1{¿1 —¿5-0} + 1 { ¿2 ^¿4-0}
{j4-j5} Zj1
+ 1{¿2-¿5-0}1{j2-j5}1{¿3-¿4-0}1{j3-j4}Zjl
Z (¿2) Z ^-¿4 )Z (¿5
{j1-j3}Zj2 Z j4 Zj5
Z XX ^-¿3 )Z ^4
{j1-j5} Zj2 Z j3 Zj4
Z (¿1)Z ^3 )Z ^5
{j2-j4} Zj1 Z j3 Zj5
Z (¿1)Z ^2 )Z ^5
{j3 - j4 } Z j 1 Z j2 Zj5 Z ^V ^2 ) Z (¿3
{j4-j5} Zj1 Zj2 Zj3 {j1-j2}1 {j1-j3} 1 {j1-j3} 1 {j1-j4} 1 {j1-j5}1 {j1-j5}1 {j2-j4} 1 (¿1)
+
{¿3 = -¿5-0} { j3 ■ iZ (¿4 =j5 } Z j4 +
{¿2 -¿4-0} { j2 Z ^-¿5 =j4 } Z j5 +
{¿4 -¿5-0} {j4 ■ T,z (¿2 =j5 } Z j2 +
{¿2 -¿5-0} { j2 ■ T,z (¿3 =j5 } Z j3 +
{¿2 -¿3-0} { j2 Z ^-¿4 =j3}Zj4 +
{¿3 = -¿4-0} { j3 Z ^-¿2 =j4 } Z j2 +
{¿3 = -¿5-0} { j3 Z (¿l =j5 } Z j1 +
(12)
where 1A is the indicator of the set A.
Let us consider a generalization of (8)-(12) to the case k £ N and also to the case of an arbitrary CONS in the space L2([t,T]) and ^(t),... ) £ L2([t,T ]).
Theorem 2 [14] (Sect. 1.11, 1.14), [43] (Sect. 15, 18), [44]. Suppose that ^1(t),... , ^(t) £ L2([t,T]) and {0j(x)}°-0 is an arbitrary CONS in the space L2([t,T]). Then the following expansion:
Pi Pk ( k [k/2]
J1=J.-.^ E ■ ■ • E j' ^(-irX
ji =0 jk=0 \/=1 r=1
r k—2r \
X S n 1{ig2s-i = ig2s=0i1{jg2s-i = jg2s} II j1 j (13)
({{si ,32},...,{s2r-i,s2r }},{9i,...,9k-2r}) s=1 /=1 /
{Si ,S2,...,S2r-i,S2r,9i,...,9k-2r }={i,2,...,k}
that converges in the mean-square sense is valid, where [x] is an integer part of a real number x, the sum in the second line of the formula (13) means the sum with respect to all possible permutations of the set
({{g1, g2}, • • • , {g2r-1, g2r}}, {tf1, • • • , qk-2r}), (14)
braces mean an unordered set, and parentheses mean an ordered set, {gi, g2, • • •, g2r-i,g2r, qi, • • •, qk-2r} = { the same as in Theorem 1.
g2r-1, g2r, q1? • • •, qk-2r} = {1, 2, • • •, k}; = 1, = 0; another notations are
2.2 Stratonovich Stochastic Integral
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
J M {(£T)2} dT< to, M {(£t)2} < to for all t G [t,T]^
t
We introduce the class Q4([t, T]) of Ito processes , t G [t, T], i = 1, • • •, m of the form
T T
= nt( 0 + J asds + y bsdw^ w. p. 1, (15)
t t
where (as)4, (bs)4 G M2([t,T]) and lim M{|bs — &t|4} = 0 for all t g [t,T]. The
s—^T
second integral on the right-hand side of (15) is the Ito stochastic integral. Here and further, w. p. 1 means with probability 1.
Let C2j1(R, [t, T]) (t > 0) be the space of functions F(x, t) : R x [t, T] ^ R with the following property: these functions are twice differentiable in x and have one derivative in t. Moreover, all these derivatives are uniformly bounded.
The mean-square limit
N-1 /1 \ f
u-m U (<• • "■•:•••) ^) Ki -= / (i6)
j-0 \ / J
is called [53] the Stratonovich stochastic integral with respect to the component
j }j-0
wT^ (/ = 1,... ,m) of the multidimensional Wiener process wT, where {rj}N
is a partition of the interval [t,T] satisfying the condition (6).
It is known [53] (also see [2]) that under proper conditions, the following relation between Stratonovich and Ito stochastic integrals holds:
* T T T
1 fdF
F{rJi\r)dw? = / F(^\r)dw^ + -l{i=i} / -—(?7t,t)MT (17)
w. p. 1, where 1A is the indicator of the set A and i, I = 1,..., m.
A possible variant of conditions under which the formula (17) is correct, for example, consists of the conditions G Q4([t,T]), F(n(i),T) G M2([t,T]), F(x,t) G C2':(R, [t,T]), where i = 1,... ,m.
Note that if F(x,t) = Fi(x)F2(r), then the smoothness condition F(x,t) G C2':(R x [t,T]) can be weakened. Namely, it suffices to replace the condition with respect to t by continuity with respect to this variable.
In Sect. 3.3, we will also consider another definition of the Stratonovich stochastic integral.
2.3 Expansion of Iterated Stratonovich Stochastic Integrals of Arbitrary Multiplicity k (k G N) Under the Condition on Trace Series
In this section, we recall Theorem 3 (see below) from [15] (Part I of this work) on the expansion of iterated Stratonovich stochastic integrals (3) of arbitrary multiplicity k (k G N) and introduce some notations.
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 (compare with (14))
({{gi, g2>,..., {g2r-1, g2r}}, [qi,•••, qk-2r}), (18)
v V ' -V-
part 1 part 2
where {g1, g2,..., g2r-1, g2r, qi, • • •, qk-2r} = {1, 2,..., k}, braces mean an unordered set, and parentheses mean an ordered set.
Let us call (18) a partition of the set {1, 2,..., k}. Further, we will consider sums with respect to all possible partitions (18) (also see (13)).
Consider the Fourier coefficient
t t2
Cjk ...j = J fa (tk) j (tk) • • J fa (*1)j (t1)dt1.. .dtk (19)
t t
corresponding to the Volterra-type kernel (4), where (x)}°=0 is a CONS in the space L2([£,T]). At that we suppose (f)o(x) = 1 /\/T — t.
Denote
C......
Cjfc -Jl + lJl JlJl-2-Jl
def
T tl+2 ti + l
= y ^ )0jfc ) ...y )x
t t t tl t2
x y ^i-2(t/-2)0ji-2(t/-2).. (ti)0ji(ti)dti •. .dt/-2dtiti+1.. .dtk = tt
T t(+2 ti+i
= y Mtk)(f>jk{tk) • • • J ^ifeijfei^i) y t t t tl t2
x y ^i-2(t/-2)0ji-2(ti-2).. (ti)dti.. .dti-2dtiti+i.. .dtk, (20)
tt
i.e. (20) is again the Fourier coefficient of type (19) but with a new shorter
multi-index jk • • • j1+i0j1-2 ... j and new weight functions ^i(r), ..., ^/-2(r),
\/Т — ^/_|_i(r), ..., ф/г(т) (also we suppose that {1, I — 1} is one
of the pairs {01,02}, -{02r-1,02r} (see (18))). Let
T
C......
Cjfc •••ji+i ji Л Л-2---Л
i+2
(ji j'i
def
i+i
def
= J ^fc (tk )фЛк (tk ) ...у (t1 M-1(t1 )флт (t1 )x
t t t tl t2
x J ^i-2(t/-2)0j,-2 (ti-2).. .J Ф1 (¿1)0ji (Î1)dÎ1... dti-2dti ti+1... dtk = (21) t t
= Cjfc•••j'i+ijmj'i-2...j'i (jm = 0, 1, 2, . . .),
i.e. Cjfc...j;+ljmj;_2---ji is again the Fourier coefficient of type (19) but with a new shorter multi-index ... j1+1jmj/-2 ... j and new weight functions ^1(t), ..., ^/-2(t), ^/—1(t(t), ^/+1(t), ..., (t) (also we suppose that {1 — 1,1} is one of the pairs {01,02},..., {g2r- 1,02r} (see (18))).
Let
C
(p)
jfc •••jq •••ji
def
00 00
def
^ S . .. S S Cjk-
jfl2r- i =p+1 jfl2r-3 =p+1 j'as =p+1 j'fli =p+1
ji
jSi jS2 '•••'jS2r-i jS2r
(22)
Introduce the following notation:
(p)
jfc •••jq •••ji
?=дъд2,---,.д2г-ь.д2г
œ œ
def 1 n V^
{921=921-1 + 1} / v / V
2
00
00
00 00
S . . . Cjk •ji
jg2i+i =P+1 jg2i-3 =P+1 js3 =P+1 j'si =P+1
jfl2r-i =P+1 jfl2r-3 =P+1
(jS2i jS2i-i jS2 '•••'jS2r-i j32r
Note that the operation S/ (l = 1, 2,..., r) acts on the value
a (p)
(j'k-j'g •••j1
(23)
q=01,02v,02r-1,02r
as follows: S/ multiplies (23) by 1{g2l=g2l-1+i}/2, removes the summation
L .
and replaces
with
/
(jk •••j1
jS1 jS2 '•••'jS2r-1 jS2r
/
jk-j1
(jS2i jS2i-1 W-)j31 jS2 '•••'jS2r-1 jS2r
Note that we write
/
= a
( jg 1 j32 ) ^ ( •) 'jS1 =jS2
jk •••j1
( jg 1 1) ^( •) ;jg 1 =jg2
/
= a
jk •••j1
( jg 1 j32 ) •) '( jS3 jS4 ) ^ ( •) 'jS 1 =jS2 'jS3 =jS4
( jg 1 1) ^( •)( jg3 jg3 ) ^( •) 1 =jg2 jg3 =jg4
(24)
(25)
Since (25) is again the Fourier coefficient, then the action of superposition S/Sm on (24) is obvious. For example, for r = 3
Q Q Q J 7(p) S3S2Sl <{ •••j1
q=ö,1,ö,2v,.s,5,.g6
3
23 n ^{s,2S=s,2S-i+i}^'ifc---ii
s=1
( jS2 1 ) •) ( jS4 j33 ) ^( •) ( jS6 jS5 ) •) jg 1 = jS2 'jS3 = jS4 'jS5 = jS6
S3 S1 < (7,
(p)
jk •••jq •••j1
q=01>02,-,g5,g6
oo
22
J'as =P+1
( j32 1 ) ^ (•) ( j36 j35 ) ^ (•) 1 =j32 'j33 =j34 >j35 =j36
(p)
oo
jfc-jq •••jl
OO
q=0i>02,-,g5,g6
-1
2
{g4=g3+1}
jgi =P+1 jg5 =P+1
(jS4 j33 ) ^ (•) 'js 1 = jS2 'j33 = jS4 'j35 =j36
1 2
Theorem 3 [15] (also see [14], [16], [49]-[51]). Assume that the continuously differentiate functions ^1(r),... (t) : [t,T] ^ R and the CONS (x)}°=0 of continuous functions (cf)o(rr) = 1/y/T — t) m L2([i,T]) are swc/i that the following conditions are satisfied:
1. The equality
S ¿2
TO „ i
$1(i1)$2(t1)dt1 = ^ / $2(i2)&fe) / ^1(t1)0j (t1)dt1dt2 (26) t j =0 t t holds for all s £ (t,T], where the nonrandom functions $1(t), $2(t) are continuously differentiate on [t,T] and the series on the right-hand side of (26) converges absolutely.
2. The estimates
(s) ' T
(t )$1(t )dt
<
J
,'1/2+a'
(t )$2(t )dt
<
*1(s)
J
_
>1/2+« '
oo
<
^2(s)
P
P
Y, J $2«^ (t) ^ (0)d0dT
j=P+1 t t hold for all s £ (t,T) and for some a,^ > 0, where (t), $2(t) are continuously differentiate nonrandom functions on [t,T], £ N, and
T T
J ^2(t)dT < oo, J |^2(t)| dT < oo. tt
s
3. The condition
p
lim \
l^Ofl < J
p^œ
Q Q Q ) 7(p)
S/1 S/2 • • • (jk•••jq•••j1
j1,...,jq,...,jk =0 q=g1>g2>...>g2r-1>g2r
= 0
q=gl,g2,-,g2r—l,g2r
holds for all possible g1, g2,..., g2r-1, g2r (see (18)) and l1,l2,...,ld such that l1, l2, • • •, G {1, 2, • • • ,r}, l1 > l2 > ... > d = 0,1, 2,... ,r - 1, where r = 1, 2,..., [k/2] and
Sl S/2 • • • sL < a
(p)
d 1 jk •••jq •••j1
def (7(p)
q=gl,g2v,g2r-l,g2r
jk •••jq •••jl
q=gl,02,-,g2r—l,g2r
for d = 0.
Then, for the iterated Stratonovich stochastic integral of arbitrary multiplicity k
= T
■M
j]Tr'k) fa v*(tk)... v(t1 )dwi;l)...dw<kk»
(27)
the following expansion:
J *
(il •••ik) T,t
lim- E a^II z
p^œ
j'l ,-,j'k =0
(il ) 'j'i
(28)
/=1
(29)
that converges in the mean-square sense is valid, where
T t2
Cjfc •••jl = j ^k (tk ) j (tk ) ...j (t1)0ji (t1)dt1 •••dtk tt
is the Fourier coefficient, l.i.m. is a limit in the mean-square sense, i1,..., ik = 0,1,..., m,
T
j = j (T )dw
T
j / j t
are independent standard Gaussian random variables for various i or j (in the case when i = 0), w[0) = t.
Futher, we will see that Condition 1 of Theorem 3 is fulfilled.
2
2.4 Expansions of Iterated Stratonovich Stochastic Integrals of Multiplicities 1 to 6. The Case of Legendre Polynomials and Trigonometric Fourier Basis
In this section, we recall several theorems on the expansion of iterated Stratonovich stochastic integrals (3) of multiplicities 3 to 6 that we obtained in [15], [16] (Parts I and II of this work) using Theorem 3. In addition, we recall the expansion of integrals (3) of multiplicity 2 (old result) [14] (Sect. 2.1.2, 2.8.1).
Theorem 4 [14] (Sect. 2.1.2, 2.8.1). Suppose that (x)}£=0 %s a CONS of Legendre polynomials or trigonometric functions in the space L2([t,T]) and fa(t),^2(t) are continuously differentiable nonrandom functions on [t,T]. Then, for the iterated Stratonovich stochastic integral J*[^(2)]Tilti2) (ii, i2 =
0,1,..., m) defined by (3) the following relations:
p
J* [W = 'pi;™- Z jij1, (30)
j1j2=0
M I (rwpy - cf'cf) J < J (31)
are fulfilled, where ii,i2 = 0,1,...,m in (30) and ii,i2 = 1,...,m in (31), constant C is independent of p; another notations are the same as in Theorem 1.
Note that an analogue of Theorem 4 for the case k = 1 follows from (8).
Theorem 5 [14], [15], [49]-[51]. Suppose that (x)}°=0 is a CONS of Legendre polynomials or trigonometric functions in the space L2([t,T]) and fa(t),fa(T),fa(T) are continuously differentiable nonrandom functions on [t,T]. Then, for the iterated Stratonovich stochastic integral J^fa31]^2^1
(ii,i2,i3 = 0,1,..., m) defined by (3) the following relations:
p
j><3>]T;r> = ^m. £ Ciljj1j1, (32)
jl ,j2 j: =0
m 1 - e ^cir'ifcf] ) < 7 (33)
[ V jU2j3 =0 /J P
are fulfilled, where ii, i2, i3 = 0,1,..., m in (32) and ii, i2, ¿3 = 1,..., m in (33), constant C is independent of p; another notations are the same as in Theorem 1.
Theorem 6 [14], [15], [49]-[51]. Let (x)}j=0 be a CONS of Legendre polynomials or trigonometric functions in the space L2([t, T]) and ^i(r),..., ^5(t) be continuously differentiable nonrandom functions on [t,T]. Then, for the iterated Stratonovich stochastic integrals J*[^(4)]y1t"M'), J*^5']^".*5' (¿i,... ,¿5 =
0,1,..., m) defined by (3) the following relations:
p
1 = u£. E c...jij11...j1 (k=4,5), (34)
ii,---,jfc=0
M I - E C,,.„cf... J (k = 4,5) (35)
are fulfilled, where ii,... ,i5 = 0,1,... ,m in (34) and ii,... ,i5 = 1,... ,m in (35), constant C does not depend on p, £ is an arbitrary small positive real number for the case of CONS of Legendre polynomials in L2([t,T]) and £ = 0 for the case of CONS of trigonometric functions in L2([t, T]); another notations are the same as in Theorem 1.
Theorem 7 [14], [16], [49]-[51]. Suppose that {^(x)}j=0 is a CONS of Legendre polynomials or trigonometric functions in the space L2([t,T]). Then, for the iterated Stratonovich stochastic integral of sixth multiplicity
* T * t2
J,1 ■ ■ ■ ;6) = / .. J dwi;11 ...dw<66) (36)
t t
the following expansion:
p
JT,;1"*61^ E Cj6..j1 zj i11 ...zj66)
j1,...,j6=0
that converges in the mean-square sense is valid, where ii,..., = 0,1,..., m,
t t2
Cj6...ji = J j (¿6).. .J j (ti)dti... dto;
t t
another notations are the same as in Theorem 1.
2.5 Connection Between Iterated Stratonovich and Ito Stochastic Integrals of Arbitrary Multiplicity k (k G N)
Introduce the following notations:
i
J>b= %s=it<+1=0} x
q=i
T ts1+3 + 2
X J fak (¿k) ...J fas;+2 (tsj +2) J fa s; (ts; + 1 + 1 (ts; + 1) X t t t t s; + 1 tSi+3 tSi + 2
X J fas,-i(ts;-1) ...J fai+2(isi+2) y fasi (isi+i)^si+l(isi+1 )x t t t
tsi + i t2
x / ^si-1 (tsi-i). . . i fa(ii)dwt(;i>... dwt(;;--ii>dtsi+idwt(;;+^2>...
...dwj;;-i>dts,+idwt(;;++2> ...dwt ;k >, (37)
where (si,... , si) G Ak,i,
Ak,i = {(si ,...,si): si > si— + 1,...,s2 > si + 1; si,..., si = 1,...,k - 1},
(38)
l = 1, 2,..., [k/2] , i1,..., ik = 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 (k G N).
Theorem 8 [54] (1997) (also see [11]-[14], [37], [52]). Suppose that ^(t), ... (t) are continuous nonrandom functions at the interval [t,T]. Then, the following relation between iterated Stratonovich and Ito stochastic integrals (3) and (2) is correct:
[k/2] 1
r=1 ( Sr ,...,si)GAfcir
(39)
where i1,..., = 0,1,..., m and is supposed to be equal to zero.
0
Note that the condition of continuity of the functions ^1(t),... (t) is related to the definition (16) of the Stratonovich stochastic integral that we use (see [14], [52] for details).
2.6 Multiple Wiener Stochastic Integral With Respect to Components of a Multidimensional Wiener Process
For further consideration, we will need the multiple Wiener stochastic integral with respect to components of a multidimensional Wiener process (generalization of the multiple stochastic integral from Ito's famous work [55] (1951)).
Consider the following step function on the hypercube [t,T]k :
N-1
$n (ti,...,tk) = ^ ah,Jk lhi ,T|1+I)(ti)... lhfc ,T|fc+1)(tk X (40) ¿i,...,/fc=0
where a/l.../k E R and such that a/l.../k = 0 if = for some p = q,
f1 if t E A
lA (T) = ,
0 otherwise
N = 2,3,..., {Tj}N=0 is a partition of [t, T] satisfying the condition (6).
Let us define the multiple Wiener stochastic integral for (t1,... ,tk) [55]
N-1
J'[$Nfr!k) = E a/l..* AwT;1'... Aw«;f, (41)
/i,...,/fc =0
where AwTj = wTj++1 — wTj, i = 0,1,...,m, wT0) = t.
It is known (see [56], Lemma 9.6.4) that for any ... ,tk) G L2([t,T]k) there exists a sequence of step functions (t1,... ,tk) of the form (40) such that
lim i ($(ti,...,tk) — (ti,...,tk ))2 dti ...dtk = 0. (42)
N-^oo /
k
We will define the multiple Wiener stochastic integral for ... ,tk) G L2([t,T]k) by the formula [55] (see [14], Sect. 1.11 for details)
J'[$]Trifc) = l.i.m. J'[$n]
(¿1...ifc ) Tt
N —1
= l.i.m
l.i.m. £ 0,1..,k Aw[;;)... Aw^, (43)
AT \ —..-v 1 k
N ' " 1'k
11 k =0
where (t1 , ...,tk) is an arbitrary function of the form (40) satisfying the condition (42), AwTj> = wTj+i — wTj>, i = 0,1,..., m, w[0> = t.
We note the following estimate for the multiple Wiener stochastic integral:
m( J^M2) < Ck ||*||L[t.™, (44)
T,t ; J < Ck H^HL2([t,T]k>
where $(t1,... ,tk) G L2([t,T]k), the constant Ck depends only on k. In [14] (Sect. 1.11) or [52] (Sect. 1.11) the following equality:
T t2
JwTr> = E /-/^(¿i—¿k)dwt(;i>...dwt(;k> w.p. 1 (45) (ti,...,tfe > t t
is proved, where permutations (t1,... ,tk) when summing are performed only in the values dwt(;i>... dwt(;k>. At the same time the indices near upper limits of integration in the iterated stochastic integrals are changed correspondently and if tr swapped with in the permutation (t1,... ,tk), then ir swapped with iq in the permutation (i1,... ,ik). In addition, the multiple Wiener stochastic
integral J'""k) is defined by (43) and
t t2
$(*1,...,tk)dw(...dwt )
is the iterated Ito stochastic integral.
Using (45) and Theorem 5 from [44], we obtain the following theorem.
Theorem 9 [14] (Sect. 1.14), [44]. Suppose that (x)}°=0 is an arbitrary CONS in the space L2([t,T]). Then the following representation:
k [A/2]
J '[j... j iTr*0 = n zf + £(-D
1=1 r=1
r k—2r
x £ ITo,,-. = "»2. =0}1{j»2,_, = ill j' (46)
({{Si,S2},...,{S2r-1,S2r }},(91,...,9fc-2r}) S=1 1 = 1
{Si,S2,...,S2r-1,S2r,91,...,9fc-2r }={1,2,.",k}
is valid w. p. 1, where i1,..., ik = 0,1,..., m, J'[ j ... j]T"1t'""fc) is defined by
(43), [x] is an integer part of a real number x, H =f 1, = 0; another notations
0 0
are the same as in Theorems 1,2.
Combining Theorems 2 and 9 we get the following theorem.
Theorem 10 [44]. Suppose that (x)}°=0 is an arbitrary CONS in the space L2([t,T]) and ^1(r),... (t) E L2([t,T]). Then the following equality:
J [^ (k)]T",rU) = l.i-m. £ ...j ffk) (47)
j1=0 jk=0
is valid w. p. 1, where i1,..., ik = 0,1,..., m, J'[ j ... j]T"1t"'"k) is defined by (43) and J[^( k)]T"1t-"k)
has the form (2); another notations are the same as in
Theorems 1, 2.
3 Main Results
3.1 Generalizations of Expansion of Iterated Stratonovich Stochastic Integrals of Arbitrary Multiplicity k (k G N) Under the Condition on Trace Series
Suppose that fai(r),... , fa(t) G L2([t,T]). Denote
[k/2]
1
+Y, r E = (48)
r=l (sr ,...,si)eAfc,r
where ^ is supposed to be equal to zero; another notations are the same as in
0
Theorem 8 (see Sect. 2.5).
Theorem 11 [14], [49]-[51]. Suppose that the CONS (fa(x)}=0 (fao(x) = 1 /y/T=t) in L2{[t,T}) and fa(r),...,fa(r) G L2{[t,T}) (fa(r)fa_!(r) G L2([t, T]) (/ = 2, 3,..., k)) are such that the following condition:
Pi Pk
lim V...V ...V
jl=0 jq =0 jfc =0
X
q=gi,02,.",g2r—i,g2r
min{psi ,Pg2 } min{Ps2r-1 'Pfl2r }
X 1 E " • E Cjfc-J'l
jgi =0 jS2r-1 =0
jS1 j32 '•"'jS2r-i jS2r
1 r x 2
2r n ^{â,2i=â'2i-l + l}Q'fc---il
1=1
= 0
(jS2 jsi ) ••• (j32r j32r-i )^(0'j3i = jg2 '•••'jS2r-i = j32r
(49)
is fulfilled for all r = 1, 2,..., [k/2]. Then, for the sum J* [fa^]^"^' of iterated Itô stochastic integrals defined by (48) the following expansion:
Pi Pk k
J7. [V,(k>iTi;-"' = J-iprn^ £... £ j j n j' (50)
ji=0 j =0 1=1
that converges in the mean-square sense is valid, where C^...^ is the Fourier coefficient (29), l.i.m. is a limit in the mean-square sense, ii,..., ik = 0,1,..., m,
T
j =| (T)dw«
t
are independent standard Gaussian random variables for various i or j (in the case when i = 0), w[0) = t.
Proof. Let us find a representation of the expression
pi pk k
... cjwi n 4*°
ji=0 jfc=0 1=1 that will be convenient for further consideration.
From (46) we obtain w. p. 1
k
n4!i) = j [j... j fr"-
i=i
[k/2] r k-2r
S(-1)r S n 1{ig2s-i = ig2s=0>1{jg2s_i = jg2s } II C •
r=1 ({{Si,S2},...,{S2r-i,S2r }},{9i,...,9fc-2r}) S = 1 1=1
{Si,S2>...>S2r-1>S2r>91>...>9fc-2r } = {i>2>...>k}
(51)
By iteratively applying the formula (51) (also see (8)-(12)), we obtain the
k ")
following representation of the product Yl Cj" ) as the sum of some constant
1=1 1
value and multiple Wiener stochastic integrals of multiplicities not exceeding
k :
k
ncj," ) = j' [j... j iTr*k)+ 1=1
[k/2] r
+ S S ni{ig2s-i = **2S =0} 1{jg2s-1 = j»2. }X
r=1 ({{Si,S2 }>...>{S2r-i>S2r }}>{9i>...>9fc-2r}) S=1 {Si,S2 >...>S2r-i>S2r>9i>...>9fc-2r }={i>2>...,k}
xJ[j ]?"'''*'"') w.p.1, (52)
where J'[ j ... j]T";t1 •••""fc-2r) =f 1 for k = 2r.
Multiplying both sides of the equality (52) by Cjk...j and summing over j1,..., jk, we get w. p. 1
P1 Pk k P1 Pk
E-E j . .,1 n zj;,) = E... E j' [ j... j ]Tr,k >+
j1=0 jfc=0 1=1 ,1 =0 jk =0
P1 Pk [k/2]
+ S . . . S Cjk. ..j1 X) S n 1{"»2s_1 = "»2s =°}X
j1=0 jk =0 r=1 ({{31,32},...,{32r_1,32r }},{91 ,...,"k_2r}) S = 1
{31,32,...,32r-1,32r,91,...,9k-2r } = {1,2,...,k}
1
{j
32s_1
j32s }
J
j 91
j"k_
2r
-|(i9r
]T,t
"9k_2r
w. p. 1.
(53)
)
Implementing the passage to the limit l.i.m. in (53) using Theorem 10,
P1v,Pk ^^
we obtain w. p. 1
P1 Pk k
J-j- »£■..£ 4 j > =J ^k>]T"r >+
1 k j1=0 jk=0 1=1
[k/2] +L
E
n
1
{"
32s_1
"32,
=0}
X
r=1 ({{31,32},...,{32r_ 1,32r }},{91,...,9k_2r}) s = 1 {31,32 ,...,32r_ 1,32r ,91 ,...,9k_2r} = {1,2,...,k}
P1 Pk r
l(i9
X, '.i;- E-Y, ^ = ,32j} j... j. jTr'"'-2''. (54)
j1=0 jk =0 s=1
Without loss of generality, let us temporarily set p1 = ... = pk = p. We have
Pr
E"T—f f ("" "" )
Cjk .. .j J! 1{j32s_1 = j32. } 1{i32s_1 = "32. =0} J [ j . . - kT"
j1 v,jk =0 s=1
= l.i.m.
p^to
E
E
C
jk •••ji
j1,...,jq,...,jfc=0 j31 ,jg3 ,
9=S1,S2,...,S2r-1,S2r
32r-1
X
jS1 j$2 '•••'jS2r-1 js'
g2r
0
r ( . . )
X II 1(ig2s-1 = ig2s =0}J' [ j '' ' 2r ]iT2r =
s=1
l.i.m.
p^to
E
E
9=S1,S2,...,S2r-1,S2r
C
r— 1
jg1 jg2
Jg,
2r—1
jg,
2r
0
Or n ^
jg2i=g2i —1 + 1}Cjfc...j1
1 = 1
X
(jg2 jg1 )^(^)...(jg2r jg2r—1 )^(^),jg1 = jg2 v-Jg.
g2r —1
jg.
2r
r (' ' )
X n 1{ig2s—1 = Ss =0} • • • jk —2r ]iT'—2r +
s=1
+l.i.m.
p^TO
^ ^ 2r^jk-jl
q=g1,g2,...,g2r—1,g2r
X
(jg2 jg1 • • .(jg2r jg2r—1 )^(0jg1 = jg2 ,• • • >jg2r—1 = jg2r
X
IP
{ig2s —1 = Ss =0} il 1(g2s=g2s —1 s=1 s=1
■/rj. 1 l(i91 •••i9fc —2r)
j91 ' ' ' —2r ]T,t
(55)
l.i.m.
p^TO
pp E ( E Cjk ••
j1 ,...,jq ,...,j'fc =0 \ jg1 ,j'g3 ,...,j'g2r — 1 =0
q=g1,g2,...,g2r—1,g2r
jg1 jg2 '...'jg2r—1 jg;
2r
Or IT ^
jg2i=g2i —1 + 1}Cjfc...j1
1 = 1
X
(jg2 jg1 )^(^)...(jg2r jg2r—1 )^(^),jg1 = j'g.
jg2r—1 jg
2 ^•^g2r—1
2r
r
xll 1{ig2s—1 = =0}J'j ■ ■ ■ j—2,ir%—"' +
s=1
-II1
{g2s=g2s_1 + 1}
J
T"1t-"k)[sr'-'s1] w. p. 1,
(56)
s=1
def
where g2i—1 = S", i = 1,2,...,r, r = 1,2,..., [k/2], (sr,...,s1) E Ak,r, J[^^r^)[sr'-'s1] is defined by (37) and Ak,r is defined by (38), g1,g2,..., g2r—1,g2r as in (18), (t),...,^k(t) E L2([t,T]), ^ (t)^i—1(t) E L2([t,T])
(/ = 2,3,..., k); another notations are the same as above.
Let us explain the transition from (55) to (56). We have for g2 = g1 + 1,
. . . , g2r = g2r—1 + 1
P
1
P^to z—' 2'
j1,...,j9,...,jk = 0 9=31,32,...,32r_1,32r
X
(j32 j31 )^(^)...(j32r j32r_ 1 )^(^)'j31 = j32 v",j32r_1 = j32r
X
n 1{»32s_1 = "32. =0} J' [j1 . . . jk_2r ]T,i
(i91 •••"9k_2r )
s=1
1P
-l.Lm. ^ r
2' p^TO z—'
j1 ,...,jq,...,jk =0
9=31,32,...,32r_1,32r
X
(j32 j31 )^0...(j32r j32r_ 1 )^0'j31 = j32 v",j32r_1 = j32r
X
1
s=1
{"32s_1 = "32. =0}
J' [0.
j91
I ]("91 •••г9k-2r )
. . 0j9k_2r ]T,t
1
1
{"32s_1 = "32. =0}
= —l.i.m.
2r p^to ^ ^ J-J- l'32s_1 "32s'
j1,...,j9,...,jk =0 jm1 ,jm3 -jm^^ =0 S = 1 9=31,32 ,...,32r_ 1,32r
X
xC
jk...j1
X
(j32 j31 )^jm1 ."C^r j32r_ 1 )^jm2r_1 'j31 = j32 v",j32r_1 = j32r
XC(0)c(0)...j J'[0j ...0j ]T"t)
'jm^Jm3 jm2r_ 1 Lrj91 r j9k_2r J T ,t
r
r
—l.i.m.
2r p—>to
E E m
j1,...,Jq,...,jfc =0 jm1 ,jm3 •••,jm2r —1 =0 s = 1
q=g1,g2,...,g2r—1,g2r
{ig2s—1 = Ss =0}
x
C
jk ...j1
x
(jg2 jg1 )^jm1 ••• (jg2
jg2r—1 )^jm2r—1 'jg1 = jg2 '•••'jg2r—1 = jg2r
XJ '[fa
m1 fajm3 ' ' ' fajm2r—1 fajq1 ' ' ' faj9k —2r
(00...0iq1 •••iqk —2r )
(57)
fa
2r
T,t w. p. 1,
(58)
where notations as the same as in (56). The transition from (57) to (58) is based on Theorem 10.
Using the estimate (44), we obtain that the condition
pp
lim
p—TO ^—' I ^—'
j1 ,...,jq ,...,jk =0 \jg1 ,jg3 ,...,jg2r — 1 =
C
jk...j1
q=g1,g2,...,g2r—1,g2r
jg1 jg2 '•••'jg2r—1 jg'
2r
Or n ^
{g2l =g2l —1 + 1}Cjk •••j1
1=1
=0
(jg2 jg1 )^(^)...(jg2r jg2r—1 )^(0>jg1 = jg2 '•••'jg2r—1 = jg2r .
(59)
implies that
l.i.m.
p—TO
pp E E Cjk....
j1 ,...,jq ,...,jk =0 \ jg1 ,j'g3 ;...;j'g2r—1 =0
q=g1,g2,...,g2r—1,g2r
jg1 jg2 '...'jg2r—1 jg'
2r
1
1=1
{g2l=g2l —1 + 1}Cjk...j1
X
(jg2 jg1 )^0)".(jg2r jg2r—1 )^(^),jg1 = jg2 '•••'jg2r—1 = jg2r .
xll 1(i
s=1
g2s—1 = ig2s =0J [fajq1
J [faq1 ' ' ' fajqk — 2r ]T,t
(i91 ...^qk—2r )
= 0,
(60)
0
2
1
2
r
r
where r = 1, 2,..., [k/2]. Obviously, we can omit the condition p1 = ... = pk = p in the above consideration.
Further, note that
E
(«31,32},...,{S2r-1,S2r }},{91,...,9fc-2r }) {31,32,...,S2r-1,S2r,91,...,9fc-2r } = {1,2,-.,k}
A
g2=g1+l,g3=g2+l,...,g2r=g2r-1+1
= E ASl,a2.........(61)
(sr ,...,S1)GÂfc,r
where Ag^...^-, AS1,S2,...,sr are scalar values, g2i-i = s», i = 1, 2,...,r, r = 1, 2,..., [k/2] , Ak,r is defined by (38).
Let us return again to the condition p1,... ,pk ^ œ instead of the condition p1 = ... = pk = p ^ œ. Using (56), (59), (60) with obvious changes and (54), (61), we have
l.i.m. V ...EC,W1TT j) =
j1 =0 jk =0 1=1
[k/2] 1
r=1 (sr ,...,S1)GÂk-
w. p. 1, where J[^^V'^)[sr'""si] is defined by (37) and Ak,r is defined by (38). Theorem 11 is proved.
Now suppose that ^ (t),..., (t) are continuous functions at the interval [t,T]. Then by Theorem 8 we have
J*^]^"^ = Jw. p. 1,
where J*[^(k)]y1t 'ifc) is the iterated Stratonovich stochastic integral (3) and tJ*[^(k)]Ti1t'ik) is the sum of iterated Ito stochastic integrals defined by (48).
Thus, we obtain the following theorem.
Theorem 12 [14], [49]-[51]. Assume that the continuous functions ),
... at the interval [t,T\ and the CONS {(f)j{x)}f=0 {<f>o{x) = 1/y/T - t)
in the space L2([t,T]) are such that the following condition:
pi
Pq
Pk
lim
pi,---,pfc
E-E-E
jl=0 jq =0 jk =0
x
minjpgi ,Pg2 } min{Ps2r-i 'Pfl2r }
*< E ■•• E
C
jsi:
0
jS2r-i =0
jgi jS2 '•"'jS2r-i jS2r
Or IT ^
{g2i=g2i-i+i}Cjfc ...ji
1=1
(jS2 jSi )^0)."(j32r jS2r-i WOfa = jS2 '•••'jS2r-i = jS2r .
= 0 (62)
,(ik ) 'tk
is satisfied for all r = 1, 2,..., [k/2]. Then, for the iterated Stratonovich stochastic integral of arbitrary multiplicity k
* T * t2
j*[fa(k)]Trk) = J fak(tk) . J fa(ti)dwt(;i)...dwt t t
the following expansion:
Pi Pk k
J=pi1^» E-ECkjllj'
ji=0 jk =0 1=1
that converges in the mean-square sense is valid, where Cjk...j is the Fourier coefficient (29), l.i.m. is a limit in the mean-square sense, i1,..., ik = 0,1,..., m,
T
j = / & (t)dwTi) t
are independent standard Gaussian random variables for various i or j (in the case when i = 0), w[0) = t.
Note that the condition of continuity of the functions fa1(T),... ,fak(t) is related to the definition (16) of the Stratonovich stochastic integral that we
2
use. Theorem 12 can be generalized (at least for k = 2 (see Sect. 3.4)) to the case ^i(t),... (t) E L2([t,T]) if instead of the definition (16) we use another definition of the Stratonovich stochastic integral (see the definition (101) below).
Theorem 13 [14], [49]-[51]. Suppose that the CONS (x)}=0 (MX = 1 /y/T=t) in L2([t,T}) and Mr),-..,Mr) e L2{[t,T}) (^(r)^-i(r) E L2([t, T]) (l = 2, 3,..., k)) are such that the condition
2
lim
p^œ
E
Su S/„... , < C4
(p)
d | jk •••jq •••ji
9=31>S2>...>S2r-1>S2r
= 0
(63)
q=gi,02v,g2r—i,g2r
holds for all possible gi, g2,..., g2r-1, g2r (see (18)) and /i,/2,...,/d such that li, 12,..., E {1, 2,... ,r}, 11 > 12 > ... > d = 0,1, 2,... ,r - 1, where r = 1, 2,..., [k/2] and
CO Q ) r(p)
Sl1 Sl2 . . Sk •••jq •••ji
=f C7(p)
jk •••jq •••ji
q=gi,g2v,g2r-i,g2r
for d = 0. Then, for the sum of iterated Ito stochastic integrals
defined by (48) the following expansion:
r
p k Tr) ^ E c^n j1
1=1
p^œ
j1 vJk =0
that converges in the mean-square sense is valid, where Cjj is the Fourier coefficient (29), l.i.m. is a limit in the mean-square sense, i1,..., ik = 0,1,..., m,
T
if = / j(T)<iw«
t
are independent standard Gaussian random variables for various i or j (in the case when i = 0), w[0) = t; another notations are the same as in Theorem 3.
Proof. Step 1. First, we prove that
C'k •••ji + 1 ji ji-1 •••js+ 1 ji js-1 •••ji
j=0
=0
(64)
or
p TO
^ ^ Cjfc..ji+lji ji-l ...js + ljjs-l-jl = ~ ^ ^ Cj + j's-l-jl , (65)
ji =0 ji =p+1
where l — 1 > s + 1.
Our proof of (64) will not fundamentally depend on the weight functions fafa),...,fa(t). Therefore, sometimes in subsequent consideration we set fafa),...,fa(t) = 1 for simplicity.
Using Fubini's Theorem, we have (see [15] (Part I of this work) for details)
C........—
Cjfc ...ji+lji ji-l ...Js + lj js-l-jl
T ti+2 tl+l tl
— y j (tk) ...J faji+l y faji (ti ^ faji-l (ti—1)... t t t t
ts+2 ts+l ts
...J fajs + l ^ / fai fas-l (ts—1) ...
t t t
t2
t
T ts + l ts t2
— j fajs+l fai fas-l (ts —1) ...J fal (t1)dt1 ...dts —1dts X t t t t
/ T T T T
X I y fajs+2 (tS+2) ...J 0ji-l (tl —1) J j (tl ^ j (tl+1) ...
\ts + l ti-2 ti-l ti
.. .J j (tk )dtk ... dti+1dti dti—1... dis+2 I dts+1 —
tfc-l / T ts + l ts t2
— y fajs+l ^ / fai (tS ^ fas-l (ts —1) ...J fal (t1)dt1 . . . dts —1 dtSX t t t t
Gjs-l"j'l (ts)
T T T
><y faji (ti ^ faji+l (ti+1) ...J j (tk )dtk ...¿¿1+1 x
ts+l ti tfc-l
X
HJfc...ji+l(ti)
( \
ti ts+3
f fai-l (tl—1) ...J fa s+2 (ts+2)dts+2 . . . dti —1 dtl
ts+l
-v-'
\ Qji-l...js+2 (ti,ts + l) /
ts+l
dts+1 —
T ts + l
— J fajs+l ^ y faji (ts)Gjs-l...jl (ts)dts X tt
T
..ji+l (tl )Qji-l...js+2 (tl , ts+1)dtl dtS + 1.
ts + l
Applying the additive property of the integral, we obtain
(66)
Qji-l...js+2 (tl, tS+1) —
ti ts+3
— y fai- (tl —1) . . . y 0js+2 (ts+2)dts+2 . . . dtl—1 —
t s + l ts+l
ti ts+4 ts+3
— y j (tl —1) ...J 0js+3 (ts+3^ 0js+2 (ts+2)dts+2dts+3 . . . dtl —1
t s + l ts+l t
ti ts+4 ts+l
— y j (tl—1) ...J 0js+3 (ts+3)dts+3 . . . dtl—1 J 0js+2 (ts+2)dts+2
t s + l ts+l t
— E h
m=1
^(m)
ji-l ."js+2 ^"'^ji-l ...js+2
(ii)«i,m)...,.+,(i«+1), d<
(67)
d
Combining (66) and (67), we have
Cjfc 1 jl jl -1 -js + lj js-1---J1
ji=0
d / T p ts+1
/ ^j's + 1 (tS + 1)qim1---js+2 / 0ji (tS )Gjs-1-j1 (tS )dtS X
ji=o {
T
(m)
X / 0ji(ti)Hjk---ji+1 (ti)hj;-1--js+2(ti)dtidts+1 . (68)
ji-1---js+2
t s + 1
Applying the generalized Parseval equality, we obtain
ts + 1 T
TO „ r,
53 / ^ (ts )Gjs-1---j1 (ts )dtW ^ji (t1 )Hj •••ji + 1 (t1 )hjmn1---js+2 (t )dt1 = ji=0 J J
t ts + 1
T
i{r<ts+1} Gjs-1-j1 (t ) • i{r>ts+1 }Hjfc-ji+1 (t jj (t )dT =0. (69)
¿s + 1}w js-1-j1V' ; {T>ts+1 rJjk•••ji + 1V' / ji — 1 •••js+2
t
From (68) and (69) we get
p
••ji+1 jiji-1 •••js + 1ji js-1•••jl
53 Cjk •••J
ji =o
d / T TO ts + 1
53 / ^j's+1 (ts+1)qjm)-js+2 C-+0 £ J j (ts )Gjs-1-j1 (ts )dts X m=1 V t ji =p+1 t
T
x y (ti)Hjk-ji+1 (ti1-js+2(ti)dtidts+1 | . (70)
ts+1
Suppose that {hj(x)}°=0 is an arbitrary CONS in L2([t,T]) and $1(t), ^2(t) E L2([t,T]). Then we have
s T
J (t)$1(t)dT J h (T)$2(t)dT <
t s
to
E
j=0
p
1
oo
It T
T
j=0
1{r<s}fa (t)$1(t)dT + / 1{T>s}fa (t)$2(t)dT
V
1 2
T
J $?(r)dr + I $2(r)dr <i(||$
2 Vl^1lli2([i,T]) + ll<^2||L2([t;T])
2
j
— C < oo. (71)
This means that the estimate (71) can be applied to the series
E
ji =p+1
ts + l T
J j (tS)Gjs-l...jl (tS)dtS ^ faji (tl)Hjfc. ..ji+l (ti )hjml...j,+, (ti w
t ts + l
Using the above result, Lebesgue's Dominated Convergence Theorem and (66)-(68), (70), we have
p
Cjfc •••jl+1 jiji-l •••js + 1jljs-1•••j1 —
ji =0
TO d / T ts+1
E El/ ^j's+1 (tS+1)qi'-)...js+2 (tS+1^ faji (ts )Gjs-l...jl (ts )dts X
ji =p+1 m=1 y t T
X y faji (ti )Hjk ...ji + l (ti )h.nl...js+2 (ti )dti dtS + 1
ts+l
TO
Cjfc 1 ji ji-1 ...js + l ji js-1 ...jl .
ji =p+1
The equality (72) implies (64), (65).
Step 2. Further, let us prove that
T t2 t
1
2
(72)
TO
E j Mh)Mt2) j = 2 j fai(r)M-r)dT, (73)
j=0 t t t
where {fa(x)}°=0 is an arbitrary CONS in the space L2([t,T]) and fa(T), fafa) G L2([t,T]).
2
s
2
2
Let us list some useful facts that we will need further in this section.
Proposition 1 ([59], Theorem 8.1). Let K : L2([t,T]) ^ L2([t,T]) be an integral operator defined by
T
(Kf) (t) = J K(T,s)f (s)ds, t
where K(t, s) is a continuous function on [t,T] x [t,T]. If, in addition, K is a trace class operator then
T
trK = J K(s,s)ds, (74)
t
where trace trK is defined as a series of singular values Sj(K) of K. Proposition 2 ([59], P. 71). Let
T
(Kf) (t) = y K(t,s)/(s)ds,
t
the kernel K(t, s) is continuous on [t,T] x [t,T] and satisfies the condition
|K(t,s2) - K(t,sI)| < C |S2 - S1 |a , (75)
where 0 < a < 1. If, in addition, K is a Hermitian operator and a > 1/2, then K is a trace class operator.
Suppose that A : H ^ H is a linear bounded operator. Recall [58] that A has a finite matrix trace if for any orthonormal basis (x)}°=0 of the space H the series
to
£<A0j )h (76)
j=0
converges, where (-, -)H is a scalar product in H. Note that the series (76) converges absolutely since its sum does not depend on the permutation of the terms of the series (76) (any permutation of basis functions (x) forms a basis in H) [58].
Proposition 3 ([59], Theorem 5.6). Let K : H ^ H be a trace class operator. Then
TO
trA — ^ (Afa >h (77)
j=0
for any orthonormal basis {fa(x)}°=0 of H.
Consider an integral operator K' : L2([t,T]) ^ L2([t,T]) defined by the equality
T
(K'f) (t) —J K'(T,s)f (s)ds,
t
where the continuous kernel K'(t, s) has the form (see [14], Sect. 2.1.2)
Kfc,« — (fa2(t1)fa1(t2)' 41 > t2 (t^ G [t,T]), [fa1(t1)fa2(t2), t1 < t2
where fa(T),fa(T) are continuously differentiable functions on [t,T]. Note that (see [14], Sect. 2.1.2)
|K'(t2, S2) — K'(t1, S1)| < L (|t2 — t1| + |S2 — S1|) , (78)
where L < to and (t1 ,s1), (t2, s2) G [t,T]2. Let us substitute t1 — t2 — t into (78)
|k'(t,s2) — k'(t,s1)| < L|s2 — S1|. (79)
Thus, the condition (75) is fulfilled (a — 1). Further, using Fubini's Theorem, we have
T t2
(K'x,y>L2([t,T]) — J fa2(t2)y(t2^ fa1 (t1)x(t1)dt1dt2 + tt
T T T T
+ J fa (t2)y(t2^ fa (t1 )x(t1)dt1dt2 — J fa1(t1)x(t1^ fa2(t2^2)^2^1 +
t t2 t tl
T t2
+ y ^1(t2)y(t2)dt2dt1 = (K/y,x)i2([t,T]) . (80)
t t
The conditions of Proposition 2 are fulfilled. Then, K' is a trace class operator. Since the kernel K'(t1,t2) is continuous, then by Propositions 1 and 3 (see (74) and (77)) we obtain
™ T T
to cp
Y, (K'j , j)L2([t,T]) = K'(s,s)ds = ^i(s)^2(s)ds. (81) j1=0 { t
Combining (80) and (81), we get
Y ( /^Mfe (t2^ ^l(tl) j (tl)dtidt2 +
j1=0 V t t
T T \ T
+ y ^l(i2)0j! (t2^ ^(tl) j (tl)dtidtJ = J ^i(s)^2(s)ds, (82)
t t2 t
where (x)}°=0 is an arbitrary CONS in the space L2([t,T]) and ^l(r),^2(t) are continuously differentiable functions on [t, T].
Let us substitute ^2(t) = (t — t)1 and ^l(r) = (t — t)m (l,m = 0,1, 2,...) into (82)
oo / t t2
£ /(t2 — t)1j fe) /(tl — t)mj (tl)dtldt2 +
T T \ T
^(t2 — t)m0j!(t2) y (tl — t)1 j(tl)dtldt J = J(t — t)1 (t — t)mdT, (83) t t2 t where l, m = 0,1, 2,...
The equality (83) was obtained in [57] using other arguments. In addition, the formula (83) was used in [57] to obtain (73).
Consider this approach [57] in more detail. Since the equality (83) is valid for monomials with respect to t — t (t E [t,T]), it will obviously also be valid
for Legendre polynomials that form a CONS in the space L2([t,T]) and finite linear combinations of Legendre polynomials.
Let fa(T),fa2(T) G L2([t,T]) and fap)(T),fa2q)(T) be approximations of the functions fa(T), fafa), respectively, which are partial sums of the corresponding Fourier-Legendre series. Then we have (see (83))
00 / t t2
El /fa2q)(t2)faj (t2 ) / fa1p) (t1)faj (t1)dt1dt2 +
j =0 \i t
T T \ T
+ J fap)(t2)faj(t2^ fa2q)(t1 (t1)dt1 dtj — J faip)(T)fa2q)(T)dT. (84) t t2 t
Let us fix q in (84). The right-hand side of (84) for a fixed q defines (as a scalar product in L2([t,T])) a linear bounded (and therefore continuous) functional in L2([t,T]), which is given by the function fa>q). The integral operator (which corresponds to the matrix trace on the left-hand side of (84)) is a trace class operator (see [57]). The matrix trace of the mentioned operator (on the left-hand side of (84)) is also a linear bounded (and therefore continuous) functional (in the space of trace class operators [58], [59]) which can be extended to the space L2([t,T]) by continuity [60].
Let us implement the passage to the limit lim in (84)
p—yTO
oo ( T t2
E / fa2q)(t2)faj(t2) / fa(t0fa(t1)dt1dt2+
+ J fa(t2)faj(t2) y fa2q)(t1 (t1)dt1 dtn — J fa (T)fa2q)(T)dT, (85) t t2 t
where q G N. Recall that fa>q)(T) is a partial sum of the Fourier-Legendre series of any function fa(t) G L2([t,T]), i.e. the equality (85) holds on a dense subset in L2([t,T]). The right-hand side of (85) defines (as a scalar product in
L2([t,T])) a linear bounded (and therefore continuous) functional in L2([t,T]), which is given by the function . On the left-hand side of (85) (by virtue of the equality (85)) there is a linear continuous functional on a dense subset in L2([t,T]). This functional can be uniquely extended to a linear continuous functional in L2([t,T]) (see [61], Theorem I.7, P. 9).
Let us implement the passage to the limit lim in (85)
q—>-to
oo / T t2
£ / Wt2)<fe (t2) / ^l(tl)0j (tl)dtldt2 +
j=0 \ t t T T \ T
+ J ^l(t2)0j (t2^ ^2(tl)0j (tl)dtldtj = J ^l(T )^2(t )dT. (86)
t t2 t
Applying Fubini's Theorem to the left-hand side of (86), we obtain (73). Step 3. Under the conditions of Theorem 13 we prove that
P 1 TO
^ ^^3k--jl+l3l3l3l-2--jl = oQ'fc-jl ~~ ^3k-3l + l3l3l3l-2--jl (87)
feWO =p+l
31 =0
or
to 1
53 ^3k-3i+iji3i3i-2-~3i = 31 =0
. (88)
j WO
Denote
tl-1 t2
tt Using Fubini's Theorem and (73), we obtain
to to T tl+2
53C3k •••31+1313131-2...31 = £ / (tk )0jfc (tk) ... ^1+1(t/+1)03i+i (t/+1)X
3l=0 3l=0 t t
^+1 t;
(t/)03; (tl )J ^-i(t/-i)03; (t/—l) Cji—2 ...ji (i/-i)di/-idi/di/+i ...dtk =
tt
oo
T ti
— E / fa (ti )faji (ti) / fa—1 (ti—1)faji (ti—1)Cji-2...jl (ti—1)dti—1 x ji=0 t t T T
X J fa+1(ti+1)fai+l (ti+1).. .J fa (tk ) j (tk )dtk .. .dti+1dti —
ti tfc-l
T
1 TO f
ji =0 t
T T
X J fa+1(ti+1)fai+l (ti+1).. .J fa (tk) j (tk )dtk .. .dti+1dti —
ti tfc-l
TO T ti+2
= ^E/Mtk)<f>jk(tk) ■ ■ ■ J fa+l(fal)fa,+ 1(fal)x ji=01 t
ti+1
X J ... dtk = ^Cfaj!
t
(ji ji WO
The equalities (88) and (87) are proved.
Step 4. Applying (65) and (87) repeatedly, we get (see [15] (Part I of this work) for details)
p r (.
E Cjk...jl n 1{jg2s-l = jg2s }1{ig2s-l = Ss =0} J' [ j . . . fajqk-2r ]T'
p—TO ' * J- -M- ^2s— 1 a2s
jl ,...,j'fc =0 S=1
1{
K^s-l = Ss =0}X
s=1
p1 xl.Lm. ^ —C'.//..,/
p—to z—' 2'
9=Sl>S2>...>S2r-l>S2r
X
(jg2 jSl )^(0".(j32r jg2r-1 WOfa = jS2 '."'far^ = far
r
X II 1te,=g2,-1 + 1} J' [ j . . . iT? ' '.!,k-2' ' +
s=1
+ n 1{ig2s-1 = ig2s =0}^
(89)
s=1
2r IT ^-{92s=g2s-i+i} J
T,t +
s=1
+ 1! 1{ig2s-i = ig2s =°>l:i-m-
(p)gi,g2,...,g2r-i,g2r(iq1 •••w-2r )
s=1
(90)
def
w. p. 1, where g2i-1 = i = 1, 2,..., r, r = 1, 2,..., [k/2] , (sr,..., s1) G Ak,r, J)[sr'•••'si] is defined by (37) and is defined by (38), and
R
(p)gi,g2,---,g2r-i,.g2r (igi •••w-2r )
T,t
E ( (-D^C^j^j, j
,=3i ,32>...>S2r-i>S2r
+
q=gi,g2v,g2r-i,g2r
'i=1
+
q=gi,g2, • • • ,g2r-i,g2r
r f
+(-1)r-2 £ S'i j j, • • j
11 2 = i I li>l2
+
q=gi,g2, • • • ,g2r-i,g2r
+(-1)1 E S'i S'2
s ) Cw
S'r-i Ï Cjfc • • j, • • ji
ii,i2,...,ir-i = i 11 > ¿2 >...>lr_i
q=gi>02, • • • ,g2r-i,g2r
X J * * * ^j,fc-2r ]T,t
(i,i • • • W-2r )
X
The transition from (89) to (90) is explained in the proof of Theorem 12 (see the derivation of (58)).
By condition (63) of Theorem 13 we have (also see the property (44) of the multiple Wiener stochastic integral)
r
r
r
P
r
i
C\
,(p)gl,g2,...,g2r-l,g2r («gl -Wo.. A
p—m M ^ (RTJ^"-tqk-2r' ] y <
p
< K lim V
p—TO ^—'
(p)
jk -jq ...jl
9=31,32> — >32r-l>32r
+
q=gl,g2,...,g2r-l,g2r ,
+ E ( ...j
il=1
+
q=gl,g2,...,g2r-l,g2r
2
2
r
l
+ E ( Sil <Cj(p)..jq ...jl
il ,i2 = 1 il>i2
+
q=gl,g2,...,g2r-l,g2r
2
r
+ E (Sil Si2 ...Sir-0 CjWq ...jl
il,i2,...,ir-l = 1 il >i2 >... >ir -1
— 0, (91)
q=gl,g2,...,g2r-l,g2r
where constant K does not depend on p. Using (90) and (91), we obtain
!p——3TO. E Cjk 1 n 1{j32s-l = j32s }1{i32s-l = i32s =0} '^ [ j . . . faqk-2r ] TT,^1 )
j'l ,-Jfc =0 s=1
1r
w. p. 1, (92)
s=1
where notations are the same sa in (90).
Applying (54) for the case p1 — ... — pk — p as well as (61), we obtain
pk l.——TO. E Cj...j,n <j*> —
j'l,...,j'fc=0 i=1
[k/2]
= J[^k)%Ak) + E ^ E = J*№k)]$t"ik) (93)
r=1 (sr ,...,sl)eAfc,r
2
w. p. 1, where notations are the same as in Theorem 8 and J*^^]^'ifc^ is defined by (48). Theorem 13 is proved.
3.2 Generalization of Theorems 11—13 to the Case When the Conditions (f)0(x) = 1/y/T^t and i(t) g L2([t,T]) (I = 2,3,..., k) are Omitted
In this section, we will consider the following generalization of Theorem 11.
Theorem 14 [14], [50]. Assume that the CONS (x)}°=0 in the space L2([t,T]) and ^i(t ),..., (t ) G L2([t,T]) are such that
P1 Pk
lim ... ...
pi,...,pfc ^œ — — — ,
j1=o jq=0 jk=0 q=gi,g2,...,g2r-i,g2r
x
min{pSl ,Pg2 } min{Pfl2r-1 'Pfl2r }
X 1 E Cjk .. .jl
jgi =0 j32r-1 =0
jS1 jS2 '...'jS2r-1 j32r
2r IT
1 = 1
= 0
(j32 jS1 )^(^)...(jS2r jS2r-1 W-),^ = j32 '...'jS2r-1 = j32r
(94)
for allr = 1, 2,..., [k/2]. Then, for the sum of iterated Ito stochas-
tic integrals defined by (48) the following expansion:
P1 Pk k
J7*^'.."1 = J.^ œ Y-£ C^II j '
j1=0 jk =0 1=1
that converges in the mean-square sense is valid, where Cjk...j is the Fourier coefficient (29), l.i.m. is a limit in the mean-square sense, i1,..., ik = 0,1,..., m,
T
j = | & (t )dw«
t
are independent standard Gaussian random variables for various i or j (in the case when i = 0), wT0) = t.
Proof. To prove Theorem 14, we need to prove that under the conditions
of Theorem 14 the following equality:
p
l.i.m. V —C«,
Z^ 2r jfc"j1
9=S1,S2,...,S2r-1,S2r
X
(j32 j31 j32r-1 MOfa = fa '•••'far-1 = far
r
(;qi •••iqfc_2r )
X II 1{ig2s-1 = ^ =0}J'[ j • • • iT,*"'tqfc-2r
s=1
= (95)
holds w. p. 1, where = gi + 1,..., = #2r-i + 1, #2i-i == S, i = 1, 2,..., r, r = 1, 2,..., [k/2], (sr,..., Si) G Ak,r, J[fa(k)]Ti)1i'"ifc)[sr"•"si] is defined by (37) and Ak,r is defined by (38); also we put pi = ... = pk = p in (95) to simplify the notation; another notations in (95) are the same as in Sect. 2.3, 2.5, 3.1.
Using the Ito formula, we obtain w. p. 1
T ti+2 ti+1 ti-i
/ fa (tk) ■■■ fa+i(fai) / fa (fai)fa-i(fal) / fal-2(t/-2) . ..
t2
" fa (ti )dwt(;1)... dwt(;fa)dti-idwt(;^1)... dwtik )
t
T ii+2 i ti + 1 \ ti+1
= J fa (tk ) - .y ^i+i(ii+iH J fa (£z-i)fa-i(iz-i)d£/_i J J fai-2(t/-2)...
t t \ t / t
t2
fa (ti)dwi1^ ...dwj;^ ...dwtfc
T tj+2 tj+1 / tj_2
-J 1k (tk).. .J 1/+i(t/+i) y 1/-2(t/-2) I J 1/ (t/-i)1/-i(t/-i)dt/-i | x
t t t \t tl-2 t2
xy^i-3(ti-a)..J^i(ti)dwt(;i}...dw^dw^Wi;^...dw(;k), (96) tt
where l > 3. Note that the formula (96) will change in an obvious way for the case t/+i = T. We will also assume that the transformation (96) is not carried out for l = 2 since the integral
t3
y ^2(ti)^i(ti)dti t
is an innermost integral on the left-hand side of (96) for this case.
It is important to note that the transformation (96) fully complies with the classical rules for replacing the order of integration (Fubini's Theorem) if we replace all differentials of the form dwt(j) with dtj in (96).
Indeed, formally changing the order of integration on the left-hand side of (96) according to the classical rules, we have
T tl+2 tl+1 tj_i
i lk (tk) ... i 1/+i(t/+i) / 1/ (tl-i)^l-i(tl-i) i ^/-2(t/-2) ... (97)
t2
r(;1) J „i;l_2U J„i;l + 1) J„i;fc)
1 (ti )dwt1^... dt/_idw^... dw
tfc
T tj+2 / tj + 1 tj+1
f 1 (tk) ...y ^/+i(t/+iM y 1 (ti)dwt;1) ...y ^/-2(t/-2)dwt(;j;2)x t t t tj-3
tl+1 \ x /1/(t/-i)i/-i(t/-i)dt/-J dw(;^1).. .dwt;k) =
tl-2
T tl+2 / ti + 1 ti+1
= J fa (tk ) ..J faz+i(tz+i) ( J fai(ti)dwt(;1) ..J faz-2(tz-2)dwt(;^2)X
t t \ t ti—3
/ti+1 *i —2\ \ x [J-J faz (tz-i)faz-i(tz-i)dtz-J dwi;^1 ) ...dwt(;k ) =
T t;+2 / t;+1 \ ti+1
= J fa (tk ) ...J fa+i(tz+i) J faz (tz-i)faz-i(tz-i)dtz-J J fa (ti)dwt(i1).. t t t t
ti+1
"7 ^-^-^fc2^'1 -dw(:k )-
ti—3
T ti+2 ti+1 ti+1
-J fa (tk ) ...y faz+i(tz+i^ fai(ti)dw(;1) ...J faz-2(tz-2)x
t t t ti-3
x ^ J fa(tz-i)faz-i(tz-i)dtz-ij dwt(;i-22)dw(;^1)... dwt(:k) =
T ti+2 i ti + 1 \ ti+1
= y fa(tk) . - y fa+i(tz+i) ( y faz(tz-i)faz-i(tz-i)dtz-H y faz-2(tz-2)...
t t t t t2
..J fai(ti)dwt(;1)... dw^dw^... dw(') t
T ti+2 ti+1 / ti —2
-J fak (tk )... y faz+i(tz+i) y faz-2(tz-2M y^z (tz-i)faz-i(tz-i)dtz-^ X t t t t ti —2 t2
X y faz-3(tz-3) ...J fai(ti)dwt(;1)... dw^dw^dwi^... dw(;k). (98)
tt
Comparing the right-hand sides of (96) and (98) we come to the conclusion that we got the same result.
The strict mathematical meaning of the transformations leading to (98) is explained in Chapter 3 [14], at least for the case when ^i(r),...,(t) are continuous functions on the interval [t,T].
Note that under the conditions of Theorem 14, the derivation of the formulas (96) and (98) will remain valid if in (96) and (98) we replace all differentials of the form dw(ij) with dtj (this follows from Fubini's Theorem).
Temporarily denote Jas I[^(k)]^:11-^«!-1^«!^2-^«^-1^«^^2-^^). Let us carry out the transformation (96) for the iterated Ito stochastic integral I[^(k)]T;it-is!-1is!+2-isr—1isr+2-ifc) iteratively for s1,...,sr. After this, apply Theorem 10 (see (47)) to each of the obtained iterated Ito stochastic integrals. As a result, we obtain w. p. 1
i [^(t)]j',1f'**1-1**1+2""'"-1*"+2'*') = n if-,=-,+1=»>x
q=1
2r
d(«1 .. . iS1-1«S1+2 ... isr-1V +2... ¿fc) fr I (k)l d(i1 ... ¿s1 —1's1+2 •.. isr — 1 i Sr +2-• • ¿k )
x ^^ ^/[^(k)]T(i1.••iSl —1iS1+2...¿sr — 1«sr +2...ifc) _ J^ ^
d=1
ll1fisq =isq + 1=0} X
q=1
xl.i.m. y y ¿f
p_~ / V / V I .1 s 1 — 1jS1 + 2 ...sr — 1jSr +2...jfc
j1,-,js1 —1,js1+2,-,jsr — 1,jsr +2,...,jfc =0 d=1
(d).....i x
j1 "j's 1 — 1 js1 +2 -isr — 1 jsr +2 •••jfc
xJ'[. ... .—1 ^....—1 .+2... .f,.-''1-1'-^-.'''-1'-+2-'k), (99) where some terms in the sum
2r
E
d=1
can be identically equal to zero due to the remark to (96).
r
r
Taking into account that the integrals /[fa^]^1"^1 1is1+21V+2 ) and the Fourier coefficients (fa(d) ■ ■ ■ ■ ■ are formed on the basis of the
,1 )))jS1 — 1js1 + 2 ...jsr-ljsr + 2)))jk
same kernels (the same applies to the integrals /[fa^]^1"^1-11is1+2)))isr—1isr+2)))ik) and the Fourier coefficients (fa(d) ■ ■ ■ ■ ■ ), as well as a remark about the relationship of the transformation (96) based on the Ito formula and on the basis of classical rules for replacing the order of integration (see the derivation of (98)), we obtain using Fubini's theorem (applying the inverse transformation from (98) to (97) in which all differentials of the form ) are replaced with dt,)
2r /
El j (d)....._ C/ (d)
I ,1))),s1 — 1,s1+2)))jsr — 1jsr + 2)))jfc ,1
■)Js1 — 1js1+2)))jsr — 1jsr + 2)))Jfc J1)))JS1 —1js1 + 2)))jsr — 1 ,sr +2))Jfc
d=1
— (
(100)
(,32,31 )^())))(,32rj32r—1 WOfa = j32 ')))j32r — 1 = j32r
where g2 — g1 +1,..., g2r — g2r-1 + 1. Combining (99) and (100), we get w. p. 1
I [^(k) ^H)))^ —1^+2)))^ — 1V+2)))ik ) —
p
p^TO z—'
j1,...,Jq,...,Jfc =0 9=31>32>...>S2r—1>32r
X
(,32j31 )^())))(,32r,32r—1 WOfa = j32 ')))j32r — 1 = j32r
X
s=1
n 1{i32s —1 = i32s =°>'J' • • • ^fc — 2r ]T,i 9fc —2r
where we use the notations from Sect. 2.3, 2.5, 3.1. The quality (95) is proved for the case when {fa (x)}°=° is an arbitrary CONS in the space L2([t, T]). Thus, the condition fa (fa = 1/y/T — t in Theorems 11-13 can be omitted.
Let us separately explain why the condition fa(t)^l-1(r) £ L2([t,T]) (l — 2,3,...,k) in Theorems 11, 13 can also be omitted. Recall that this condition appeared due to the direct application of Theorem 10 to the iterated Ito
stochastic integral J[iM]^1,"'*^''•••'sl] defined by (37) (see the transition from (57) to (58)).
It is easy to see that the kernels Kd(t1,..., tk_2r) and KKd(t1,..., tk_2r) of the iterated Ito stochastic integrals /[1(k)]T(t1".is1—11is1+2".isr—11isr+2--.ik) and J[1(k)]T(;1--.is1—1is1+2---isr—1isr+2".ifc) have the same structure as (4) but with new wight functions 1(t),..., 1k_2r(t) and 1^t),...,1k_2r(T), some of which possibly coincide with ^1(t),..., 1k(t) (see (96)). Moreover, the conditions ^1(t),..., 1k(T) G L2([t,T]) and 1/(t)1i_1(t) £ ¿1([t,T]) (l = 2,3,...,k) guarantees that Kd(t1,..., tk_2r), ..., tk_2r) G L2([t,T]) (see (96)). This
means that the formula (99) is true if ^1(t),...,1k(t) £ L2([t,T]) and 1(t)^/_1(t) G L1([t, T]) (l = 2,3,..., k). Furthermore, the formula (100) holds under the conditions 11(t),..., 1k(T) G L2([t,T]) and 1^t)^/_1(t) G L1([t,T]) (l = 2,3,..., k).
Since the condition ^1(t),..., 1k(t) G L2([t,T]) implies the condition 1(t)1/_1(t) G L1([t, T]) (l = 2, 3,..., k), then the condition ^/(t)^/_1(t) G L1([t,T]) (/ = 2,3,..., k) can be omitted in the above reasoning.
Thus, the equalities (99) and (100) are satisfied under the condition ^1(t),..., 1k(t) G L2([t,T]) and the condition 1/(t)1/_1(t) G ¿2([t,T]) (l = 2,3,..., k) can be omitted in Theorems 11, 13. Theorem 14 is proved.
3.3 Another Definition of the Stratonovich Stochastic Integral
def
Let (Q, F, P) be a complete probability space and let w(t, w) = : [0, T] x Q _ R be the standard Wiener process defined on the probability space (Q, F, P).
Let us consider the family of a-algebras {Ft, t G [0, T]} defined on (Q, F, P) and connected with the Wiener process in such a way that
1. Fs C F, c F for s < t.
2. The Wiener process is Ft-measurable for all t G [0,T].
3. The process wt+A — wt for all t > 0, A > 0 is independent with the events of a-algebra Ft.
Let ^(t, u) =f fa : [0,T] x Q ^ R be some random process, which is measurable with respect to the pair of variables (t, u) and satisfies to the following condition:
T
J |fa|dT < oo w. p. 1 (t > 0).
t
Let {Tj}N=o is a partition of [t,T] such that the condition (6) is fulfilled. Consider the definition of the Stratonovich stochastic integral, which differs from the definition given in Sect. 2.2.
The mean-square limit (if it exists)
N—1 i tj+i T
l.i.m. y^- / isd.s (wTj+1 - wTj) = /fa ° dwT (101)
j=0 Tj+i _ Tj J J
J Tj t
is called [63], [64] the Stratonovich stochastic integral of the process fa, t E [t,T]. We also denote by
T
J O dws
t
the Stratonovich stochastic integral like (101) (if it exists) of 1{sE[t,T]} for t E [t,T], t > 0.
It is known [64] (Lemma A.2) that the following iterated Stratonovich stochastic integral:
T t2
JS fak) ]Tit-ik) = J fa (tk) ..J fai(ti) O dwt(il)... O ) (102) tt
exists for the case i1 = ... = ik = 0, where t E [t,T], 'fafa),...,fa(t) E L2([t, T]), fa ..., ik = 0,1,..., m, wTi) (i = 1,..., m) are independent standard Wiener processes defined as above in this section and wT0) = t.
Note that in [65] (2021) an analogue of Theorem 8 (1997) is proved for the integral JS[^k)]T;t-ifc) (ii = ... = ik = 0, 1i(t),...,1k(t) G L2([t,T])).
3.4 Expansion of Iterated Stratonovich Stochastic Integrals of Multiplicity 2. The Case of an Arbitrary CONS in the Space
L2([t,T]) and 1i(t),12(t) G L2([t,T])
Consider the special case k = 2 of Theorems 12, 14 in more detail. In this case, the conditions (62), (94) (p1 = p2 = p) takes the following form (compare with
(73)):
^ t t2 t
J2 J Mh)(t>j(t2) J Mti)^(ti)dtidt2 = ^ J 1i(r)12(r)dr, (103) j=0 t t t
where {1j(x)}°=0 is an arbitrary CONS in L2([t,T]) and 11 (t),12(t) G L2([t,T]) (Theorem 14) or 11(t),12(t) are continuous functions on [t,T] (Theorem 12).
Thus, from Theorem 12 (the case k = 2) we obtain the following theorem (recall that the conditions (f)o(x) = 1 /\/T — t can be omitted in Theorem 12).
Theorem 15 [14], [49]-[51]. Suppose that {1j(x)}°=0 is an arbitrary CONS in L2([t,T]) and 11(t),12(t) are continuous functions on [t,T]. Then, for the iterated Stratonovich stochastic integral
ST * Î2
r^/.(2n(i1i2) _ / „/, \ I J, (+
tl
t t
J*[1(2)]T;^ = J ^2) / 1i(ti)dw|l dw^ (ii,i2 = 1,... ,m) t
the following expansion:
,/><2)]Tf = e Ec*.<j;°cj:2)
jl=0 j2=0
Pl P2
that converges in the mean-square sence is valid, where the notations are the same as in Theorems 1,2.
The condition of continuity of the functions fa (t),fafa) is related to the definition (16) of the Stratonovich stochastic integral that we use.
Theorem 15 can be generalized to the case fafa),fa2(T) E L2([t,T] (see below) if instead of the definition (16) we use another definition of the Stratonovich stochastic integral (see the definition (101)).
From Proposition 3.1 [65] for the case k = 2 we obtain
T t2 T t2
J fa(t2) J fafa) ◦ W ◦ w = J fa(t2) j fai(ti)dwt(;)dwt(;)+ t t t t
T
J fa(ii)fa(iifaii (104)
t
w. p. 1, where fafa),fafa) E L2([t, T]), i = 1,..., m,
t t2
J fa(t2)^ fa1(t1) ◦ dwt ;) ◦ dwi;) tt is defined by (101), (102) and
T t2
j ^2(t2^ fa1(t1)dwt(;)dwi;) tt
is the iterated Ito stochastic integral of the form (2) (k = 2, i1 = i2 = i).
On the other hand, it is not difficult to show that
T t2 T t2
j fa(t2)^ fa(t1) ◦ dwt( ;) ◦ dwj =J fa^) j fa1(t1)dwt( jWj (105) t t t t
w. p. 1, where fafa),fa2(T) E L2([t,T]), i = j (i,j = 1,...,m), another notations are the same as in (104).
Combining (104) and (105), we get (see (48)) T t2 T t2
J fafa)/ fafa) ◦ dwi;i) ◦ dwi;2) = J fafa)/ fa1(t1)dwi;i)dwt(;2)+ t t t t
T
+ii{il=i2} J Mti)Mti)dt! = r[^2)t:2) (ice)
t
w. p. 1, where fa(t), fafa) G L2([t, T]), ii, i2 = 1,..., m.
Summing up the above arguments, we obtain from Theorem 14 (k = 2) the following generalization of Theorem 15 to the case fafa),fa(T) G L2([t,T]).
Theorem 16 [14], [49]-[51]. Suppose that {fa (x)}°=0 is an arbitrary CONS in the space L2([t,T]) and fa(t),fa(T) G L2([t,T]). Then, for the iterated Stratonovich stochastic integral
t t2
JS[fa2^^ = /fafa)/fafa) ◦ dwi(;i) ◦ dwt(i2) (ii,i2 = 1,..., m) t t
the following expansion:
Pi P2
js [fa(2)]Tiit'2) = i.i.m^ Ej jj (107)
jl =0 j2=0
that converges in the mean-square sence is valid, where the notations are the same as in Theorems 1, 2 and JS[fa(2)]T;it;2) is defined by (102).
3.5 Expansion of Iterated Stratonovich Stochastic Integrals of Multiplicity 3. The Case of an Arbitrary CONS in the Space
L2([t,T]) and fafa),fafa),fafa) = 1
In this section, we will prove the following theorem.
Theorem 17 [14], [49], [50]. Suppose that {fa(x)}°=0 is an arbitrary CONS in the space L2([t,T]). Then, for the iterated Stratonovich stochastic integral of third multiplicity
* T * ts * t2
J*[fa(3)]T;,1t;2;3) = // / dwt(;i)dwt(;2)dwt(ss) (ii,i2,i3 = 0,1,...,m) t t t
the following expansion:
J'i#>]<rs) = £ ,,C'cfC» (108)
jl,j2 ,,3=0
that converges in the mean-square sense is valid, where
T t3 t2
C,3j2ji = J 1,3 ^ 1,2 (t2")J , (t1 )dt1dt2dt3 t t t
and
T
cf = J(T )dw<:>
t
are independent standard Gaussian random variables for various i or j (in the case when i = 0), w[0) = t.
Proof. First, note that under the conditions of Theorem 17 the equality
* T * t3 * t2
j'iv^'iTf*0=¡J I dw<:i)dw«:2>dw<33»
t t t
is true w. p. 1 (see Theorem 8), where J*^3)]^2^ is defined by (48).
According to Theorem 14, we come to the conclusion that Theorem 17 will
be proved if we prove the following equalities (see (94)):
/ \ 2 P f P 1
E a«.* - 2C>
¿3=0 \ ji=0
J3JU1
= 0, (109)
(ii ii WO,
. 2
P P 1
Ji=0 V ,3=0
= 0, (110)
(,3 J3W0,
2
P P 2
= 0 (m)
p^œ
,2=0 \ =0
Let us prove (109). Using Fubini's Theorem and Parseval's equality, we have
. 2
p / p \ 2
¿is, E ( ~ E a
,3=0 V OmWO ,i=0
p
V
j3jiji
p
T
jSsX yj^ \\Jds-i2f few/J dr
p /t /
js =0 \ t
t j1=01 p
2
JlS,E /Ifds~^l\J
/t
V
oo
s )ds
j3=0 ^ t \ '' =" '
T/
j =0
2
t j =0 t
fyj 1 (s)ds
dT
7 7
22
dT / )
dT.
<
(112)
/
Applying the Parseval equality, we have
T
j1=0 2 t j1=0 2 t
¡,(s)a^ =
T
1 f 2 1 2 / i1!^})
(113)
Moreover,
11
ji=0
j(s)ds
11
<~(r-t)<-(T-t)<oo.
(114)
Using (113), (114) and applying Lebesgue's Dominated Convergence Theorem in (112), we obtain the equality (109).
Let us prove (110). Using Fubini's Theorem and Parseval's equality, we obtain
j1=0
Cj3j37l
(j3j3W0 j3=0
2
s
2
2
2
p
p l T t p T « t \2
pl^E I 2 / / ^i(s)dsdr / feW / / ^(s^sdrdtf =
P ji=0 \ t t j3=0 t t t /
lim >
p^X —' \ 2
ji =0 \
T T
2
j (s)(T — s)ds — 53 J j (s) J j (t)J j (#)d#dTdsj =
S T /
¿3=0 '
/T
P
2
1
¿SsZ y^w
ji=° Vt V j =0
/T /
I I <f>h(T)dr
¿3 =0
\
T
ds
y y
22
<
< lim V /
P—T^OO ^-' I
ji=0 \ t
j(s
\
¿3 =0
T
ds
/
/
t \ j3=0 \s /
2
/
ds.
Using the Parseval equality, we get
2
T \ 2 / T
oo h / ^ \ co
El / few^l =E^ 11 MsKr^jsi^dr
¿3=0
¿3=0
T
1 /*,. .2 , 1
2 / v1^}) dr = -(T-s).
(115)
(116)
Moreover, 1
T
2
(T — s) — 53 2 ( I Mr)dr
¿3=0
11
<-(T-s)<-(T-t)< oo. (117)
Combining (115)-(117) and using the same reasoning as in the proof of (109), we obtain
2
ds = 0.
1™jhlT-s)-±\(JMT)d^
J
P
2
2
2
P
The equality (110) is proved.
Let us prove (111). Applying Fubini's Theorem and Parseval's equality, we have
p p 2
P™ E ^ Cji
j2=0 \ ji=0
t e
pip ^ v T \ l^E E / j (°)f j (t)/ j (s)dsdTd^
p^to z
j2=0 ji=0 t t t
= e i e/ j (t ) j j (s)d^ j (^)d^dT
P °°j2=^ ji=0 t t T
to / t p t t
< p^E I j (t)£ I j (s)dW j (0)dfldT
P °°J'2=^ t ji=0 t T
T / p T T
lim i ( E / j (s)ds / fa (0)d0 1 dT.
T
T
<
p^TO
t \ji=0 ( T
(118)
Using (71), we obtain
T
£ / j (s)ds / j (0)d0
ji=0 t T
p T T f f
<E / j (s)dW j (0)d6> <
ji=0 tT
TO
<E
ji=0
T
j (s)ds / j (0)d#
<I(T-i)<oo.
(119)
Applying the generalized Parseval equality, we get
p T t ^^ T T
limE / j(s)dW j(#)d# = ^ / 1{s<t}j(s)ds / 1{s>t}j(s)ds =
ji=0 t T ji=0 t t
T
= 1{s<T}1{s>T}ds =
(120)
2
2
T
2
Taking into account (119), (120) and using Lebesgue's Dominated Convergence Theorem in (118), we obtain the equality (111). Theorem 17 is proved.
3.6 Expansion of Iterated Stratonovich Stochastic Integrals of Multiplicity 4. The Case of an Arbitrary CONS in the Space
L2([t,T]) and fafa),...,fafa) = 1
In this section, we will prove the following theorem.
Theorem 18 [14], [49], [50]. Suppose that {fa(x)}°=0 is an arbitrary CONS in the space L2([t,T]). Then, for the iterated Stratonovich stochastic integral of fourth multiplicity
>jc T ^ t4 ^13 ^ t2
J * [#>]<:f3!4) = J J J J dwi;i»dw<;2)dwi:3»dw<44)
t t t t
the following expansion:
p
J.[fa<4)]T;r3;4)=1^. ^ j j>jjj
ji j2 j3j4=0
that converges in the mean-square sense is valid, where fa i2, i3, i4 = 0,1,..., m,
T t4 t3 t2
Cj2ji = j j y j ^ 0j2 (t2^ fai (ti )dt1dt2dt3dt4 (121)
tttt
and
T
j = / </j(T)dw<->
t
are independent standard Gaussian random variables for various i or j (in the case when i = 0), wT0) = t.
Proof. First, note that under the conditions of Theorem 18 the equality
>jc T ^ t4 ^13 ^12
j-. fa4)]^*.) = J J J J dw«;i>dw<;2>dwi;3>dw<44>
t t t t
is valid w. p. 1 (see Theorem 8), where J*[1(4)]T,ti2i3M) is defined by (48).
It is easy to see that Theorem 18 will be proved if we prove the following equalities (see Theorem 14 and (94)):
p / p
1 \
= 0,
.¡is, £ ( £ - ¡cHnn]1
¿3,j4=0 \ ji=0
\ 2
p x 2
(j'ij'i WO,
pfeE E Cj4ji¿2ji = 0,
¿2,¿4 =0 \ ji=0 /
P / P \ 2
Pfe E E Cjij3j2ji =0,
¿2,¿3 =0 \ ji=0 /
/ \ 2
P / P 1
l^o 53 ( 53 ^^ihhji 2 ^^ihhji
¿i,¿4 =0 \ ¿2=0
/ \ 2
p / p \ 2
= 0,
(¿2,2 WO,
Am E ( E =0,
¿U3 =0 \ ¿2=0 '
/ \ 2 P / P 1
«s £ £ c»«i -
¿U2 =0 \ ¿3=0
= 0,
(J'3J3 WO,
P 1
lim / Cjsjsjdi = 7^¿з¿з¿l¿l P^TO z—» 4
¿U3=0
(¿¿^WOO'i ¿'iWO
P
lim ^¿'¿¿'i = 0,
1—vno ' »
P^to
¿i,j'3=0
P
lim ¿¿^¿i =
¿ij2=0
122)
123)
124)
125)
126)
127)
128)
129)
130)
Let us prove the equalities (122)-(127). Using Fubini's Theorem and Par-seval's equality, we obtain the following relations for the prelimit expressions on the left-hand sides of (122)-(127):
p / p
V'; ; ; ; -
2
53 ( E cy^jmoiji
¿3,¿4=0 \ ¿i=0
¿mWO,
p
1
T t4
53 9 / / <t>hfa)(U-t)dUdt4-
2
j3,j4=0 \ t t
T t4 t3 t2
53 / j (t4 ) / j (t3M j (t2M j (ti)dtidt2dt3dt4
ji=01 t t t p / T t4
1
53 I J J fete) (-
j3,j4 =0 \ t t
t3 t2
53 J j(t2W j(ti)dtidtfa dt3dt4 ji=01 t /
/ T t4 /.
t3
E
j3 ,j4=0
<
j3,j4=0
I j ^ J j (t3) tt
/ T t4
I j (U) J j (t3) tt
/
1
1
2^3-*)- E 9 I I feMd
\ I
ji=0
2
11
2
y
2
dt3dt4
<
/
2
V
ji=0
t3
dt3dt4
/ /
1
{t3<t4}
[t,T
1
V
ji =0
2
2
y
dt3dt4, (131)
pp
pp
\ Cj4jij2ji
j2,j4=0 \ ji=0
T t4 t3 t2
53 I 53 J j(t4^ j(t3^ j(t2) y j(ti)dtidt2dt3dt4 I
j2,j4=0 \ji=0 t t t t y
T t4 t2
t4
53 |53 /fe (t4^ fe (t2^ fe (ti)dt^y j (t3)dt3dt2dt4 I =
j2 j4=0 \ji=° t t t t2 y
2
2
p
2
s
2
2
p
2
2
2
P fT ) P ) ) \2
= 53 J ^¿4 (^4) J ^¿2 (^2^ J (t1)dt^ (t3)dt3dt2dtj <
¿2 ,¿4 =0 ^ t t ¿i=0 t t2 /
< E ( j fa (t4^ ^¿2 (t2) (t1 )dt1 y (t3)dt3dt2dtj =
¿2,¿4 =0 \ t t ¿i=0 t t2 /
/ P t2 t4 \ 2
= 1{t2<t4} 53 y ^¿i (t1)dt^ (t3)dt3 dt2dt4, (132)
[t,T]2 \¿1=0 t t2 /
2
p / p N 2
( 53 ^'¿^¿3¿^i
¿2,¿3=0 \ ¿i=0
P / P T / / / V
= 53 53 / ^¿'i (t4M ^¿3 (t3) / fe (¿2) / (¿1)dt1dt2dt3dt4 =
¿2j3=° V"i=0 t t t t /
P / P T / / T \2
= 53 53 / ^¿'3 (t3M fe (t2 ) / (t1)dt1 / (t4)dt4dt2dt3 =
¿2j3=° N^1=° t t t t3 /
P /T ) P ) T \2
= 53 I I ^¿3 (t3W ^¿2 (¿2)^ I hi (t1)dtW ^¿1 (t4)dt4dt2dtJ <
¿2 ,¿3 =0 \ t t ¿1=0 t t3 /
< E I /^¿3 (*)/ ^¿2 (t2) ^ ^¿1 (t1 )dt^ ^¿1 (t4)dt4dt2dtJ =
¿2 ,¿3 =0 \ t t ¿1=0 t t3 /
f / P '/ T \2
= 1{t2<t3} 53 / ^¿'1 (t1)dt1 / ^¿1 (¿4^4 dt2dt3, (133)
[t,T]2 \¿1=0 t t3 /
P / P 1
53 ( 53 ^3*323231 ~ 2^34323231 ¿1 ,¿4=0 \ ¿2=0
¿272 WO,
2
p
1
T t4 t2
E I § I I I ^ji{ti)dtidt2dtA-
jij4 =0 \ t t t
T t4 t3 t2
E / j(t4) / jte) / jte) / j(ti)dtidt2dt3dt4
j2 =01 t t t
T t4 t4
p
= E I 2 j fete) j fete) j dt2dtidtA jj =0 \ t t ti
p T t4 t4 t4 \
E J j (t4)J j (ti^ j (t2^ fe (t3)dt3dt2dtidt4 I
j2 =0 t t ti t2 /
/T
t4
E
ji j4=0
/
j (t4 ) / fe (t1)
V'
i4 - ¿1 Ai 2 ^ 2
t4
2
V
E 2 fe(s)ds
\
<
ji j4=0
TO / T t4 / p 1
/ fete) / fete) 4 1 ^
t4
v
tt
Eo /fe(s)d'
v
j2=0
ti
dt1dt4 < / /
22 dtidt4 y )
A p / t4 X 2\
1
{ti<t4}
[t,T ]2
11 ote-ii)-Eo ( I fete^
v
j2=0
dtidt4, (134)
ti
/
pp
pp
E E Cj2j3 j2 ji
ji,j3=0 \ j2=0 T t4 t3 t2
E IE/ j (t4M j (t3 ) / fe (t2M j (t1 )dt1dt2dt3 dt4 I
ji,j3=0 \j2=° t t t t /
pp
T t3
t2
T
El E/ j (t3^ fe (t2 ^ fe (t1)dt1dt^ j (t4 )dt4dt3
ji,j3=0 \j2=0 t t t t3
2
2
2
2
2
2
2
T t3
t3
E IE/ &3 ^ ^¿i (t1^ ^¿2 (t2)dt^ ^¿2 (t4)dt4dt1 dt3 I
¿1j3=° N^2=° t t ti t3 /
p / T t3 P t3 T
53 ( J MJ hi (^)E J fe (t2)dt^ ^¿2 (t4)dt4dt1dtJ <
¿i^0 \ t t ¿2=0 ti t3 /
2
TO / T t3 P t3 T \
53 / ^¿3 (i3M ^¿1 (t^E / ^¿2 (t2)dt2 / ^¿2 (t4)dt4dt1dt3
<
¿1j3 =0 \ t t ¿2=0
t3
ti
t3
T
1{ti<t3} ( £ J ^¿2(t2)dt^ ^¿2(¿4^4 ) dt1 dt3, (135)
ti t3
[t,T]2 \¿2=0
P / P
( 53^3,3,2,1
¿1 ,¿2=0 \ ¿3=0 P
1
— I 2
t2
53 o / / ^2(^2) / ^(¿i^mm^-
¿1 ,¿2 =0 \ t t t T t4 t3 t2
53 J ^¿'3 (t4) J ^¿3 (t3^ ^¿2 (¿2)^ ^¿1 (t1)dt1 dt2dt3dt4 I
¿3 =0 t t t t /
P
1
T T
T
= 53 ( 2 j ^^ j ^(¿2) j dtzdt2dti-
¿1 ,¿2 =0 \ t ti t2
P T T T T \
E /^¿1 (t1 ) / ^¿2 (t2) / ^¿3 (¿3)/ ^¿3 (t4)dt4dt3dt2dt1 I
¿3=0
t ti t2 t3
P / T T / P
[ 1 , ^ [ 1 , J t —12 A1
2
E
¿1,¿2=0
t ti
Eo /
V
¿3=0
dt2 dt1
/ /
<
2
2
2
2
2
P
2
2
TO
<
ji j2=0
/ T T /
/ fete) / fete) 2 ^
V'
V
22
j3=0
fe(s)ds 1 j3=02 Vt2 y
2
y
dt2dt1
/
1
{ti<t2}
[t,T ]2
V j3=0 Vt2 y
2
y
dt2 dti. (136)
Using Parseval's equality, generalized Parseval's equality and Lebesgue's Dominated Convergence Theorem, as well as applying the same reasoning as in the proof of Theorem 17, we obtain that the right-hand sides of (131)-(136) tend to zero when p ^ to. The equalities (122)-(127) are proved.
Let us prove the equalities (128)-(130). First, let us show that
Cj4j3j2ji + Cjij2j3j4 = Cj4 Cj3j2ji Cj3j4 Cj2ji + Cj2j3 j4 Cji ,
where Cj4j3j2ji has the form (121). Using Fubini's Theorem, we have
T t4 t3 t2
jj'i = j j (t4) y fe te^ j (t2^ y fe (t1 )dt1dt2dt3dt4 =
tttt
T T t3 t2
= / j (t4^ fe (t3) J j te) y fe (ti)dtidt2dt3dt4-
tttt T T t3 t2
-/ j (t4^ fe (*)/ fe te) y j (ti)dtidt2 dt3dt4 =
t t4 t t
T T T t2
= Cj4Cjjji ^y j (t4) y j ^ fe (t2y j (t1 )dt1dt2dt3dt4 +
t t4 t t T T T t2
+ J j (t4^ fe (*)/ j (t2^ j (ti )dtidt2dt3dt4 =
t t4 t3 t
(137)
2
2
T T T T
= Cj4 jj - Cj3j4 j'i +J j (i4) j j ^J j (t2^ j (t1 )dt1dt2dt3dt4
t t4 t3 t
T T T T
-J j (t4^ j te)/ j te) / j (ti)dtidt2dt3dt4 =
t t4 t3 t2
= Cj4 Cj3j2j'i — Cj3j4 Cj2j'i + Cj2j3j4 Cj'i — Cj'i j2j3j4 . (138)
From (138) we get (137). Recall that in [16] (Part II of this work) we obtained an analogue of (137) for the case k = 6.
It is easy to see that we can consider the following generalization of (137) for the case k = 2r (r = 2, 3, 4,...):
jk jk-1 ---jl + jlj2'''jk — jk ^ jk-1 jk-2 ---jl jk-ljk ^ jk-2 jk-3 ---jl +
+ Cjk-2jk-ljk ^ Cjk-3jk-4-'-jl ' ' ' Cj3j4'-'jk ^ Cj2jl + Cj2j3'-'jk ^ Cjl ' (
where
T t2
jj — i fa (tk ) j (tk )... i fa (ti ) j (ti)di!.. .dtk (140)
t t
and fa (t),..., fa(t) e L2([t, T]). Further, we will write Cjk...j-i instead of if this does not cause misunderstandings.
In principle, using (139), we can calculate any expressions of the form
pp
lim E ... E Cjk-J i
p—>-TO z-' z-'
p
1 , (141)
j3l j32 ''"'j32r-l j32r
j3l — 0 j92r-l — 0
where gi, g2, • • •, g2r-1, g2r are as in (18) and the following symmetry condition:
fafa ) — fa (t ), fa (t )— fak-i(T ), •••, fa (t )— far+i(T ) (142)
is fulfilled for k — 2r, r — 2, 3, 4,...
Obviously, the case fa (t),..., fa (t) = 1 is possible since it is a special case of the symmetry condition (142). This case is important because it covers the
mean-square approximation of iterated Stratonovich stochastic integrals from the classical Taylor-Stratonovich expansions [2]-[7].
Let us prove (128). Substitute j4 = j3, j2 = j into (137)
Cj3j3jljl + Cjljlj3j3 = Cj3 Cj3jl jl — Cj3j3 Cjljl + Cjlj3j3 Cjl . (143)
From (143) we obtain
p p p
53 j3jljl + Cjljlj3j3) = 53 Cj3Cj3jljl - E Cj3j3 Cjljl +
jl,j3=0 jl ,j3=0 jl,j3 =0
P
^ E Cjlj3 j3 Cjl.
jl,j3=0
Then
2
P \ 2
2 53 ^¿^¿i¿i =2 53 ^¿3 ¿m E ^¿¿1) ■ (144)
¿1,¿3 =0 ¿U3=0 N¿1=0
Using (144) and Fubini's Theorem, we get
p p 1 / p \2
53 ^hhhh = 53 ^3?Phh3\ ~ 2 ( 53 ^¿i^i
¿U3=0 ¿l,¿з =0 N¿1=0
p 1 / p 1 2\2 p 1 / p
E ^3?Phhh ~ 2 ( Ea^) J = E ^3?Phhh ~ g ( E(Q'i)
¿^,¿3 =0 N¿1=0 / ¿l,¿з=0 N¿1=0
(145)
Applying Parseval's equality, we have
T
p T
lim V (Cn)2 = 12dT = T — t. (146)
P^TO ^—' J
¿1=0 t
Combining (145) and (146), we get
P P (T — t)2
lim 53 f '/3.':,/./ = iim 53 Qs^m--«-• (147)
p^TO z—' p^TO z—' 8
¿i ,¿3=0 ¿U3=0
P
P
Further, we have
Cj3C
p—TO
j3jiji
ji,j3=0
1
- lim > Cj3Cj3jljl 2p
j3=0
^ E Ci> (
(j'ij'iWO p j3=0
j3jiji
53 Cj3jiji
(jijiWO ji=0
(148)
Using the generalized Parseval equality, we obtain
t T
lim E Cj3C
1—vno ' »
p—TO
j3jiji
j3=0
(j'ijiW0 p—TO j3=0 t
l—roE J j (t)dT/ j (T) J ^T
T T
= / 1 • / d^dT =
(T -1)2
tt
From (148) and (149) we have
(149)
lim / v Cj3 jjiji
p—TO
jij3 =0
(T - i):
M:"E'' ./
p
j3=0
j3jiji
53 Cj3jiji
(j'ijiW0 ji=0
(150)
Combining (147) and (150), we obtain
lim V C
p
p—TO —
ji j3=0
j3j3jiji
(T -
8
p
j3=0
j3jiji
53 Cj3j'^i
(j'ij'iW0 ji=0
(151)
Due to Cauchy-Bunyakovsky's inequality and (109), (146), we get
li:" (V <' (./ •
p—TO
j3=0
j3jiji
53 Cj3j'm
(j'ij'iW0 ji=0
<
p
j3=0
j3jiji
j3=0
Ec
j3jiji I <
(j'ij'iW0 ji=0
p
p
p
p
2
p
2
P
¿3=0 ¿3=0
p
2
__¿3jljl
(¿'lj'i WO ¿1=0
^-^IsXI "E =0. (152)
p TO¿3=0 V (¿mW-) ¿1=0 J
p
From (152) and (151) we obtain (128).
Let us prove (129). Substitute j4 = ji, j2 = j3 into (137)
^¿^¿^¿^¿1 + ^¿^¿^¿^¿1 = ^¿1 ^¿^¿3¿1 — ¿1 ¿"1 + ^¿^¿3¿1 ^¿1 • (153)
Using (153), we get
p p p
2 ^^'¿^¿^¿^¿1 = 2 53 ^¿^¿3¿1 - E (¿0 • (154)
¿1j3=0 ¿1,¿3 =0 ¿1j3=0
Applying (154), we obtain
p p 1 p 2
53 ^¿^^¿1= 53 QiQ^ _ 2 53 (^^¿1) ■ (155)
¿^,¿3=0 ¿1 ,¿3=0 ¿1 ,¿3=0
Parseval's equality gives
. \2
lim E CO 2 = lim E / 1 {t 1 <¿2}^¿1 (t1 ^¿3 (t2)dt1 dt2
?—to z—* v p—to z—* /
\[t,T ]2 /
p—>-to
¿'1 ,¿3 =0 ' ¿^,¿3=0
2 (T - t)2
= J (l{ti<t2}) dt\dt2 = - —. (156)
[t,T ]2
Combining (155) and (156), we have
p p (T -1)2
lim 53 ^ './ ./"../:■,/ — lim 53 ^ './ ^ '/"../:■,/ 1 • (157)
p—TO z-' p—TO z-' ¿L
¿'1 ,¿3 =0 ¿^,¿3=0
Further, we obtain
p
lim / v CJ1 ¿¿¿1
p—TO z-'
¿1j3 =0
1
- lim > CjlCj3j3j1 2p
ji=0
^ E (
(j3j3)^(^) p ji=0
j3j3ji
53 C333331
(j3j3W0 j3=0
(158)
Applying Fubini's Theorem and the generalized Parseval equality, we have
p p T T t2
lim V CjC
pi
p ji=0
j3j3ji
(j3j3W0 p—TO j1=0
lim ^^ J j (t)dT J J j (t)dTdt2
tt
T
ji=0 t
T T T T
lim ^^ J j (t)dT J j (t) J dt2dT = J 1 •/ d^dT =
p ji=01
(T -1)2
From (158) and (159) we get
(159)
lim L Cj. c
p—TO
j3j3ji
jij3=0
(T"i)2 '-Vr (V
ji=0
j3j3ji
Cj3j3ji (j3j3)^(^) j3=0
(160)
Combining (157) and (160), we obtain
lim C
p
p—TO '
ji,j3=0
jij3j3ji
p ji=0
j3j3ji
ECj3j3ji) . (161)
(j3j3W0 j3=0
Due to Cauchy-Bunyakovsky's inequality and (110), (146), we get
2
p
j3j3ji
53 Cj3j3ji (j3j3W0 j3=0
<
p p A
ji=0 ji=0 V
Z C
(j3j3W0 j3=0
j3j3ji I <
p
1
ji=0 ji=0
53 C333331 (j3 j3 W-) j3=0
p
p
p
p
2
p
2
p
p
p
(T " *) ü™ £ =0. (162)
p TO ¿^=0 V C^WO ¿3=0
From (161) and (162) we obtain (129).
Let us prove (130). Substitute j3 = j15 j4 = j2 into (137)
C¿2¿1¿2¿1 + ^¿¿'¿2 = ^2 ^¿¿2¿1 — ^¿¿2 ^^ + ^^ ¿2 • (163)
Then
p
53 ¿1 + ¿2 ¿1 ¿2 ^ = 53 ^¿2 C¿1¿2¿1 + ^1)
¿1j2=0 ¿1 ,¿2=0
53 ^¿2 CJ2J1 • (164)
¿1 ,¿2=0
Using (164), we have
p p 1 p 53 ^^¿^¿1 = 53 ^h^jlhjl ~~ 2 53 • (165)
¿1 ,¿2=0 ¿1 ,¿2=0 ¿1 ,¿2=0
Fubini's Theorem and the generalized Parseval equality give
p
j2j1
lim 53 ¿2 C
n—ViYl ' *
p—TO
¿1 ,¿2 =0
p T T T t2
piTO £ J ^¿2 (t2^ ^¿1 (t1 )dt1dt^y ^¿2 (¿2^ y ^¿1 (t1 )dt1dt2 ¿-1^^2=0 t ¿2 t t
pll|TO S3 / 1{t2<t1} ^¿1 (t0fe (t2 )dt1dt^ I{t1<t2>0j1 (t1 ) ^¿2 (t2)dt1dt2 =
¿1,¿2=0[t T]2
2
[t,T]2 [t,T]
= y 1{t2<t1}1{t1<t2} dt1dt2 = 0. (166)
[t,T ]2
The equalities (165) and (166) imply the relation
pp
lim 53 CJ2J1J2J1 = lim 53 ^2 C¿1¿2¿1 • (167)
p—TO z-' p—TO z-'
¿1 ,¿2=0 ¿1 ,¿2=0
2
p
Further, we have (see the derivation of (152))
/ p p \ 2 p p/p \2
p—to E E c3.fa < p—to E (■ )2 E E ■Cj.j2* <
Vj2 =0 ji=0 / j2=0 j2=0 3=0 J
TO p/p \2 p/p \ 2
< p1—nTOE (Cj2 )2E ECjij2jJ =(T - i) p1—TOE E323i =0,
j2 =0 32=0 \ji=0 / 32=0 3=0 J
(168)
where (168) follows from (111).
Using (167) and (168), we obtain (130). The equalities (122)-(130) are proved. Theorem 18 is proved.
3.7 Generalization of the Results from Section 3.5 to the Case
fafa),fafa),fafa) e fa([t,T])
In this section, we will prove the following two theorems.
Theorem 19 [14], [50], [51]. Suppose that {fa(x)}°=0 is an arbitrary CONS in the space L2([t,T]) and fafa),fafa),fafa) e L2([t,T]). Then, for the sum j*[fa3) ]Tiiti2i3) (ii, ¿2, ¿3 = 0,1,..., m) of iterated Ito stochastic integrals defined by (48) the following expansion:
p
^(3)]Tr3) = ip—£ E jj№Zj33)
j1,j2,j3=0
that converges in the mean-square sense is valid, where T t3 t2
Cj3j2ji = J ^3^)3 to) J ^2^)3 / fai(ti)0ji (ti)dtidt2dt3
ttt
and
T
3 = / & (T )dwT*>
t
are independent standard Gaussian random variables for various i or j (in the
(0)
case when i = 0), wT ) = t.
Theorem 20 [14], [50], [51]. Suppose that (x)}°=0 is an arbitrary CONS in the space L2([t,T]) and 1i(r),12(t),13(t) are continuous functions on [t,T]. Then, for the iterated Stratonovich stochastic integral of third multiplicity
* T * is * ¿2
j*[v>(:i)]Trs) = / wt3) / ^i(ii)dw<:'»dw«:2)dw<is»
¿1
t t t
the following expansion:
p
J'^iTr" = >im- £ c,sJ2Jicj;i)cj22)cjss)
p^to
jl,j2,j3=0
that converges in the mean-square sense is valid, where ii, i2, = 0,1,..., m; another notations are the same as in Theorem 19.
Note that Theorem 20 is a simple consequence of Theorem 19 and Theorem 8 (k = 3). Let us prove Theorem 19.
Proof. First, let us note some facts that follow from Monotone Convergence Theorem ([60], Theorem 3.5.1). Suppose that {gj(x)}°=0 is an arbitrary sequence of real-valued measurable functions such that the series
to
£gj (x) (169)
j=0
converges absolutely almost everywhere on X (with respect to Lebesgue's measure) to some function f (x). From Monotone Convergence Theorem, in particular, it follows the following equality (see [60], Theorem 3.5.2):
/to to „
(x)dx = y / gj(x)dx. (170)
X j=0 j =0 X
It is easy to see that under the above conditions the following equality:
p- A2 ff to V
x) dx = / Tgj (x) dx (171)
A™/(Eg(x)) = /(E
is true (further, we will use the equality (171)). Indeed, we have g3-(x) =
j(x) - gj-(x) |gj(x)| = gj+(x) + (x), where gj+(x) = max{gj(x),0} > 0,
g-(x) = - min{g3-(x), 0} > 0. Moreover,
TO TO TO
Egj(x) = E g+(x) -E gj-(x^
j =0 3=0 3=0
TO TO TO
Eigj (x)i = E g+(x) + E g-(x). (172)
3=0 3=0 3=0
Since the series (169) converges absolutely, then by virtue of the equality (172) the series (with non-negative terms) on the right-hand side of (172) converge (to some functions f1(x) and f2(x), respectively). Further, using Monotone Convergence Theorem, we obtain
\ 2 / \ 2
p 2 p p 2
/p \2 / p p \ 2 pW Egj(x) dx = pW Eg+(x) -Eg-(x) dx
X \j=0 / X \3=0 3=0 /
X Xj=0 7 X
/p \2 i p p \ / Eg+(xH dx - 2/1 Egj+(x^E gj-(x)i dx+ Vj'=0 / X ^3=0 3=0 y
2
+J ^E gj(x)^ dx
X
// p \2 ^ p p
lim I g+ (xM dx - 2 lim g+(x)53g3-(x)dx+ X p—\ 3=0 / X p—TO j =0 3=0
2
+/p—m (Eg-(x)) dx
p
-
x)
X Vj=0 22
= y (fi(x)) dx - 2J fi(x)f2(x)dx + J (/2(x)) dx =
X X X
2 Jx / Egj(x) dx. j=0
= (/1(x) - f2(x)) dx = / } faj(x X X Vj
According to Theorem 14, we come to the conclusion that Theorem 19 will be proved if we prove the following equalities (see (94)):
p
1
js=0
53 Cj3jm
(jijiW0 ¿1=0
= 0,
(173)
jSs, 53 ( ¿1=0 x
53C¿sjsj i (jsjsW0 js=0
= 0,
pp
iimE ECjij2ji = 0
j2 =0 \ji=0
Let us prove (173). Using Parseval's equality, we have
(174)
(175)
p
1
j3=0
T
53 Cjsjiji
(jijiW0 jl=0
1(T )dT
53 I 12(T)j (T) / (0)d0dT ) ds I <
ji =0 t t
T
/ ^OO&sM [I J Mr)Mr)dr
j3 =0 V t V t
,t \ t
s T
531 mt)j(t)l ds
ji=01 t
T
1
= J I 2 J (r)dr-tt
2
p
531 mt)j(t)i 11 (0)iji(0)d0d^ ds =01 t
(176)
2
p
2
p
2
2
p
2
s
s
2
s
T f s
■.2, — ' 1
= / 2 / fa(r)fa(r)dr~
t \ t s T
" E I )fai (T) / fa(0)fai(0)d0dr ds = 0, (177)
ji=01 t /
where (177) follows from (73) and the transition from (176) to (177) is based on (171) and the absolute convergence of series on the left-hand side of (73) (the sum of this series does not depend on the order of terms since the sum is equal to the integral on the right-hand side of (73) for any basis {fa(x)}°=0 (we mean the order of numbering of the functions fa(x))). The equality (173) is proved.
Let us prove (174). Using Fubini's Theorem and Parseval's equality, we obtain
p
i™ E (- E a
p2 ji =0
T
p '1
p
_ j3j3ji
(j3j3W-) j3=0
= i}lS)E I 2 / J fa(s)<f>h(s)dsdr
P °°ji=0 V t t
p t e t
E /fa(Wj3(0)/fa(T)j(t) Jfa(s)fai(s)dsdrd0
j3=0 t t t )
p / T T
= iJlS)E / fa(s)3(s) Jfa(fafa(fadrds-
P °°j'i=0 \ t s
p t t t \
£ J fa(s)fai (s)/ fafa )j (t )J fa(0)fa-3 (0)d0dTds
j3=0 t s t /
p (T (1 T
= ^E / fa(s)3(s) 2 / fa(r)fa(r)dr~
p T T \ 2
E / ^2(T)j(t)J fa(0)fa-3(0)d0dH ds I <
j3 =0 . t /
2
p
2
2
P TO ji=A t \ s
p t t \
E / ^2(T) j (T)J «A) j (0)d0dr ds
j3=0 s t )
T / T
ts
T
pT
T t ^ 2
E J ^2(t) j(t)J 1(0) j(0)d0dr ds = (178)
j3 =0 s t /
T / T
= J I I J {r)ip2{r)dT-
ts T T \ 2
E / 12(T) j(t) J 1s(0) j(0)d0dr j ds = 0, (179)
j3 =0 s t /
where (179) follows from (73) and the transition from (178) to (179) is based on (171) and the absolute convergence of series on the left-hand side of (73). The equality (174) is proved.
Let us prove (175). Applying Fubini's Theorem and Parseval's equality, we have
pp
Cj1j2j1
j2=0 \ji=0
PliTO E (E / ^Wj(0)j 12(T) j(T) J (s)^ji(s)dsdrd0 j2=0 \j'l=0 t t t
p / p T T T A^E E i 12(T) j(t) / 1i(s)1ji(s)ds / ^(0) j(0)d0dr ) <
p TO j2=0 \ji=0 t t T
2
where (181) follows from the equality
t T
to t «
I fa(s)fai (s)ds / fa(tf)fai (0)d0 =
ji=° t t
T
= J fai(s)l{s<T|fa3(s)l{s>T|ds = 0 (182)
t
(the relation (182) follows from the generalized Parseval equality) and the transition from (180) to (181) is based on (171) and the absolute convergence of series on the left-hand side of (182) (see the derivation of (71)). The equality (175) is proved. Theorem 19 is proved.
3.8 Generalization of the Results from Section 3.6 to the Case
fafa),...,fafa) e L2([t,T])
Let us develop the approach discussed in the previous section.
Theorem 21 [14], [50], [51]. Suppose that {fa(x)}°=° is an arbitrary CONS in the space L2([t,T]) and fa(t),...,fafa) e L2([t,T]). Then, for the sum J*[fa4) ]yfaM) (fa ... ,i4 = 0,1,..., m) of iterated Ito stochastic integrals defined by (48) the following expansion:
p
[fa^Tr1 = E Cj4..ji... zj44)
j1 ,...J4=°
that converges in the mean-square sense is valid, where
T t2
Cj4...ji = J fafafaj (t4)...J fa (t1 (t1)dt1 . . .dt4 (183)
tt
and
T
cf = J j (t)dwT*)
t
are independent standard Gaussian random variables for various i or j (in the case when i = 0), wT° = t.
Theorem 22 [14], [50], [51]. Suppose that {fa(x)}°=° is an arbitrary CONS in the space L2([t,T]) and fafa),... , fafa) are continuous functions on [t,T]. Then, for the iterated Stratonovich stochastic integral of fourth multiplicity
* T * t2
./•[fa4)]Tri4) = / fa4(t4)...J fa1(t1)dwi;i)...dw':4)
tt
the following expansion:
p
j* [fa4)]Tr4)=l^ e cj:°... cj44)
j1 ,...J4=°
that converges in the mean-square sense is valid, where i1,..., i4 = 0,1,..., m; another notations are the same as in Theorem 21.
Note that Theorem 22 is a simple consequence of Theorem 21 and Theorem 8 (k = 4). Let us prove Theorem 21.
Proof. It is easy to see that Theorem 21 will be proved if we prove that
(see Theorem 14 and (94))
pp
. i »
= 0, (184)
53 ( 53 ^^ijajiji 2Cidsjdi
P 33J4 =° V j1=° (j1j1 )^fa
/ \ 2
p p 2
^ E ECj4j, »1 = °, (185)
j2 ,j4 =° \ j1=° '
p p 2
Am E E Cjij3j2ji =
j2,j3=0 \ ji=0 / pp
E E - f
ji,j4 =0 \ j2=0
pp
2
= 0,
(j2j2 WO,
Jim E E jj2ji =0,
p—>-TO
ji,j3 =0 \ j2=0 pp
^ E E -
ji ,j2 =0 \ j3=0
p 1 T *
2
= 0,
(j3 j3 WO,
lim V" Cnnnn = - ^4(^3)^3(^3) / (¿1)^1(^1)^1^3,
p—TO 4 J y
ji,j3=0 t t
p
lim ¿v Cjij3j3 ji = 0, p—TO z-'
ji,j3 =0 p
lim Cj2jij2 ji = 0,
1—ViYl ' *
p—TO
ji,j2 =0
186)
187)
188)
189)
190)
191)
192)
where Cj4...ji has the form (183), {0j(x)}TO=0 is an arbitrary CONS in the space L2 ([t, T]), and 11 (t),..., 14 (t) e L2 ([t, T]).
To prove (184)-(189) we modify the proof of (122)-(127). More precisely, the proof of (184)-(189) is carried out by analogy with the proof of (122)-(127) using the equality (171) instead of Lebesgue's Dominated Convergence Theorem (see the proof of Theorem 19 for details) and adjusted for the fact that in the proof of (122)-(127) the functions 11 (t),..., 14(t) = 1 are replaced by 11(t),... ,14(t) e L2([t,T]). Thus, the equalities (184)-(189) are proved.
In [57] an efficient method is proposed for proving equalities similar to (190)-(192). In particular, the equality (190) is proved in [57]. The above method [57] is based on the equality of the matrix and integral traces of trace class operators ([62], Theorem 3.1). In the next section, the equalities (190)-(192)
are proved using the generalized Parseval equality and (73). At that, we use some ideas from [57]. Theorem 21 is proved.
3.9 On the Calculation of Matrix Traces of Volterra—Type Integral Operators
It is easy to see that the function (4) for even k = 2r (r e N) forms a family of integral operators K : L2([£,T]r) ^ L2([t,T]r) (with the kernel (4)) of the form
(Kf)(fa ,...,fa)= J K(t1,...,tk)/(tgr+1 ,...,tgfc)dtgr+1 ...dfa, (193)
[t,T ]r
where {g1,..., } = {1,..., k}, the kernel K(fa..., £k) is defined by (4), i.e.
fafa) ...fa (tk), £1 <...<tk
K(£1,..., £k) = <
, (194)
0, otherwise
where fa (t),..., fa(t) e L2([£,T]), £1,...,£k e [£,T] (k > 2) and K(£1) = fafa) for £1 e [£, T].
For example,
T t2
(Kf) (£2) = i K(£1, £2)/(£1)d£1 = fa(£2) / fa (£1)/(fadfa (195)
(Kf) (£2, £3) = J K(£1,..., £4)/(£1, £4)d£1d£4 = [t,T ]2
t2 T
fa2(£2)fa3(£3)1{t2<t3^ fa (£1^ fa (£4)/(£1,£4)d£4d£1, (196)
tt3
where K(£1,..., £4) is defined by (194).
The simplest representative of the family (193) has the form
x
(Vf) (x) = J f (t)dT (197)
0
and is called the Volterra integral operator, where V : L2([0,1]) — L2([0,1]), f (t) e L2([0,1]). The kernel of the Volterra integral operator is determined by the relation: K(t, x) = 1{T<x} (t, x e [0,1]). It is well known that the Volterra integral operator (197) is not a trace class operator since its singular values are equal to [62]
2
=
On the other hand, it is known [62] that for trace class operators the equality of matrix and integral traces holds (recall that the matrix trace of a linear bounded operator is defined by (76)). It turns out that for the Volterra integral operator (197) (although it is not a trace class operator), the equality of matrix and integral traces is also true [62].
Thus, one cannot count on the fact that operators of the more general form (193) (from the same family of operators as the Volterra integral operator (197)) are operators of the trace class. Nevertheless, the proof of the equalities of matrix and integral traces for Volterra-type integral operators (193) (which is obviously a problem) provides a way to calculate the matrix traces of these operators.
Why do we talk so much in this section about matrix traces of operators from the family (193)? The point is that matrix traces of operators of the form (193) are of great importance for obtaining of expansions of iterated Stratonovich stochastic integrals.
Let us consider some illustrative examples. We have
TO
, j >i2([t,T]) =
ji=0
to
T t
to „ 2 to
E / (¿2) / (ti)dtidi2 = E Cm, (198)
ji=o { { ji=o
to
E <k j, ^
'1,j2=0
T t4 t2 t2
■ jij2/L2([t,T ]2) j1,j2=0
J MU)j(toy fajfe) j(ti^ fa^fa(¿2^ fai(ii)j(¿i)x
j1,j2=0 t t t t
xdtidt2dt3dt4 =
TO
= 53 Cj2j2j1j1 , (199) j1,j2=0
where {^j1j2(x y)}TOT,j2=0 = i^ji(x) j(y)}TOT,j2=0 ' (x)}TO=0 is an arbitrary CONS in L2([t,T]), (Kf) (¿2) in (198) is defined by (195), and (Kf) (¿2, ¿3) in (199) has the form (196).
The expressions on the right-hand sides of (198) and (199) were considered earlier in this article (see (73), (190)).
Let us prove the equalities (190)-(192) using a method based on generalized Parseval's equality and (73). At that, we will use some ideas from [57].
First we prove (190). Using (73), we have
p T t4 T t2
lim E / M / fa(t3)3 (t3M fa2(t2)0j1 (t2M fai(ti)0j1 (ti)x
j1 'j2=0 t t t
xdtidt2dt3 dt4 =
p T t4
lim£ / fa (¿4) j (¿4) / fa (¿3) j (¿3)dt3dt4 x
p—T^OO
j2=0 t t p T ^
x lim V / fa (¿2 ) j (¿2) / fa (¿i ) j (¿i )diidi2
p
p
j1=0 t
T T
= \f ^4(^4)^3(^4)^4 J ^2(^2)^1(^2)^2 = t t
= \ J MU)MU)Mh)Mh)dt2dU, (200)
[t,T ]2
where ),...,) G L2([t,T]).
Suppose that (t) and ^3(t) are polynomials of finite degrees. For example, (t) and (t) can be Legendre polynomials that form a CONS in L2([t,T]). Denote
q
Sq(t2,t3) = E 4 (t2)4 (t3), (201)
1l,l2=0
where {((x)}j=0 is a CONS of Legendre polynomials in L2([t,T]) and C/2/l are Fourier-Legendre coefficients for the function g(t2,t3) = (t2)%53(t3) 1{t2<t3} (Wt),$j(t) G L2([t,T])), i.e.
T t3
C/2/1 =J fofo)^ (t3^ ^2(t2)(//i (t2)dt2dt3.
tt 2
Further, we have lim ||sq — g||L2([tT]2) = 0. From (200) we obtain (the sum on the right-hand side of (201) is finite)
E / 1{ti<t2}1|t3<t4}^4(t4)(2 (t4)sq(t2,t3)(j2 (t3)(ji (t2(t1)(ji (t1)X
j1j2=0[t,T ]4
xdtidt2dt2,dti = i I V;4(^4)sq(^2, £4}^1(^2)dt2dU. (202)
[t,T ]2
Note that the equality (202) remains true when Sq is a partial sum of the Fourier-Legendre series of any function from L2([t,T]2), i.e. the equality holds on a dense subset in L2([t,T]2). The right-hand side of (202) defines (as a scalar product of sq(t2, t4) and (t4)^1(t2) in L2([t, T]2)) a linear bounded (and
therefore continuous) functional in L2([t,T]2), which is given by the function fafe)fafe). On the left-hand side of (202) (by virtue of the equality (202)) there is a linear continuous functional on a dense subset in L2([t, T]2). This functional can be uniquely extended to a linear continuous functional in L2([t,T]2) (see [61], Theorem I.7, P. 9). Let us implement the passage to the limit lim in
q—>-to
the equality (202) (at that we suppose that sq is defined by (201))
to „
E / 1{ti<i2<t3<t4} fa fe)fafe )fafe)fafe) j (fafa2 (fafai (fafai (faX
j'ij'2"V ]4
T t4
xdtidt2dhdti = fafa)fafa) f fafe)fafe)oMfa (203)
where fafa),fa(r),fa(r),fa(r) G fa([i,T]). Rewrite the equality (203) in the form
p
Cj2j2ji ji
lim V C
p—TO ^^ j'ij2 =0
00 T ^4 i3 ¿2
= E / fafe) j fe) ^ fa(fafa2 fe) ^ fafe)fai fe) ^ fa (fafai fefa ji'j2=0 t t t t
T t4
xdtidt2dhdU = fa(i4)fafe) J fafe)fafe)oMfa (204)
tt where fafa),..., fafa) G L2([t,T]).
Note that the series on the left-hand side of (204) converges absolutely since its sum does not depend on permutations of basis functions (here the basis in
L2([t,T]2) is {fai(fafa-2(y)}TOTO,j2=0)- The equality (190) is proved-
Let us prove (192). Using the generalized Parseval equality, we obtain
p T t4 T t2
lim E / fa4(t4)0j2 (t4M fa(t3)j feW fafe)fa2 (t2) / fafe^ji (t1)x
jij2=0 t
oo
T t4
= E / (Î4W 03 (¿3) j(Î3)dt3dt4x
ji,j2=0 i i T t2
x^ 02 (Î2) j (¿2) ^ 0i (ti)0ji (ti)dtidt2 =
t t œ „
= E / 1{t3<t4>03(¿3)^4(t4)j (¿3)0j2 (¿4)dt3dt4X
j1 j2=0[t,T ]2
X J 1{t3<t4}^i (¿3)^2(^4)0^1 (t3)j (¿4)dt3dt4 =
[t,TT]2
= J 1{t3<t4}^3(^3)^2(^4)04^4)0i (¿3)dt3dt4 =
[t,T]2
= J 1{i3<i2}^3(t3)^2(t2)^4(t2)^i(t3)dt3dt2, (205)
[t,T ]2
where 0i(t),02(t),03(t),04(t) G L2([t,T]).
Suppose that 02(t) and 03(t) are Legendre polynomials of finite degrees. Denote
q
Sq (¿2,t3) = E Cl2li 4 (¿2 )4 (¿3), (206)
1l,l2=0
where |0j (x)}j=0 is a CONS of Legendre polynomials in L2([t,T]) and C/2/l are Fourier-Legendre coefficients for the function g(t2,t3) = 02(i2)03(i3)1{t2<t3}
$2(t),03(t) G L2([t,T])), i.e.
T t3
C/2/1 = J ^3(¿3)0/2 (¿3^ 02 (¿2)<?/i (¿2)dt2dt3 t t
and
Jim K - g^L2([t,T]2) = 0-
From (205) we obtain (the sum on the right-hand side of (206) is finite)
oo
E J 1{ii<t2>1{i3<i4>^4(t4)sq(t2,t3)^1(t1)0j2 (t4)0j1 (t3)0j2 (t2)0j1 (t1)x j1j2=0[t,T ]4
xdMMM^ = J 1{t3<t2}Sq (fafa'^fafafe^Mfa (207)
[t,T ]2
The right-hand side of (207) defines (as a scalar product of sq(t2,t3) and 1{t3<t2}fa (t3)'4(t2) in L2([t,T]2)) a linear bounded (and therefore continuous) functional in L2([t,T]2), which is given by the function 1{t3<t2}'1(i3)'4(i2). On the left-hand side of (207) there is also a linear continuous functional in L2([t,T]2) (see note below the formula (202)).
Let us implement the passage to the limit lim in (207) ^ „
E / 1{ti<i2<t3<i4}fa(t4)'fa(t3)^/^2(t2)^1(t1)0j2 (t4)0ji (t3)0j2 (t2)0ji (t1)x j1'j2=0[t,T ]4
xdMM^ dt4 = J 1{t2>t3} 1{i2<is}'?3(t3)'?2(t2)'1(t3)'4(t2)dt3di2 = 0. (208) [t,T ]2
Rewrite the equality (208) in the form
p
Cj2j1j2 jl
lim V Cj
j'lj2 =0
OO T t4 ^2
= E / faj (t^y fafe) j (*)/ fafe) j (t2^ fa (tl)0ji (tl)x jl'j2=° t t t t
xdt1dt2dt3 dt4 = 0, (209)
where fafa),...,fafa) G fa([i,T]).
Note that the series on the left-hand side of (209) converges absolutely since its sum does not depend on permutations of basis functions (here the basis in
L2([t,T]2) is {j(x)0j2(y)}TOJ2=0). The equality (192) is proved-
Let us prove (191). Using Fubini's Theorem and generalized Parseval's equality, we get
p T T is t2
t t t xdt1dt2dt3 dt4 =
j'l J2=0 t
1
lim V cf'Cfff1 n .....
p—>■ to ' J j1 ,2j2j1 2 p—to
,1,,2=0 ,1=0
- lim V cfc?
2 j1 j
fsf2f ,2,2,1
(,2 J2WO
p
lim y cf -( /;
p—TO
,1=0 T
1 —
2~ ,2,2,1
T
Ec
(,2,2)^(') j2=0
,2,2,1
1
lim£ / 04 (s), (s)ds / 03(t)02(t) / 0,1 (s)0i(s)dsdr-
2 p—to
,1=0
p—TO^-' \ 2^ ,2,2,1
lim y Of4 -( /;
p , 1 2 ,
,1=0
t p
Ec
(,2,2)^(') ,2=0
,2,2,1
1
T
T
T
I 04(s)0,i(s)ds I l(s) I 03(^)02(r)drds-
,1=0
lim V^1 r7^4 I ^ n'^'Ml
Am z^ 0,i 2o,2,2,1 ,1=0
EC
(,2,2)^(-) ,2=0
,2,2,1
T
T
1 2
04(s)0l(sW 03(T)02(T)drds-
lim y C?4 ( -Cfff1 ,1=0
C
(,2,2 WO ,2=0
,2,2,1
(210)
where Cf4 and C,^1 are defined by (140).
«71 2^/ 2^/1
Due to Cauchy-Bunyakovsky's inequality, Parseval's equality and (174), we
get
p
p
p
p
lim I f1'1^ I ^ n'^'Ml jïïfazLS. \2L'nnn
Ec
(j2j2)^(-) j2=0
j2j2j1
<
< lim Y (cf*) V -fa
- Z^ V j1 J I 2 j:
jl=0
ji=0
2 ~ j2j2jl
Ec
(j2j2)^(') j2=0
j2j2j1
<
œ 2 P / i
< inn V (fa1) V fa/; ;
- V j1 / ^ \ 2 j2j:
P—T^OO z
j1=0
T
j2j2 j1
j2=0
j2j2j1
(j2j2)^(') j2=0
2
J ^ds lim t fa
/ j2=0 V
1 ""¡/>3^2
2 j2j2j1
(j2j2)^(') j2=0
EC ^3^2 Cj2j2j1
| = Q
(211)
Combining (21Q) and (211), we obtain
p T T t3 t2
lim Ê / ^4(t4)0j1 (Î4M fa(t3)fa2 (t3M ^2(t2)0j2 (Î2M fal(t1)^j1 (t1)x
j1 j2=0 t
T
T
xdtidt2dtsdti = - / fa(s)fa(s) / fa(r)fa(r)<ir(is =
2
fa(fafa (Î4)l{i4<i3}^l(t4)^2(t3)dt4dt3,
(212)
[t,T
where '1 (t),...,fafa) G fafa,T]).
Suppose that '3(t) and '4(t) are Legendre polynomials of finite degrees. Denote
q
Sq(fafa= E C12I14(t3)4(t4), (213)
l1,l2=0
where {fa(x)}j=0 is a CONS of Legendre polynomials in L2([t,T]) and C/2/1 are Fourier-Legendre coefficients for the function g(t3,t4) = (t3)^/^4(t4) 1{t3<t4}
(fafa),fa(r) G fa([t,T])), i.e.
T t4
C/2/1 = fafefa^ (¿4) / fa^fa (¿3)^3^4
2
2
p
2
p
1
and
Um K - =0.
From (212) we obtain (the sum on the right-hand side of (213) is finite)
oo
£ J 1{tl<t2<t3>^jl (t4) j (t3)sq(t3,t4)02(t2)01(t1 )0j2 (t2)0j1 (t1)x jl,j2 =0[i)T ]4
xdtidt2dtzdU = ^ J sq(h,tA)l{u<t3}Mt^2(h)dtAdh. (214)
[t,T ]2
The right-hand side of (214) defines (as a scalar product of sq(t3,t4) and l{t4<t3}01(t4)02(t3) in L2([t,T]2)) a linear bounded (and therefore continuous) functional in L2([t,T]2), which is given by the function l{t4<t3}01(i4)02(i3). On the left-hand side of (214) there is also a linear continuous functional in L2([t,T]2) (see note below the formula (202)).
Let us implement the passage to the limit lim in (214)
q—>-to
to „
£ / 1{ii<i2<i3<i4}^/54(t4)0ji (t4)03(t3)0j2 (t3)02(t2)0j2 (t2)01(tO0jl (t1)x
jl'j2=0[t,T ]4
xdtidt2dt3du = ^ J 03(i3)04(i4)l{t3<t4}l{t4<t3}0i(i4)02(i3)^4^3 = o. (215)
[t,T ]2
Rewrite the equality (215) in the form
p
Cjlj2j2 jl
lim V Cj
p—TO
jlj2 =0
CO T ^4 ¿3 ¿2
= £ J 04(¿4)0jl (¿4^ 03(¿3)0j2 (*)/ 02(¿2) j (^ 01 (tx)^jl fa)x jl'j2=0 t t t t
xdt1dt2dt3 dt4 = 0, (216)
where 01 (t),...,04(t) £ L2([t,T]).
Note that the series on the left-hand side of (216) converges absolutely since its sum does not depend on permutations of basis functions (here the basis in L2([t,T]2) is {j(x)j(y)}TOj2=0). The equality (191) is proved. The equalities (190)-(192) are proved.
4 Conclusion
Recall that this article is Part III of the work devored to a new approach to the series expansion of iterated Stratonovich stochastic integrals of arbitrary multiplicity with respect to components of a multidimensional Wiener process ([15] and [16] are Parts I and II of this work, respectively). The results of the work make it possible to construct efficient procedures for the mean-square approximation of iterated Stratonovich stochastic integrals that appear in strong methods with orders 1.0, 1.5, 2.0, 2.5, and 3.0 of convergence for Ito SDEs with multidimensional non-commutative noise (approach based on the Taylor-Stratonovich expansion). The above procedures based on multiple Fourier-Legendre series have been successfully implemented as part of the software package in the Python programming language in [66].
Acknowledgement. I would like to thank Dr. Konstantin A. Rybakov for useful discussion of some presented results.
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