Jan 2, 2018 - Hereafter, we will identify the following records: a(j1,...,jk) = a(j1...jk) ..... Ã(Φ(t1,...,ts1â1,ts1+1,ts1+1,...,tsrâ1,tsr +1,tsr +1,...,tk) Ã. Ãdw. (i1) t1 ...dw.
arXiv:1801.00784v1 [math.PR] 2 Jan 2018
EXPANSION OF MULTIPLE STRATONOVICH STOCHASTIC INTEGRALS OF ARBITRARY MULTIPLICITY, BASED ON GENERALIZED REPEATED FOURIER SERIES, CONVERGING POINTWISE DMITRIY F. KUZNETSOV
Abstract. The article is devoted to the expansion of multiple Stratonovich stochastic integrals of arbitrary multiplicity k, based on the generalized repeated Fourier series. The case of Fourier-Legendre series and the case of trigonotemric Fourier series are considered. The obtained expansion provides a possibility to represent the multiple Stratonovich stochastic integral in the form of repeated series of products of standard Gaussian random variables. Convergence in the mean of degree 2n, n ∈ N of the expansion is proven. The results of the article can be applied to numerical solution of Ito stochastic differential equations.
1. Introduction In this article we consider the idea connected with representation of multiple Stratonovich stochastic integrals in the form of multiple stochastic integrals from specific nonrandom functions of several variables and following expansion of these functions using generalized repeated Fourier series in order to get effective mean-square approximations of mentioned stochastic integrals. For the first time this approach is considered in [1]. Usage of Fourier-Legendre series for approximation of multiple Stratonovich stochastic integrals took place for the first time in [1] (see also [2] - [6]). The results of [1] and this work (see also [2] - [6]) convincingly testify, that there is a doubtless relation between multiplier factor 21 , which is typical for Stratonovich stochastic integral and included into the sum, connecting Stratonovich and Ito stochastic integrals, and the fact, that in point of finite discontinuity of sectionally smooth function f (x) its Fourier series converges to the value 12 (f (x − 0) + f (x + 0)). In addition, as it is demonstrated, final formulas for expansions of multiple Stratonovich stochastic integrals, based on the Fourier-Legendre series is simpler than its analogues, based on trigonometric Fourier series. 2. Theorem on expansion of multiple Stratonovich stochastic integrals of arbitrary multiplicity k Let (Ω, F, P) be a complete probubility space, let {Ft , t ∈ [0, T ]} be a nondecreasing right-continous family of σ-subfields of F, and let f t be a standard m-dimensional Wiener stochastic process, which (i) is Ft -measurable for any t ∈ [0, T ]. We assume that the components ft (i = 1, . . . , m) of this process are independent. Hereafter we call stochastic process ξ : [0, T ] × Ω → ℜ1 as non-anticipative when def
it is measurable according to the family of variables (t, ω) and function ξ(t, ω) = ξt is Ft -measurable for all t ∈ [0, T ] and ξτ independent with increments ft+∆ − f∆ for ∆ ≥ τ, t > 0. Consider Z∗T Z∗t2 (i ) (i ) ∗ (k) (1) J [ψ ]T,t = ψk (tk ) . . . ψ1 (t1 )dwt11 . . . dwtkk , t
t
Mathematics Subject Classification: 60H05, 60H10, 42B05. Keywords: Multiple Stratonovich stochastic integral, Repeated Fourier series, Approximation, Expansion. 1
2
(2)
D.F. KUZNETSOV
J[ψ
(k)
]T,t =
ZT
ψk (tk ) . . .
t
Zt2
(i )
(i )
ψ1 (t1 )dwt11 . . . dwtkk ,
t
(i)
where every ψl (τ ) (l = 1, . . . , k) is a continuous non-random function on [t, T ]; wτ (0) i = 1, . . . , m and wτ = τ ; i1 , . . . , ik = 0, 1, . . . , m; and Z∗
and
(i)
= fτ
for
Z
denote Stratonovich and Itˆ o stochastic integrals, respectively. Further we will denote complete orthonormal systems of Legendre polynomials or trigonometric functions in the space L2 ([t, T ]) as {φj (x)}∞ j=0 . We will also pay attention on the following well-known facts about these two systems of functions. Suppose that f (x) is a bounded at the interval [t, T ] and sectionally smooth function at the open interval (t, T ). Then the Fourier series ∞ X
Cj φj (x), Cj =
j=0
ZT
f (x)φj (x)dx
t
converges at any internal point of the interval [t, T ] to the value 12 (f (x − 0) + f (x + 0)) and converges uniformly to f (x) in any closed interval of continuity of the function f (x), laying inside [t, T ]. At the same time the Fourier series obtained using Legendre polinomials converges if x = t and x = T to f (t + 0) and f (T − 0) correspondently, and the trigonometric Fourier series converges if x = t and x = T to 12 (f (t + 0) + f (T − 0)) in the case of periodic continuation of function. Define the following function on a hypercube [t, T ]k : (3)
K(t1 , . . . , tk ) =
k Y
ψl (tl )
l=1
k−1 Y l=1
1{tl 0 and choosen for all ε > 0 and doesn’t depend on points of the interval [t, T ]. Then the right part of the formula (40) tends to zero when N → ∞. Considering this fact, as well as (39), we come to (38). (i ) If for some l ∈ {1, . . . , k} : wtl l = tl , then proving of this lemma becomes obviously simpler and it is performed similarly. The lemma 5 is proven. Using lemmas 2, 3 w. p. 1 we obtain: [k/2]
(41)
J ∗ [ψ (k) ]T,t = J[ψ (k) ]T,t +
X 1 2r r=1
X
(k)
r ,...,s1 = J[K ∗ ]T,t , J[ψ (k) ]sT,t
(sr ,...,s1 )∈Ak,r
(k)
where stochastic integral J[K ∗ ]T,t defined in accordance with (31). Let’s subsitute the relation pk p1 pk p1 k k Y Y X X X X φjl (tl ) Cjk ...j1 ... φjl (tl ) + K ∗ (t1 , . . . , tk ) − Cjk ...j1 ... K ∗ (t1 , . . . , tk ) = j1 =0
jk =0
j1 =0
l=1
jk =0
into (41). Here p1 , . . . , pk < ∞. Then using the lemma 5 we obtain: (42)
J ∗ [ψ (k) ]T,t =
p1 X
j1 =0
...
pk X
jk =0
Cjk ...j1
k Y l=1
(i )
(k)
ζjl l + J[Rp1 ...pk ]T,t w. p. 1,
l=1
12
D.F. KUZNETSOV (k)
where stochastic integral J[Rp1 ...pk ]T,t defined in accordance with (31) and Rp1 ...pk (t1 , . . . , tk ) = K ∗ (t1 , . . . , tk ) −
(43)
(i ) ζjl l
=
ZT
p1 X
...
pk X
Cjk ...j1
jk =0
j1 =0
k Y
φjl (tl ),
l=1
φjl (s)dws(il ) .
t
At that, the following equation is executed pointwise in (t, T )k in accordance with the lemma 1: (44)
lim
p1 →∞
. . . lim Rp1 ...pk (t1 , . . . , tk ) = 0. pk →∞
Lemma 6. In the conditions of the theorem 1 2n (k) = 0, n ∈ N. lim . . . lim M J[Rp1 ...pk ]T,t p1 →∞
pk →∞
Proof. At first let’s analize in details the case k = 2. In this case w. p. 1 we have: N −1 N −1 X X
(2)
J[Rp1 p2 ]T,t = l.i.m.
N →∞
= l.i.m.
N →∞
N −1 lX 2 −1 X
Rp1 p2 (τl1 , τl2 )∆wτ(il1 ) ∆wτ(il2 ) = 1
Rp1 p2 (τl1 , τl2 )∆wτ(il1 ) ∆wτ(il2 ) + l.i.m. 1
2
N →∞
l2 =0 l1 =0
+l.i.m.
N →∞
=
ZT Zt2 t
2
l2 =0 l1 =0
N −1 X
N −1 lX 1 −1 X
Rp1 p2 (τl1 , τl2 )∆wτ(il1 ) ∆wτ(il2 ) + 1
2
l1 =0 l2 =0
Rp1 p2 (τl1 , τl1 )∆wτ(il1 ) ∆wτ(il2 ) = 1
1
l1 =0
(i ) (i ) Rp1 p2 (t1 , t2 )dwt11 dwt22
+
t
ZT Zt1 t
+1{i1 =i2 6=0}
ZT
(i )
(i )
Rp1 p2 (t1 , t2 )dwt22 dwt11 +
t
Rp1 p2 (t1 , t1 )dt1 ,
t
where Rp1 p2 (t1 , t2 ) = K ∗ (t1 , t2 ) − Using lemma 4 we obtain
p2 p1 X X
j1 =0 j2 =0
Cj2 j1 φj1 (t1 )φj2 (t2 ); p1 , p2 < ∞.
ZT Zt2 2n (2) ≤ Cn M J[Rp1 p2 ]T,t (Rp1 p2 (t1 , t2 ))2n dt1 dt2 + t
(45)
+
ZT Zt1 t
t
(Rp1 p2 (t1 , t2 ))
2n
t
dt2 dt1 + 1{i1 =i2 6=0}
ZT t
2n
(Rp1 p2 (t1 , t1 ))
dt1 ,
where Cn < ∞ — is a constant which depend on n and T − t; n = 1, 2, . . . Note, that due to assumptions proposed earlier, the function Rp1 p2 (t1 , t2 ) is continuous in the domains of integrating of integrals in the right part of (45) and it is bounded at the boundary of square [t, T ]2 .
EXPANSION OF MULTIPLE STRATONOVICH STOCHASTIC INTEGRALS
13
Let’s estimate the first integral in the right part of (45): 0≤ ≤
ZT Zt2 t
(Rp1 p2 (t1 , t2 ))
i=0 j=0
max
(t1 ,t2 )∈[τi ,τi+1 ]x[τj ,τj+1 ]
N −1 X i X
≤
Z
+
Dε
N −1 X i X
i=0 j=0
(46)
dt1 dt2 =
t
≤ +
2n
N −1 X i X
Z
2n
(Rp1 p2 (t1 , t2 ))
dt1 dt2 ≤
Γε
(Rp1 p2 (t1 , t2 ))
(Rp1 p2 (τi , τj ))
2n
2n
∆τi ∆τj + M SΓε ≤
∆τi ∆τj +
i=0 j=0
2n 2n Rp1 p2 (t(p1 p2 ) , t(p1 p2 ) ) − (Rp1 p2 (τi , τj )) ∆τi ∆τj + M SΓε ≤ j i
N −1 X i X
2n
(Rp1 p2 (τi , τj ))
i=0 j=0
1 1 2 ∆τi ∆τj + ε1 (T − t − 3ε) 1 + + M SΓε , 2 N
where Dε = {(t1 , t2 ) : t2 ∈ [t + 2ε, T − ε], t1 ∈ [t + ε, t2 − ε]}; Γε = D\Dε ; D = {(t1 , t2 ) : t2 ∈ [t, T ], t1 ∈ [t, t2 ]}; ε — is any sufficiently small positive number; SΓε is area of Γε ; M > 0 — is a (p p ) (p p ) 2n positive constant limiting (Rp1 p2 (t1 , t2 )) ; (ti 1 2 , tj 1 2 ) is a point of maximum of this function, when (t1 , t2 ) ∈ [τi , τi+1 ]x[τj , τj+1 ]; τi = t + 2ε + i∆ (i = 0, 1, . . . , N ); τN = T − ε; ∆ = (T − t − 3ε)/N ; ∆ < ε; ε1 > 0 — is any sufficiently small positive number. Getting (46), we used well-known properties of integrals, the first and the second Weierstrass theorems for the function of two variables, as well as the continuity and as a result the uniform 2n continuity of function (G √ p1 p2 (t1 , t2 )) in the domain Dε (∀ε1 > 0 ∃δ(ε1 ) > 0, which doesn’t depend on t1 , t2 , p1 , p2 and if 2∆ < δ, then the following inequality takes place: 2n 2n Rp1 p2 (t(p1 p2 ) , t(p1 p2 ) ) (τ , τ )) − (R p1 p2 i j j i < ε1 ). Considering (11) let’s write down: lim
lim (Rp1 p2 (t1 , t2 ))2n = 0 when (t1 , t2 ) ∈ Dε
p1 →∞ p2 →∞
and execute the repeated passage to the limit lim
lim
lim in inequality (46). Then according to
ε→+0 p1 →∞ p2 →∞
arbitrariness of ε1 we have (47)
lim
lim
p1 →∞ p2 →∞
ZT Zt2 t
(Rp1 p2 (t1 , t2 ))
2n
dt1 dt2 = 0.
2n
dt2 dt1 = 0,
t
Similarly to arguments given above we have: (48)
lim
lim
p1 →∞ p2 →∞
ZT Zt1 t
(49)
lim
lim
p1 →∞ p2 →∞
(Rp1 p2 (t1 , t2 ))
t
ZT
(Rp1 p2 (t1 , t1 ))2n dt1 = 0.
t
From (45), (47) – (49) we get lim
2n (2) = 0; n ∈ N. lim M J[Rp1 p2 ]T,t
p1 →∞ p2 →∞
14
D.F. KUZNETSOV
Let’s consider the case k = 3. W. p. 1 we have: N −1 N −1 N −1 X X X
(3)
J[Rp1 p2 p3 ]T,t = l.i.m.
N →∞
= l.i.m.
N →∞
Rp1 p2 p3 (τl1 , τl2 , τl3 )∆wτ(il1 ) ∆wτ(il2 ) ∆wτ(il3 ) = 1
2
3
l3 =0 l2 =0 l1 =0
N −1 lX 3 −1 lX 2 −1 X
Rp1 p2 p3 (τl1 , τl2 , τl3 )∆wτ(il1 ) ∆wτ(il2 ) ∆wτ(il3 ) + 1
2
3
l3 =0 l2 =0 l1 =0
+Rp1 p2 p3 (τl1 , τl3 , τl2 )∆wτ(il11 ) ∆wτ(il23 ) ∆wτ(il32 ) + Rp1 p2 p3 (τl2 , τl1 , τl3 )∆wτ(il12 ) ∆wτ(il21 ) ∆wτ(il33 ) + +Rp1 p2 p3 (τl2 , τl3 , τl1 )∆wτ(il1 ) ∆wτ(il2 ) ∆wτ(il3 ) + Rp1 p2 p3 (τl3 , τl2 , τl1 )∆wτ(il1 ) ∆wτ(il2 ) ∆wτ(il3 ) + 3 2 1 2 3 1 (i1 ) (i2 ) (i3 ) +Rp1 p2 p3 (τl3 , τl1 , τl2 )∆wτl3 ∆wτl1 ∆wτl2 + +l.i.m.
N →∞
N −1 lX 3 −1 X
Rp1 p2 p3 (τl2 , τl2 , τl3 )∆wτ(il12 ) ∆wτ(il22 ) ∆wτ(il33 ) +
l3 =0 l2 =0
+Rp1 p2 p3 (τl2 , τl3 , τl2 )∆wτ(il1 ) ∆wτ(il2 ) ∆wτ(il3 ) +Rp1 p2 p3 (τl3 , τl2 , τl2 )∆wτ(il1 ) ∆wτ(il2 ) ∆wτ(il3 ) + 2
+l.i.m.
N →∞
3
3
2
N −1 lX 3 −1 X
2
2
Rp1 p2 p3 (τl1 , τl3 , τl3 )∆wτ(il1 ) ∆wτ(il2 ) ∆wτ(il3 ) + 1
3
3
l3 =0 l1 =0
+Rp1 p2 p3 (τl3 , τl1 , τl3 )∆wτ(il1 ) ∆wτ(il2 ) ∆wτ(il3 ) +Rp1 p2 p3 (τl3 , τl3 , τl1 )∆wτ(il1 ) ∆wτ(il2 ) ∆wτ(il3 ) + 3
+l.i.m.
N →∞
=
ZT Zt3 Zt2
+
ZT Zt3 Zt2
+
ZT Zt3 Zt2
t
t
t
t
t
t
1
N −1 X
3
3
3
1
Rp1 p2 p3 (τl3 , τl3 , τl3 )∆wτ(il13 ) ∆wτ(il23 ) ∆wτ(il33 ) =
l3 =0
+
ZT Zt3 Zt2
Rp1 p2 p3 (t1 , t3 , t2 )dwt11 dwt23 dwt32 +
Rp1 p2 p3 (t2 , t1 , t3 )dwt12 dwt21 dwt33 +
ZT Zt3 Zt2
Rp1 p2 p3 (t2 , t3 , t1 )dwt13 dwt21 dwt32 +
ZT Zt3 Zt2
Rp1 p2 p3 (t3 , t1 , t2 )dwt12 dwt23 dwt31 +
(i ) (i ) (i ) Rp1 p2 p3 (t1 , t2 , t3 )dwt11 dwt22 dwt33
t
t
(i )
(i )
(i )
t
t
(i )
(i )
(i )
Rp1 p2 p3 (t3 , t2 , t1 )dwt13 dwt22 dwt31 +
t
t
+1{i1 =i2 6=0}
ZT Zt3
+1{i2 =i3 6=0}
ZT Zt3
t
t
+1{i1 =i3 6=0}
t
t
(i )
(i )
(i )
(i )
(i )
(i )
(i )
(i )
t
t
t
+ 1{i1 =i3 6=0}
ZT Zt3
Rp1 p2 p3 (t2 , t3 , t2 )dt2 dwt32 +
Rp1 p2 p3 (t3 , t2 , t2 )dt2 dwt31 + 1{i2 =i3 6=0}
ZT Zt3
Rp1 p2 p3 (t1 , t3 , t3 )dwt11 dt3 +
(i ) Rp1 p2 p3 (t2 , t2 , t3 )dt2 dwt33
t
t
(i )
t
ZT Zt3 t
t
(i )
t
(i )
Rp1 p2 p3 (t3 , t1 , t3 )dwt12 dt3 + 1{i1 =i2 6=0}
t
(i )
t
t
ZT Zt3 t
(i )
(i )
Rp1 p2 p3 (t3 , t3 , t1 )dwt13 dt3 .
t
Using the lemma 4 we obtain: ZT Zt3 Zt2 2n (3) 2n 2n ≤ Cn M J[Rp1 p2 p3 ]T,t (Rp1 p2 p3 (t1 , t2 , t3 )) + (Rp1 p2 p3 (t1 , t3 , t2 )) + t
+ (Rp1 p2 p3 (t2 , t1 , t3 ))
2n
t
t
+ (Rp1 p2 p3 (t2 , t3 , t1 ))2n + (Rp1 p2 p3 (t3 , t2 , t1 ))2n +
EXPANSION OF MULTIPLE STRATONOVICH STOCHASTIC INTEGRALS
2n
+ (Rp1 p2 p3 (t3 , t1 , t2 ))
15
dt1 dt2 dt3 +
ZT Zt3 1{i1 =i2 6=0} (Rp1 p2 p3 (t2 , t2 , t3 ))2n + (Rp1 p2 p3 (t3 , t3 , t2 ))2n + + t
t
2n 2n + +1{i1 =i3 6=0} (Rp1 p2 p3 (t2 , t3 , t2 )) + (Rp1 p2 p3 (t3 , t2 , t3 )) 2n 2n dt2 dt3 . +1{i2 =i3 6=0} (Rp1 p2 p3 (t3 , t2 , t2 )) + (Rp1 p2 p3 (t2 , t3 , t3 ))
(50)
It is important, that integands functions in the right part of (50) are continuous in the domains of integration of multiple integrals and bounded at the boundaries of these domains. Moreover, everywhere in (t, T )3 the following formula takes place: (51)
lim
lim
lim Rp1 p2 p3 (t1 , t2 , t3 ) = 0.
p1 →∞ p2 →∞ p3 →∞
Further, similarly to (46) (two dimensional case) we realize the repeated passage to the limit lim
p1 →∞
lim
lim
p2 →∞ p3 →∞
under the integral signs in the right part of (50) and we get: 2n (3) = 0; n ∈ N. lim lim lim M J[Rp1 p2 p3 ]T,t p1 →∞ p2 →∞ p3 →∞
Let’s consider the case of arbitrary k. Let’s analyze the stochastic integral of type (31) and find its representation, convenient for following verbal proof. In order to do it we introduce several denotations. Suppose that (k) SN (a)
=
N −1 X
...
jX 2 −1
X
a(j1 ,...,jk ) ,
j1 =0 (j1 ,...,jk )
jk =0
(k)
Csr . . . Cs1 SN (a) = =
N −1 X
jk =0
where
js1 +2 −1 js1 +1 −1
jsr +2 −1 jsr +1 −1
...
X
X
...
jsr +1 =0 jsr −1 =0
r Y
X
X
...
js1 +1 =0 js1 −1 =0
jX 2 −1
r j1 =0 Q l=1
Ijs
l
X
,js +1 (j1 ,...,jk ) l
r aQ l=1
Ijs
l
,js +1 (j1 ,...,jk ) l
,
def
Ijsl ,jsl +1 (j1 , . . . , jk ) = Ijsr ,jsr +1 . . . Ijs1 ,js1 +1 (j1 , . . . , jk ),
l=1
(k)
(k)
Cs0 . . . Cs1 SN (a) = SN (a),
0 Y
Ijsl ,jsl +1 (j1 , . . . , jk ) = (j1 , . . . , jk ),
l=1
def
Ijl ,jl+1 (jq1 , . . . , jq2 , jl , jq3 , . . . , jqk−2 , jl , jqk−1 , . . . , jqk ) = def
= (jq1 , . . . , jq2 , jl+1 , jq3 , . . . , jqk−2 , jl+1 , jqk−1 , . . . , jgk ).
Here l = 1, 2, . . . ; l 6= q1 , . . . , q2 , q3 , . . . , qk−2 , qk−1 , . . . , qk = 1, 2, . . . ; s1 , . . . , sr = 1, . . . , k − 1; sr > . . . > s1 ; a(jq1 ,...,jqk ) — scalars; q1 , . . . , qk = 1, . . . , k; expression X (jq1 ,...,jqk )
means the sum according to all possible derangements (jq1 , . . . , jqk ).
16
D.F. KUZNETSOV
Using induction it is possible to prove the following equality: N −1 X
(52)
...
N −1 X
a(j1 ,...,jk ) =
k−1 X
(k)
Csr . . . Cs1 SN (a),
r=0 sr ,...,s1 =1
j1 =0
jk =0
k−1 X
sr >...>s1
where k = 1, 2, . . . ; the sum according to empty set supposed as equal to 1. Hereafter, we will identify the following records: a(j1 ,...,jk ) = a(j1 ...jk ) = aj1 ...jk . In particular, from (52) when k = 2, 3, 4 we get the following formulas N −1 N −1 X X
(2)
(2)
a(j1 ,j2 ) = SN (a) + C1 SN (a) =
N −1 jX 2 −1 X
X
a(j1 j2 ) +
=
N −1 jX 2 −1 X
(aj1 j2 + aj2 j1 ) +
aj2 j2 ,
j2 =0
j2 =0 j1 =0 N −1 N −1 N −1 X X X
N −1 X
a(j2 j2 ) =
j2 =0
j2 =0 j1 =0 (j1 ,j2 )
j2 =0 j1 =0
N −1 X
(3)
(3)
(3)
(3)
a(j1 ,j2 ,j3 ) = SN (a) + C1 SN (a) + C2 SN (a) + C2 C1 SN (a) =
j3 =0 j2 =0 j1 =0
=
N −1 jX 3 −1 j2 −1 X X
X
a(j1 j2 j3 ) +
N −1 jX 3 −1 X
X
a(j1 j3 j3 ) +
N −1 jX 3 −1 j2 −1 X X
a(j2 j2 j3 ) +
N −1 X
a(j3 j3 j3 ) =
j3 =0
j3 =0 j1 =0 (j1 ,j3 ,j3 )
=
X
j3 =0 j2 =0 (j2 ,j2 ,j3 )
j3 =0 j2 =0 j1 =0 (j1 ,j2 ,j3 )
+
N −1 jX 3 −1 X
(aj1 j2 j3 + aj1 j3 j2 + aj2 j1 j3 + aj2 j3 j1 + aj3 j2 j1 + aj3 j1 j2 ) +
j3 =0 j2 =0 j1 =0
+
N −1 jX 3 −1 X
(aj2 j2 j3 + aj2 j3 j2 + aj3 j2 j2 ) +
N −1 jX 3 −1 X
(aj1 j3 j3 + aj3 j1 j3 + aj3 j3 j1 ) +
j3 =0 j1 =0
j3 =0 j2 =0
(53)
+
N −1 X
aj3 j3 j3 ,
j3 =0 N −1 N −1 N −1 N −1 X X X X
(4)
(4)
(4)
a(j1 ,j2 ,j3 ,j4 ) = SN (a) + C1 SN (a) + C2 SN (a)+
j4 =0 j3 =0 j2 =0 j1 =0 (4)
(4)
(4)
(4)
(4)
+C3 SN (a) + C2 C1 SN (a) + C3 C1 SN (a) + C3 C2 SN (a) + C3 C2 C1 SN (a) = =
N −1 jX 4 −1 j3 −1 j2 −1 X X X
X
a(j1 j2 j3 j4 ) +
+
X
a(j1 j3 j3 j4 ) +
N −1 jX 4 −1 X
X
a(j3 j3 j3 j4 ) +
N −1 jX 4 −1 X
N −1 jX 4 −1 j2 −1 X X
X
N −1 jX 4 −1 X
X
a(j1 j2 j4 j4 ) +
a(j2 j2 j4 j4 ) +
j4 =0 j2 =0 (j2 ,j2 ,j4 ,j4 )
j4 =0 j3 =0 (j3 ,j3 ,j3 ,j4 )
+
a(j2 j2 j3 j4 )
j4 =0 j2 =0 j1 =0 (j1 ,j2 ,j4 ,j4 )
j4 =0 j3 =0 j1 =0 (j1 ,j3 ,j3 ,j4 )
+
X
j4 =0 j3 =0 j2 =0 (j2 ,j2 ,j3 ,j4 )
j4 =0 j3 =0 j2 =0 j1 =0 (j1 ,j2 ,j3 ,j4 ) N −1 jX 4 −1 j3 −1 X X
N −1 jX 4 −1 j3 −1 X X
X
j4 =0 j1 =0 (j1 ,j4 ,j4 ,j4 )
a(j1 j4 j4 j4 ) +
N −1 X
j4 =0
aj4 j4 j4 j4 =
EXPANSION OF MULTIPLE STRATONOVICH STOCHASTIC INTEGRALS
=
N −1 jX 2 −1 3 −1 jX 4 −1 jX X
17
(aj1 j2 j3 j4 + aj1 j2 j4 j3 + aj1 j3 j2 j4 + aj1 j3 j4 j2 +
j4 =0 j3 =0 j2 =0 j1 =0
+aj1 j4 j3 j2 + aj1 j4 j2 j3 + aj2 j1 j3 j4 + aj2 j1 j4 j3 + aj2 j4 j1 j3 + aj2 j4 j3 j1 + aj2 j3 j1 j4 + +aj2 j3 j4 j1 + aj3 j1 j2 j4 + aj3 j1 j4 j2 + aj3 j2 j1 j4 + aj3 j2 j4 j1 + aj3 j4 j1 j2 + aj3 j4 j2 j1 + +aj4 j1 j2 j3 + aj4 j1 j3 j2 + aj4 j2 j1 j3 + aj4 j2 j3 j1 + aj4 j3 j1 j2 + aj4 j3 j2 j1 ) + +
N −1 jX 3 −1 4 −1 jX X
(aj2 j2 j3 j4 + aj2 j2 j4 j3 + aj2 j3 j2 j4 + aj2 j4 j2 j3 + aj2 j3 j4 j2 + aj2 j4 j3 j2 +
N −1 jX 4 −1 j3 −1 X X
(aj3 j3 j1 j4 + aj3 j3 j4 j1 + aj3 j1 j3 j4 + aj3 j4 j3 j1 + aj3 j4 j1 j3 + aj3 j1 j4 j3 +
N −1 jX 4 −1 j2 −1 X X
(aj4 j4 j1 j2 + aj4 j4 j2 j1 + aj4 j1 j4 j2 + aj4 j2 j4 j1 + aj4 j2 j1 j4 + aj4 j1 j2 j4 +
j4 =0 j3 =0 j2 =0
+aj3 j2 j2 j4 + aj4 j2 j2 j3 + aj3 j2 j4 j2 +aj4 j2 j3 j2 + aj4 j3 j2 j2 + aj3 j4 j2 j2 ) +
+
j4 =0 j3 =0 j1 =0
+aj1 j3 j3 j4 + aj4 j3 j3 j1 + aj4 j3 j1 j3 +aj1 j3 j4 j3 + aj1 j4 j3 j3 + aj4 j1 j3 j3 ) +
+
j4 =0 j2 =0 j1 =0
+aj1 j4 j4 j2 + aj2 j4 j4 j1 + aj2 j4 j1 j4 + aj1 j4 j2 j4 + aj1 j2 j4 j4 + aj2 j1 j4 j4 ) + +
N −1 jX 4 −1 X
(aj3 j3 j3 j4 + aj3 j3 j4 j3 + aj3 j4 j3 j3 + aj4 j3 j3 j3 ) +
j4 =0 j3 =0
+
N −1 jX 4 −1 X
(aj2 j2 j4 j4 + aj2 j4 j2 j4 + aj2 j4 j4 j2 + aj4 j2 j2 j4 + aj4 j2 j4 j2 + aj4 j4 j2 j2 ) +
j4 =0 j2 =0
(54)
+
N −1 jX 4 −1 X
(aj1 j4 j4 j4 + aj4 j1 j4 j4 + aj4 j4 j1 j4 + aj4 j4 j4 j1 ) +
N −1 X
aj4 j4 j4 j4 .
j4 =0
j4 =0 j1 =0
Possibly, the formula (52) for any k was founded by the author for the first time. The relation (52) will be used frequently in the future. Assume, that k Y ∆wτ(ijl ) , a(j1 ,...,jk ) = Φ (τj1 , . . . , τjk ) l
l=1
where Φ (t1 , . . . , tk ) — is a nonrandom function of k variables. Then from (31) and (52) we have (k)
J[Φ]T,t =
k−1 X
X
r=0 (sr ,...,s1 )∈Ak,r
×l.i.m.
N →∞
N −1 X
jk =0
js1 +2 −1 js1 +1 −1
jsr +2 −1 jsr +1 −1
...
X
X
×
X
X
...
js1 +1 =0 js1 −1 =0
jsr +1 =0 jsr −1 =0
...
jX 2 −1
r j1 =0 Q l=1
Ijs
l
X
,js +1 (j1 ,...,jk ) l
× Φ τj1 , . . . , τjs1 −1 , τjs1 +1 , τjs1 +1 , . . . , τjsr −1 , τjsr +1 , τjsr +1 , . . . , τjk × (i
)
(i
)
(i
)
×∆wτ(ij11) . . . ∆wτjss1 −1 ∆wτjss1 +1 ∆wτjss1 +1 ... 1 −1 1 1 +1 ! (i
)
(i
)
(i
)
. . . ∆wτjssrr−1 ∆wτjssrr +1 ∆wτjssrr+1 . . . ∆wτ(ijk ) −1 +1 k
=
×
18
D.F. KUZNETSOV
(55)
=
k−1 X
(k)s1 ,...,sr
X
I[Φ]T,t
w.p.1,
r=0 (sr ,...,s1 )∈Ak,r
where (k)s ,...,sr I[Φ]T,t 1
=
ZT
...
t
tZ tsr sr +3 tZ sr +2 Z t
t
ts1 +3 ts1 +2 ts1
...
Z
Z
t
t
t
Z
...
t
Zt2 t
r Q l=1
Its
l
X
×
,ts +1 (t1 ,...,tk ) l
× Φ (t1 , . . . , ts1 −1 , ts1 +1 , ts1 +1 , . . . , tsr −1 , tsr +1 , tsr +1 , . . . , tk ) × (i
(i )
(i
)
)
(i
(i
)
)
+2 +1 −1 ... dwtss11+2 dwtss11+1 dwtss11+1 ×dwt11 . . . dwtss11−1
(i +2 ) (i +1 ) (i ) (i −1 ) dwtssrr+2 dwtssrr+1 dwtssrr+1 . . . dwtssrr−1
(56) and
(i ) . . . dwtkk
!
,
P def = 1, k ≥ 2; the set Ak,r is defined by relation (16); we suppose, that right part of (56) exists ∅
as Ito stochastic integral. Remark 2. The summands in the right part of (56) should be understood as follows: for each derangement from the set r Y Itsl ,tsl +1 (t1 , . . . , tk ) l=1
it is necessary to perform replacement in the right part of (56) of all pairs (their number is r) of (j) (i) differentials with similar lower indexes of type dwtp dwtp by values 1{i=j6=0} dtp . Using the lemma 4 we get: k−1 2n 2n X X (k) (k)s1 ,...,sr (57) M J[Φ]T,t ≤ Cnk , M I[Φ]T,t r=0 (sr ,...,s1 )∈Ak,r
where
≤
s1 ...sr Cnk
ZT t
2n (k)s ,...,sr ≤ M I[Φ]T,t 1
...
tZ tsr sr +3 tZ sr +2 Z t
t
t
ts1 +3 ts1 +2 ts1
...
Z t
Z t
Z t
...
Zt2 t
r Q l=1
Its
l
X
×
,ts +1 (t1 ,...,tk ) l
×Φ2n (t1 , . . . , ts1 −1 , ts1 +1 , ts1 +1 , . . . , tsr −1 , tsr +1 , tsr +1 , . . . , tk ) × (58)
× dt1 . . . dts1 −1 dts1 +1 dts1 +2 . . . dtsr −1 dtsr +1 dtsr +2 . . . dtk ,
where derangements in the course of summation in (58) are performed only in Φ2n (t1 , . . . , ts1 −1 , ts1 +1 , s1 ...sr ts1 +1 , . . . , tsr −1 , tsr +1 , tsr +1 , . . . , tk ); Cnk , Cnk < ∞. Consider (57), (58) for Rp1 ...pk (t1 , . . . , tk ): 2n (k) ≤ M J[Rp1 ...pk ]T,t
EXPANSION OF MULTIPLE STRATONOVICH STOCHASTIC INTEGRALS
(59)
≤ Cnk
≤
s1 ...sr Cnk
ZT t
...
k−1 X
X
r=0 (sr ,...,s1 )∈Ak,r
2n (k)s ,...,sr , M I[Rp1 ...pk ]T,t 1
2n (k)s ,...,sr ≤ M I[Rp1 ...pk ]T,t 1
tZ tsr sr +3 tZ sr +2 Z t
t
ts1 +3 ts1 +2 ts1
...
Z t
t
19
Z
Z
...
t
t
Zt2 t
r Q l=1
Its
l
X
×
,ts +1 (t1 ,...,tk ) l
×Rp2n1 ...pk (t1 , . . . , ts1 −1 , ts1 +1 , ts1 +1 , . . . , tsr −1 , tsr +1 , tsr +1 , . . . , tk ) × (60)
× dt1 . . . dts1 −1 dts1 +1 dts1 +2 . . . dtsr −1 dtsr +1 dtsr +2 . . . dtk ,
where derangements in the course of summation in (60) are performed only in Rp2n1 ...pk (t1 , . . . , ts1 −1 , s1 ...sr ts1 +1 , ts1 +1 , . . . , tsr −1 , tsr +1 , tsr +1 , . . . , tk ); Cnk , Cnk < ∞. According to (7) we have the following in all internal points of the hypercube [t, T ]k : k−1 k k−1 k−1 k−1 r Y Y X 1 X Y Y Rp1 ...pk (t1 , . . . , tk ) = ψl (tl ) 1{tl s1
−
p1 X
...
pk X
Cjk ...j1
jk =0
j1 =0
k Y
l=1
l6=s1 ,...,sr
φjl (tl ).
l=1
Due to (61) the function Rp1 ...pk (t1 , . . . , tk ) is continuous in the domains of integration of stochastic integrals in the right part of (60) and it is bounded at the boundaries of these domains (let’s remind, that the repeated series k ∞ ∞ Y X X φjl (tl ) Cjk ...j1 ... j1 =0
jk =0
l=1
converges at the boundary of hypercube [t, T ]k ). Then performing the repeated passage to the limit lim . . . lim under the integral signs in these p1 →∞
pk →∞
estimations (like it was performed for the two-dimensional case), considering (44), we get the required result. The theorem 1 is proven. It easy to note, that if we expand the function K ∗ (t1 , . . . , tk ) into the Fourier series at the interval (t, T ) at first according to the variable tk , after that according to the variable tk−1 , etc., then we will have the expansion: (62)
K ∗ (t1 , . . . , tk ) =
∞ X
jk =0
...
∞ X
Cjk ...j1
j1 =0
φjl (tl )
l=1
instead of the expansion (8). Let’s prove the expansion (62). Similarly with (12) we have: (63)
k Y
X ∞ ZT 1 ψk (tk ) 1{tk−1
