Expansion of Multiple Ito Stochastic Integrals of Arbitrary Multiplicity

arXiv:1712.09746v1 [math.PR] 28 Dec 2017

EXPANSION OF MULTIPLE ITO STOCHASTIC INTEGRALS OF ARBITRARY MULTIPLICITY, BASED ON GENERALIZED MULTIPLE FOURIER SERIES, CONVERGING IN THE MEAN DMITRIY F. KUZNETSOV

Abstract. The article is devoted to expansions of multiple Ito stochastic integrals, based on generalized multiple Fourier series converging in the mean. The method of generalized multiple Fourier series for expansions and mean-square approximations of multiple Ito stochastic integrals is derived. In the article it is also obtained generalization of this method for discontinuous basis functions. The comparison of derived method with well-known expansions of multiple Ito stochastic integrals based on Ito formula and Hermite polynomials is given. The proof of convergence in the mean of degree 2n, n ∈ N of considered method is obtained.

1. Introduction The idea of representing of multiple Ito stochastic integrals in the form of multiple stochastic integrals from specific nonrandom functions of several variables and following expansion of these functions using Fourier series in order to get effective mean-square approximations of mentioned stochastic integrals was represented in several works of the author (another approaches to expansions of multiple stochastic integrals see in [1] - [3]). Specifically, this approach appeared for the first time in [4] (1994). In that work the mentioned idea is formulated more likely at the level of guess (without any satisfactory grounding), and as a result the work [4] contains rather fuzzy formulations and a number of incorrect conclusions. Note, that in [4] we used multiple Fourier series according to the trigonometric system of functions converging in the mean. It should be noted, that the results of work [4] are true for a sufficiently narrow particular case when numbers i1 , ..., ik are pairwise different; i1 , . . . , ik = 1, . . . , m. Usage of Fourier series according to the system of Legendre polynomials for approximation of multiple stochastic integrals took place for the first time in [5] (see also [6] - [11]). The question about what integrals (Ito or Stratonovich) are more suitable for expansions within the frames of distinguished direction of researches has turned out to be rather interesting and difficult. On the one side, theorem 1 conclusively demonstrates, that the structure of multiple Ito stochastic integrals is rather convenient for expansions into multiple series according to the system of standard Gaussian random variables regardless of their multiplicity k. On the other side, the results of [5], [8] - [11] 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 1 2 (f (x − 0) + f (x + 0)). In addition, as it is demonstrated in [8] - [11], final formulas for expansions of multiple Stratonovich stochastic integrals (of second multiplicity in the common case and of third and fourth multiplicity in some particular cases) are more compact than their analogues for Ito stochastic integrals. Mathematics Subject Classification: 60H05, 60H10, 42B05. Keywords: Multiple Ito stochastic integral, Multiple Stratonovich stochastic integral, Multiple Fourier series, Approximation, Expansion. 1

2

D.F. KUZNETSOV

2. Theorem on expansion of multiple Ito 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. Let’s consider the following multiple Ito stochastic integrals:

(1)

J[ψ

(k)

]T,t =

ZT

ψk (tk ) . . .

t

Zt2

(i )

(i )

ψ1 (t1 )dwt11 . . . dwtkk ,

t

(i)

(i)

where every ψl (τ ) (l = 1, . . . , k) is a non-random function on [t, T ]; wτ = fτ (0) wτ = τ ; i1 , . . . , ik = 0, 1, . . . , m. Suppose that every ψl (τ ) (l = 1, . . . , k) is a continuous on [t, T ] function. Define the following function on a hypercube [t, T ]k : (2)

K(t1 , . . . , tk ) =

k Y

ψl (tl )

l=1

k−1 Y l=1

when i = 1, . . . , m;

1{tl 0 (δ > 0 and chosen for all ε > 0 and it doesn’t depend on mentioned points), then the corresponding oscillation of function Φ(t1 , t2 , t3 ) for these two points of domain D3 is less than ε.

6

D.F. KUZNETSOV

If we assume, that ∆τj < δ (j = 0, 1, . . . , N − 1), then the distance between points (t1 , t2 , τj3 ), (t1 , τj2 , τj3 ) is obviously less than δ. In this case |Φ(t1 , t2 , τj3 ) − Φ(t1 , τj2 , τj3 )| < ε. Consequently, when ∆τj < δ (j = 0, 1, . . . , N − 1) the expression (16) is evaluated by the following value: N −1 jX 3 −1 j2 −1 X X (T − t)3 ∆τj1 ∆τj2 ∆τj3 < ε2 ε2 . 6 j =0 j =0 j =0 1

2

3

Because of this, the first limit in the right part of (15) equals to zero. Similarly we can prove equality to zero of the second limit in the right part of (15). Consequently, the equality (13) is proven when k = 3. The cases when k = 2 and k > 3 are analyzed absolutely similarly. It is necessary to note, that proving of formula (13) rightness is similar, when the nonrandom function Φ(t1 , . . . , tk ) is continuous in the open domain Dk and bounded at its border. Assume, that N −1 X

l.i.m.

N →∞

Φ (τj1 , . . . , τjk )

k Y

def

(k)

∆wτ(ijl ) = J ′ [Φ]T,t . l

l=1

j1 ,...,jk =0 jq 6=jr ; q6=r; q,r=1,...,k

Then we will get according to (13) (17)

J

′

(k) [Φ]T,t

=

ZT

...

t

Zt2 t

 X  (ik ) (i1 ) Φ(t1 , . . . , tk )dwt1 . . . dwtk ,

(t1 ,...,tk )

where summation according to derangements (t1 , . . . , tk ) is performed only in the expression, which is enclosed in parentheses, and the nonrandom function Φ(t1 , . . . , tk ) is assumed to be continuous in the corresponding domains of integration. Not difficult to see, that (17) may be rewritten in the form: J

′

(k) [Φ]T,t

=

X

ZT

(t1 ,...,tk ) t

...

Zt2

(i )

(i )

Φ(t1 , . . . , tk )dwt11 . . . dwtkk ,

t

(i )

(i )

where derangements (t1 , . . . , tk ) for summing are performed only in the values dwt11 . . . dwtkk , at the same time the indexes near upper limits of integration in the multiple stochastic integrals are changed correspondently and if tr changed places with tq in the derangement (t1 , . . . , tk ), then ir changes places with iq in the derangement (i1 , . . . , ik ). Let’s consider the class M2 ([0, T ]) of functions ξ : [0, T ] × Ω → ℜ1 , which satisfy the conditions: (i). The function ξ(t, ω) is a measurable in accordance with the collection of variables (t, ω); (ii). The function ξ(t, ω) is Ft -measurable for all t ∈ [0, T ] and ξ(τ, ω) independent with increments ft+∆ − f∆ for ∆ ≥ τ, t > 0; RT  (iii). M ξ 2 (t, ω) dt < ∞; 0

(iv). M{ξ 2 (t, ω)} < ∞ for all t ∈ [0, T ]. It is well known [1], that Ito stochastic integral exists in the mean-square sence for any ξ ∈ M2 ([0, T ]) Lemma 2. Suppose that the following condition is met ZT t

...

Zt2 t

Φ2 (t1 , . . . , tk )dt1 . . . dtk < ∞,

EXPANSION OF MULTIPLE ITO STOCHASTIC INTEGRALS

7

where Φ(t1 , . . . , tk ) — is a nonrandom function. Then 2   ZT Zt2 (k) M I[Φ]T,t ≤ Ck . . . Φ2 (t1 , . . . , tk )dt1 . . . dtk , Ck < ∞. t

t

Proof. Using standard properties and estimations of stochastic integrals for ξτ ∈ M2 ([t0 , t]) we have 2 ) ( Zt 2  Zt  Zt Zt 2 ≤ (t − t0 ) Mkξτ |2 }dτ. (18) M ξτ dfτ = M{|ξτ | }dτ, M ξτ dτ t0

t0

t0

t0

Let’s denote

(l) ξ[Φ]tl+1 ,...,tk ,t

tZl+1

=

...

t def

(0)

Zt2

(i )

(i )

Φ(t1 , . . . , tk )dwt11 . . . dwtl l ,

t

where l = 1, . . . , k − 1 and ξ[Φ]t1 ,...,tk ,t = Φ(t1 , . . . , tk ).

(l)

In accordance with induction it is easy to demonstrate, that ξ[Φ]tl+1 ,...,tk ,t ∈ M2 ([t, T ]) using the variable tl+1 . Subsequently, using the estimations (18) repeatedly we can be led to confirmation of the lemma. Not difficult to see, that in the case i1 , . . . , ik = 1, . . . , m from lemma 2 we have: 2  ZT  Zt2 (k) (19) M I[Φ]T,t = . . . Φ2 (t1 , . . . , tk )dt1 . . . dtk . t

t

Lemma 3. Suppose that every ϕi (s) (i = 1, . . . , k) is a continuous function on [t, T ]. Then k Y

(20)

(k)

J[ϕl ]T,t = J[Φ]T,t w. p. 1,

l=1

where J[ϕl ]T,t =

ZT

ϕl (s)dws(il ) , Φ(t1 , . . . , tk ) =

k Y

ϕl (tl )

l=1

t

(k)

and the integral J[Φ]T,t is defined by the equality (12). Proof. Let at first il 6= 0; l = 1, . . . , k. Let’s denote def

J[ϕl ]N =

N −1 X

ϕl (τj )∆wτ(ijl ) .

j=0

Since k Y l=1

J[ϕl ]N −

k Y

J[ϕl ]T,t =

l=1

k  l−1 X Y l=1

g=1

  Y  k J[ϕg ]T,t (J[ϕl ]N − J[ϕl ]T,t ) J[ϕg ]N , g=l+1

then because of the Minkowsky inequality and inequality of Cauchy-Bunyakovsky 2 1/2 1/4   Y k   k X Y k 4 ≤ Ck M |J[ϕl ]N − J[ϕl ]T,t | , Ck < ∞. (21) M J[ϕl ]N − J[ϕl ]T,t l=1

l=1

l=1

Note, that

J[ϕl ]N − J[ϕl ]T,t =

N −1 X g=0

J[∆ϕl ]τg+1 ,τg , J[∆ϕl ]τg+1 ,τg =

τZg+1

τg

(ϕl (τg ) − ϕl (s)) dws(il ) .

8

D.F. KUZNETSOV

Since J[∆ϕl ]τg+1 ,τg are independent for various g, then [12] 4  4  NX  NX −1  −1 M J[∆ϕl ]τj+1 ,τj + M J[∆ϕl ]τj+1 ,τj = j=0

j=0

(22)

+6

N −1 X j=0

2  j−1  2   X M J[∆ϕl ]τj+1 ,τj M J[∆ϕl ]τq+1 ,τq . q=0

Because of gaussianity of J[∆ϕl ]τj+1 ,τj we have

τZj+1  τZj+1 2 n n 4 o 2 o 2 2 =3 (ϕl (τj ) − ϕl (s)) ds, M J[∆ϕl ]τj+1 ,τj = M J[∆ϕl ]τj+1 ,τj (ϕl (τj ) − ϕl (s)) ds . τj

τj

Using this relations and continuity and as a result the uniform continuity of functions ϕi (s), we get 4    NX  N j−1 N −1 −1 −1 X X X
∆τj ∆τq < 3ε4 δ(T − t) + (T − t)2 , (∆τj )2 + 6 M J[∆ϕl ]τj+1 ,τj ≤ ε4 3 j=0

j=0

j=0

q=0

where ∆τj < δ, δ > 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 (22) tends to zero when N → ∞. Considering this fact, as well as (21), we come to (20). (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 3 is proven. According to lemma 1 we have J[ψ (k) ]T,t = l.i.m.

N →∞

= l.i.m.

N →∞

= l.i.m.

N →∞

N −1 X lk =0

N −1 X

N −1 X

lX 2 −1

ψ1 (τl1 ) . . . ψk (τlk )∆wτ(il11 ) . . . ∆wτ(ilk ) = k

l1 =0

lX 2 −1

K(τl1 , . . . , τlk )∆wτ(il11 ) . . . ∆wτ(ilk ) = k

l1 =0

...

lk =0

N −1 X

K(τl1 , . . . , τlk )∆wτ(il11 ) . . . ∆wτ(ilk ) = k

l1 =0

N −1 X

N →∞

=

...

lk =0

= l.i.m.

(23)

...

K(τl1 , . . . , τlk )∆wτ(il1 ) . . . ∆wτ(ilk ) = 1

k

j1 ,...,jk =0 jq 6=jr ; q6=r; q,r=1,...,k

ZT

...

t

Zt2 t

X

(t1 ,...,tk )



(i )

(i )

K(t1 , . . . , tk )dwt11 . . . dwtkk



,

where derangements (t1 , . . . , tk ) for summing are executed only in the expression, enclosed in parentheses. It is easy to see, that (23) may be rewritten in the form: J[ψ

(k)

]T,t =

X

ZT

(t1 ,...,tk ) t

...

Zt2

(i )

(i )

K(t1 , . . . , tk )dwt11 . . . dwtkk ,

t

(i )

(i )

where derangements (t1 , . . . , tk ) for summing are performed only in the values dwt11 . . . dwtkk , at the same time the indexes near upper limits of integration in the multiple stochastic integrals are

EXPANSION OF MULTIPLE ITO STOCHASTIC INTEGRALS

9

changed correspondently and if tr changed places with tq in the derangement (t1 , . . . , tk ), then ir changes places with iq in the derangement (i1 , . . . , ik ). Note, that since integration of bounded function using the set of null measure for Riemann integrals gives zero result, then the following formula is reasonable for these integrals: Z

G(t1 , . . . , tk )dt1 . . . dtk =

ZT

X

...

(t1 ,...,tk ) t

[t,T ]k

Zt2

G(t1 , . . . , tk )dt1 . . . dtk ,

t

where derangements (t1 , . . . , tk ) for summing are executed only in the values dt1 , . . . , dtk , at the same time the indexes near upper limits of integration are changed correspondently and the function G(t1 , . . . , tk ) is considered as integrated in hypercube [t, T ]k . According to lemmas 1 – 3 and (17), (23) w. p. 1 we get the following representation J[ψ

(k)

]T,t = p1 X

...

pk X

Cjk ...j1 l.i.m.

...

pk X

jk =0

p1 X

−l.i.m.

N →∞

=

j1 =0

where (24)

...

pk X

Cjk ...j1

jk =0

N →∞

...

Zt2 t

X

(t1 ,...,tk )



N −1 X

(i ) ζjl l

N →∞

N −1 X

l1 ,...,lk =0

φj1 (τl1 ) . . . φjk (τlk )∆wτ(il11 ) . . . ∆wτ(ilk ) − k

1

X

ZT

...

(t1 ,...,tk ) t

N →∞

+

k

φj1 (τl1 )∆wτ(il1 ) . . . φjk (τlk )∆wτ(ilk )

− l.i.m.



φj1 (τl1 ) . . . φjk (τlk )∆wτ(il11 ) . . . ∆wτ(ilk ) +

k

(l1 ,...,lk )∈Gk k Y

(i )

(i )

φj1 (t1 ) . . . φjk (tk )dwt11 . . . dwtkk

j1 ,...,jk =0 jq 6=jr ; q6=r; q,r=1,...,k

Cjk ...j1 l.i.m.

X

l=1

p1 ,...,pk RT,t =

t

jk =0

j1 =0

p1 X

Cjk ...j1

jk =0

j1 =0

p1 ,...,pk +RT,t =

ZT

...

j1 =0

p1 ,...,pk = +RT,t

pk X

p1 X

X

!

p1 ,...,pk + RT,t =

φj1 (τl1 )∆wτ(il1 ) . . . φjk (τlk )∆wτ(ilk ) 1

k

(l1 ,...,lk )∈Gk

!

p1 ,...,pk + RT,t ,

 Zt2 pk p1 k Y X X (i ) (i ) φjl (tl ) dwt11 . . . dwtkk , Cjk ...j1 ... K(t1 , . . . , tk ) − j1 =0

t

jk =0

l=1

(i )

(i )

where derangements (t1 , . . . , tk ) for summing are performed only in the values dwt11 . . . dwtkk , at the same time the indexes near upper limits of integration in the multiple stochastic integrals are changed correspondently and if tr changed places with tq in the derangement (t1 , . . . , tk ), then ir changes places with iq in the derangement (i1 , . . . , ik ). p1 ,...,pk Let’s evaluate the remainder RT,t of the series. According to the lemma 2 we have M



(25)

p1 ,...,pk RT,t

2 

= Ck

≤ Ck

X

ZT

(t1 ,...,tk ) t

2 Zt2 pk p1 k Y X X φjl (tl ) dt1 . . . dtk Cjk ...j1 ... K(t1 , . . . , tk ) − ... j1 =0

t

jk =0

l=1

2 Z  pk p1 k Y X X φjl (tl ) dt1 . . . dtk → 0 Cjk ...j1 ... K(t1 , . . . , tk ) −

[t,T ]k

j1 =0

jk =0

l=1

if p1 , . . . , pk → ∞, where the constant Ck depends only on the multiplicity k of multiple Ito stochastic integral. The theorem is proven.

10

D.F. KUZNETSOV

Not difficult to see, that for the case of pairwise different numbers i1 , . . . , ik = 1, . . . , m from the theorem 1 we get: pk p1 X X (i ) (i ) Cjk ...j1 ζj11 . . . ζjkk . ... J[ψ (k) ]T,t = l.i.m. p1 ,...,pk →∞

jk =0

j1 =0

In order to evaluate significance of the theorem 1 for practice we will demonstrate its transformed particular cases for k = 1, . . . , 6 [6] - [11]: J[ψ (1) ]T,t = l.i.m.

(26)

(27)

p1 →∞

J[ψ

(2)

]T,t = l.i.m.

p1 ,p2 →∞

J[ψ (3) ]T,t =

(28)

p2 p1 X X

Cj2 j1

j1 =0 j2 =0

l.i.m.

p1 ,...,p3 →∞

(i )

Cj1 ζj11 ,

j1 =0 (i ) (i ) ζj11 ζj22

p3 p2 X p1 X X

j1 =0 j2 =0 j3 =0

 − 1{i1 =i2 6=0} 1{j1 =j2 } ,

 (i ) (i ) (i ) Cj3 j2 j1 ζj11 ζj22 ζj33 − (i )

(i )

(i )

− 1{i1 =i2 6=0} 1{j1 =j2 } ζj33 − 1{i2 =i3 6=0} 1{j2 =j3 } ζj11 − 1{i1 =i3 6=0} 1{j1 =j3 } ζj22 J[ψ (4) ]T,t =

l.i.m.

p1 ,...,p4 →∞

p1 X

...

p4 X

Cj4 ...j1

j4 =0

j1 =0

(i ) (i ) −1{i1 =i2 6=0} 1{j1 =j2 } ζj33 ζj44 (i ) (i ) −1{i1 =i4 6=0} 1{j1 =j4 } ζj22 ζj33 (i ) (i ) −1{i2 =i4 6=0} 1{j2 =j4 } ζj11 ζj33

(29)



p1 X

Y 4 l=1

 ,

(i )

ζjl l − (i ) (i )

− 1{i1 =i3 6=0} 1{j1 =j3 } ζj22 ζj44 − (i ) (i )

− 1{i2 =i3 6=0} 1{j2 =j3 } ζj11 ζj44 − (i ) (i )

− 1{i3 =i4 6=0} 1{j3 =j4 } ζj11 ζj22 + +1{i1 =i2 6=0} 1{j1 =j2 } 1{i3 =i4 6=0} 1{j3 =j4 } + 1{i1 =i3 6=0} 1{j1 =j3 } 1{i2 =i4 6=0} 1{j2 =j4 } +  + 1{i1 =i4 6=0} 1{j1 =j4 } 1{i2 =i3 6=0} 1{j2 =j3 } , J[ψ (5) ]T,t =

l.i.m.

p1 ,...,p5 →∞

p1 X

j1 =0

(i ) (i ) (i ) −1{i1 =i2 6=0} 1{j1 =j2 } ζj33 ζj44 ζj55 (i ) (i ) (i ) −1{i1 =i4 6=0} 1{j1 =j4 } ζj22 ζj33 ζj55 (i ) (i ) (i ) −1{i2 =i3 6=0} 1{j2 =j3 } ζj11 ζj44 ζj55 (i ) (i ) (i ) −1{i2 =i5 6=0} 1{j2 =j5 } ζj11 ζj33 ζj44 (i ) (i ) (i ) −1{i3 =i5 6=0} 1{j3 =j5 } ζj11 ζj22 ζj44 (i ) +1{i1 =i2 6=0} 1{j1 =j2 } 1{i3 =i4 6=0} 1{j3 =j4 } ζj55 (i ) +1{i1 =i2 6=0} 1{j1 =j2 } 1{i4 =i5 6=0} 1{j4 =j5 } ζj33 (i ) +1{i1 =i3 6=0} 1{j1 =j3 } 1{i2 =i5 6=0} 1{j2 =j5 } ζj44 (i ) +1{i1 =i4 6=0} 1{j1 =j4 } 1{i2 =i3 6=0} 1{j2 =j3 } ζj55 (i ) +1{i1 =i4 6=0} 1{j1 =j4 } 1{i3 =i5 6=0} 1{j3 =j5 } ζj22 (i ) +1{i1 =i5 6=0} 1{j1 =j5 } 1{i2 =i4 6=0} 1{j2 =j4 } ζj33

...

p5 X

j5 =0

Cj5 ...j1

5 Y l=1

(i )

ζjl l − (i ) (i ) (i )

− 1{i1 =i3 6=0} 1{j1 =j3 } ζj22 ζj44 ζj55 − (i ) (i ) (i )

− 1{i1 =i5 6=0} 1{j1 =j5 } ζj22 ζj33 ζj44 − (i ) (i ) (i )

− 1{i2 =i4 6=0} 1{j2 =j4 } ζj11 ζj33 ζj55 − (i ) (i ) (i )

− 1{i3 =i4 6=0} 1{j3 =j4 } ζj11 ζj22 ζj55 − (i ) (i ) (i )

− 1{i4 =i5 6=0} 1{j4 =j5 } ζj11 ζj22 ζj33 +

(i )

+ 1{i1 =i2 6=0} 1{j1 =j2 } 1{i3 =i5 6=0} 1{j3 =j5 } ζj44 + (i )

+ 1{i1 =i3 6=0} 1{j1 =j3 } 1{i2 =i4 6=0} 1{j2 =j4 } ζj55 + (i )

+ 1{i1 =i3 6=0} 1{j1 =j3 } 1{i4 =i5 6=0} 1{j4 =j5 } ζj22 + (i )

+ 1{i1 =i4 6=0} 1{j1 =j4 } 1{i2 =i5 6=0} 1{j2 =j5 } ζj33 + (i )

+ 1{i1 =i5 6=0} 1{j1 =j5 } 1{i2 =i3 6=0} 1{j2 =j3 } ζj44 + (i )

+ 1{i1 =i5 6=0} 1{j1 =j5 } 1{i3 =i4 6=0} 1{j3 =j4 } ζj22 +

EXPANSION OF MULTIPLE ITO STOCHASTIC INTEGRALS (i )

(i )

(30)

11

+1{i2 =i3 6=0} 1{j2 =j3 } 1{i4 =i5 6=0} 1{j4 =j5 } ζj11 + 1{i2 =i4 6=0} 1{j2 =j4 } 1{i3 =i5 6=0} 1{j3 =j5 } ζj11 + ! (i )

+ 1{i2 =i5 6=0} 1{j2 =j5 } 1{i3 =i4 6=0} 1{j3 =j4 } ζj11

J[ψ (6) ]T,t =

l.i.m.

p1 ,...,p6 →∞

p1 X

j1 =0

(i ) (i ) (i ) (i ) −1{i1 =i6 6=0} 1{j1 =j6 } ζj22 ζj33 ζj44 ζj55 (i ) (i ) (i ) (i ) −1{i3 =i6 6=0} 1{j3 =j6 } ζj11 ζj22 ζj44 ζj55 (i ) (i ) (i ) (i ) −1{i5 =i6 6=0} 1{j5 =j6 } ζj11 ζj22 ζj33 ζj44 (i ) (i ) (i ) (i ) −1{i1 =i3 6=0} 1{j1 =j3 } ζj22 ζj44 ζj55 ζj66 (i ) (i ) (i ) (i ) −1{i1 =i5 6=0} 1{j1 =j5 } ζj22 ζj33 ζj44 ζj66 (i ) (i ) (i ) (i ) −1{i2 =i4 6=0} 1{j2 =j4 } ζj11 ζj33 ζj55 ζj66 (i ) (i ) (i ) (i ) −1{i3 =i4 6=0} 1{j3 =j4 } ζj11 ζj22 ζj55 ζj66

...

p6 X

j6 =0

Cj6 ...j1

6 Y l=1

,

(i )

ζjl l − (i ) (i ) (i ) (i )

− 1{i2 =i6 6=0} 1{j2 =j6 } ζj11 ζj33 ζj44 ζj55 − (i ) (i ) (i ) (i )

− 1{i4 =i6 6=0} 1{j4 =j6 } ζj11 ζj22 ζj33 ζj55 − (i ) (i ) (i ) (i )

− 1{i1 =i2 6=0} 1{j1 =j2 } ζj33 ζj44 ζj55 ζj66 − (i ) (i ) (i ) (i )

− 1{i1 =i4 6=0} 1{j1 =j4 } ζj22 ζj33 ζj55 ζj66 − (i ) (i ) (i ) (i )

− 1{i2 =i3 6=0} 1{j2 =j3 } ζj11 ζj44 ζj55 ζj66 − (i ) (i ) (i ) (i )

− 1{i2 =i5 6=0} 1{j2 =j5 } ζj11 ζj33 ζj44 ζj66 − (i ) (i ) (i ) (i )

− 1{i3 =i5 6=0} 1{j3 =j5 } ζj11 ζj22 ζj44 ζj66 − (i ) (i ) (i ) (i )

−1{i4 =i5 6=0} 1{j4 =j5 } ζj11 ζj22 ζj33 ζj66 + (i ) (i )

(i ) (i )

(i ) (i )

(i ) (i )

(i ) (i )

(i ) (i )

(i ) (i )

(i ) (i )

(i ) (i )

(i ) (i )

(i ) (i )

(i ) (i )

(i ) (i )

(i ) (i )

(i ) (i )

(i ) (i )

(i ) (i )

(i ) (i )

(i ) (i )

(i ) (i )

(i ) (i )

(i ) (i )

(i ) (i )

(i ) (i )

(i ) (i )

(i ) (i )

(i ) (i )

(i ) (i )

(i ) (i )

(i ) (i )

(i ) (i )

(i ) (i )

(i ) (i )

(i ) (i )

(i ) (i )

(i ) (i )

(i ) (i )

(i ) (i )

(i ) (i )

(i ) (i )

(i ) (i )

(i ) (i )

+1{i1 =i2 6=0} 1{j1 =j2 } 1{i3 =i4 6=0} 1{j3 =j4 } ζj55 ζj66 + 1{i1 =i2 6=0} 1{j1 =j2 } 1{i3 =i5 6=0} 1{j3 =j5 } ζj44 ζj66 + +1{i1 =i2 6=0} 1{j1 =j2 } 1{i4 =i5 6=0} 1{j4 =j5 } ζj33 ζj66 + 1{i1 =i3 6=0} 1{j1 =j3 } 1{i2 =i4 6=0} 1{j2 =j4 } ζj55 ζj66 + +1{i1 =i3 6=0} 1{j1 =j3 } 1{i2 =i5 6=0} 1{j2 =j5 } ζj44 ζj66 + 1{i1 =i3 6=0} 1{j1 =j3 } 1{i4 =i5 6=0} 1{j4 =j5 } ζj22 ζj66 + +1{i1 =i4 6=0} 1{j1 =j4 } 1{i2 =i3 6=0} 1{j2 =j3 } ζj55 ζj66 + 1{i1 =i4 6=0} 1{j1 =j4 } 1{i2 =i5 6=0} 1{j2 =j5 } ζj33 ζj66 + +1{i1 =i4 6=0} 1{j1 =j4 } 1{i3 =i5 6=0} 1{j3 =j5 } ζj22 ζj66 + 1{i1 =i5 6=0} 1{j1 =j5 } 1{i2 =i3 6=0} 1{j2 =j3 } ζj44 ζj66 + +1{i1 =i5 6=0} 1{j1 =j5 } 1{i2 =i4 6=0} 1{j2 =j4 } ζj33 ζj66 + 1{i1 =i5 6=0} 1{j1 =j5 } 1{i3 =i4 6=0} 1{j3 =j4 } ζj22 ζj66 + +1{i2 =i3 6=0} 1{j2 =j3 } 1{i4 =i5 6=0} 1{j4 =j5 } ζj11 ζj66 + 1{i2 =i4 6=0} 1{j2 =j4 } 1{i3 =i5 6=0} 1{j3 =j5 } ζj11 ζj66 + +1{i2 =i5 6=0} 1{j2 =j5 } 1{i3 =i4 6=0} 1{j3 =j4 } ζj11 ζj66 + 1{i6 =i1 6=0} 1{j6 =j1 } 1{i3 =i4 6=0} 1{j3 =j4 } ζj22 ζj55 + +1{i6 =i1 6=0} 1{j6 =j1 } 1{i3 =i5 6=0} 1{j3 =j5 } ζj22 ζj44 + 1{i6 =i1 6=0} 1{j6 =j1 } 1{i2 =i5 6=0} 1{j2 =j5 } ζj33 ζj44 + +1{i6 =i1 6=0} 1{j6 =j1 } 1{i2 =i4 6=0} 1{j2 =j4 } ζj33 ζj55 + 1{i6 =i1 6=0} 1{j6 =j1 } 1{i4 =i5 6=0} 1{j4 =j5 } ζj22 ζj33 + +1{i6 =i1 6=0} 1{j6 =j1 } 1{i2 =i3 6=0} 1{j2 =j3 } ζj44 ζj55 + 1{i6 =i2 6=0} 1{j6 =j2 } 1{i3 =i5 6=0} 1{j3 =j5 } ζj11 ζj44 + +1{i6 =i2 6=0} 1{j6 =j2 } 1{i4 =i5 6=0} 1{j4 =j5 } ζj11 ζj33 + 1{i6 =i2 6=0} 1{j6 =j2 } 1{i3 =i4 6=0} 1{j3 =j4 } ζj11 ζj55 + +1{i6 =i2 6=0} 1{j6 =j2 } 1{i1 =i5 6=0} 1{j1 =j5 } ζj33 ζj44 + 1{i6 =i2 6=0} 1{j6 =j2 } 1{i1 =i4 6=0} 1{j1 =j4 } ζj33 ζj55 + +1{i6 =i2 6=0} 1{j6 =j2 } 1{i1 =i3 6=0} 1{j1 =j3 } ζj44 ζj55 + 1{i6 =i3 6=0} 1{j6 =j3 } 1{i2 =i5 6=0} 1{j2 =j5 } ζj11 ζj44 + +1{i6 =i3 6=0} 1{j6 =j3 } 1{i4 =i5 6=0} 1{j4 =j5 } ζj11 ζj22 + 1{i6 =i3 6=0} 1{j6 =j3 } 1{i2 =i4 6=0} 1{j2 =j4 } ζj11 ζj55 + +1{i6 =i3 6=0} 1{j6 =j3 } 1{i1 =i5 6=0} 1{j1 =j5 } ζj22 ζj44 + 1{i6 =i3 6=0} 1{j6 =j3 } 1{i1 =i4 6=0} 1{j1 =j4 } ζj22 ζj55 + +1{i6 =i3 6=0} 1{j6 =j3 } 1{i1 =i2 6=0} 1{j1 =j2 } ζj44 ζj55 + 1{i6 =i4 6=0} 1{j6 =j4 } 1{i3 =i5 6=0} 1{j3 =j5 } ζj11 ζj22 + +1{i6 =i4 6=0} 1{j6 =j4 } 1{i2 =i5 6=0} 1{j2 =j5 } ζj11 ζj33 + 1{i6 =i4 6=0} 1{j6 =j4 } 1{i2 =i3 6=0} 1{j2 =j3 } ζj11 ζj55 + +1{i6 =i4 6=0} 1{j6 =j4 } 1{i1 =i5 6=0} 1{j1 =j5 } ζj22 ζj33 + 1{i6 =i4 6=0} 1{j6 =j4 } 1{i1 =i3 6=0} 1{j1 =j3 } ζj22 ζj55 + +1{i6 =i4 6=0} 1{j6 =j4 } 1{i1 =i2 6=0} 1{j1 =j2 } ζj33 ζj55 + 1{i6 =i5 6=0} 1{j6 =j5 } 1{i3 =i4 6=0} 1{j3 =j4 } ζj11 ζj22 + +1{i6 =i5 6=0} 1{j6 =j5 } 1{i2 =i4 6=0} 1{j2 =j4 } ζj11 ζj33 + 1{i6 =i5 6=0} 1{j6 =j5 } 1{i2 =i3 6=0} 1{j2 =j3 } ζj11 ζj44 +

12

D.F. KUZNETSOV (i ) (i )

(i ) (i )

+1{i6 =i5 6=0} 1{j6 =j5 } 1{i1 =i4 6=0} 1{j1 =j4 } ζj22 ζj33 + 1{i6 =i5 6=0} 1{j6 =j5 } 1{i1 =i3 6=0} 1{j1 =j3 } ζj22 ζj44 + (i ) (i )

+1{i6 =i5 6=0} 1{j6 =j5 } 1{i1 =i2 6=0} 1{j1 =j2 } ζj33 ζj44 −

−1{i6 =i1 6=0} 1{j6 =j1 } 1{i2 =i5 6=0} 1{j2 =j5 } 1{i3 =i4 6=0} 1{j3 =j4 }

−1{i6 =i1 6=0} 1{j6 =j1 } 1{i2 =i4 6=0} 1{j2 =j4 } 1{i3 =i5 6=0} 1{j3 =j5 }

−1{i6 =i1 6=0} 1{j6 =j1 } 1{i2 =i3 6=0} 1{j2 =j3 } 1{i4 =i5 6=0} 1{j4 =j5 }

−1{i6 =i2 6=0} 1{j6 =j2 } 1{i1 =i5 6=0} 1{j1 =j5 } 1{i3 =i4 6=0} 1{j3 =j4 }

−1{i6 =i2 6=0} 1{j6 =j2 } 1{i1 =i4 6=0} 1{j1 =j4 } 1{i3 =i5 6=0} 1{j3 =j5 }

−1{i6 =i2 6=0} 1{j6 =j2 } 1{i1 =i3 6=0} 1{j1 =j3 } 1{i4 =i5 6=0} 1{j4 =j5 }

−1{i6 =i3 6=0} 1{j6 =j3 } 1{i1 =i5 6=0} 1{j1 =j5 } 1{i2 =i4 6=0} 1{j2 =j4 }

−1{i6 =i3 6=0} 1{j6 =j3 } 1{i1 =i4 6=0} 1{j1 =j4 } 1{i2 =i5 6=0} 1{j2 =j5 }

−1{i3 =i6 6=0} 1{j3 =j6 } 1{i1 =i2 6=0} 1{j1 =j2 } 1{i4 =i5 6=0} 1{j4 =j5 }

−1{i6 =i4 6=0} 1{j6 =j4 } 1{i1 =i5 6=0} 1{j1 =j5 } 1{i2 =i3 6=0} 1{j2 =j3 }

−1{i6 =i4 6=0} 1{j6 =j4 } 1{i1 =i3 6=0} 1{j1 =j3 } 1{i2 =i5 6=0} 1{j2 =j5 }

−1{i6 =i4 6=0} 1{j6 =j4 } 1{i1 =i2 6=0} 1{j1 =j2 } 1{i3 =i5 6=0} 1{j3 =j5 }

−1{i6 =i5 6=0} 1{j6 =j5 } 1{i1 =i4 6=0} 1{j1 =j4 } 1{i2 =i3 6=0} 1{j2 =j3 }

(31)

−1{i6 =i5 6=0} 1{j6 =j5 } 1{i1 =i2 6=0} 1{j1 =j2 } 1{i3 =i4 6=0} 1{j3 =j4 } !

−1{i6 =i5 6=0} 1{j6 =j5 } 1{i1 =i3 6=0} 1{j1 =j3 } 1{i2 =i4 6=0} 1{j2 =j4 } ,

where 1A is the indicator of the set A. Let’s generalize formulas (26) – (31) for the case of any arbitrary multiplicity of J[ψ (k) ]T,t . In order to do it we will introduce several denotations. Let’s examine the unregulated set {1, 2, . . . , k} and separate it up in two parts: the first part consists of r unordered pairs (sequence order of these pairs is also unimportant) and the second one — of the remains k − 2r numbers. So, we have: ({{g1 , g2 }, . . . , {g2r−1 , g2r }}, {q1 , . . . , qk−2r }), {z } | {z } |

(32)

part 1

part 2

where {g1 , g2 , . . . , g2r−1 , g2r , q1 , . . . , qk−2r } = {1, 2, . . . , k}, curly braces mean an unordered set, and the round braces mean an ordered set. Let’s call (32) as partition and examine the sum using all possible partitions: X (33) ag1 g2 ,...,g2r−1 g2r ,q1 ...qk−2r . ({{g1 ,g2 },...,{g2r−1 ,g2r }},{q1 ,...,qk−2r }) {g1 ,g2 ,...,g2r−1 ,g2r ,q1 ,...,qk−2r }={1,2,...,k}

Let’s give an example of sums in the form (33): X ag1 g2 = a12 , ({g1 ,g2 }) {g1 ,g2 }={1,2}

X

ag1 g2 g3 g4 = a1234 + a1324 + a2314 ,

({{g1 ,g2 },{g3 ,g4 }}) {g1 ,g2 ,g3 ,g4 }={1,2,3,4}

X

({g1 ,g2 },{q1 ,q2 }) {g1 ,g2 ,q1 ,q2 }={1,2,3,4}

ag1 g2 ,q1 q2 = a12,34 + a13,24 + a14,23 + a23,14 + a24,13 + a34,12 ,

EXPANSION OF MULTIPLE ITO STOCHASTIC INTEGRALS

X

13

ag1 g2 ,q1 q2 q3 = a12,345 + a13,245 + a14,235 + a15,234 + a23,145 + a24,135 +

({g1 ,g2 },{q1 ,q2 ,q3 }) {g1 ,g2 ,q1 ,q2 ,q3 }={1,2,3,4,5}

+a25,134 + a34,125 + a35,124 + a45,123 , X

ag1 g2 ,g3 g4 ,q1 = a12,34,5 + a13,24,5 + a14,23,5 + a12,35,4 + a13,25,4 + a15,23,4 +

({{g1 ,g2 },{g3 ,g4 }},{q1 }) {g1 ,g2 ,g3 ,g4 ,q1 }={1,2,3,4,5}

+a12,54,3 + a15,24,3 + a14,25,3 + a15,34,2 + a13,54,2 + a14,53,2 + a52,34,1 + a53,24,1 + a54,23,1 . Now we can formulate the theorem 1 (formula (6)) using alternative and more comfortable form. Theorem 2 (see [7] - [11]). In conditions of the theorem 1 the following converging in the meansquare sense expansion is valid:

J[ψ (k) ]T,t = [k/2]

(34) +

X r=1

p1 ,...,pk →∞

X

r

(−1) ·

l.i.m.

p1 X

...

k Y

Cjk ...j1

jk =0

j1 =0

({{g1 ,g2 },...,{g2r−1 ,g2r }},{q1 ,...,qk−2r }) {g1 ,g2 ,...,g2r−1 ,g2r ,q1 ,...,qk−2r }={1,2,...,k}

pk X

r Y

s=1

(i )

ζjl l +

l=1

1{ig

2s−1

= ig2s 6=0} 1{jg

2s−1

= jg2s }

k−2r Y

(iq ) ζjq l l

l=1

!

.

In particular from (34) if k = 5 we obtain:

J[ψ (5) ]T,t =

∞ X

Cj5 ...j1

j1 ,...,j5 =0

+

X

5 Y l=1

(i )

ζjl l −

X

= ig2 6=0} 1{jg1 = jg2 } 1

1{ig

({g1 ,g2 },{q1 ,q2 ,q3 }) {g1 ,g2 ,q1 ,q2 ,q3 }={1,2,3,4,5}

(i ) 1{ig = ig 6=0} 1{jg = jg } 1{ig = ig 6=0} 1{jg = jg } ζjqq1 1 4 3 4 3 2 1 2 1

({{g1 ,g2 },{g3 ,g4 }},{q1 }) {g1 ,g2 ,g3 ,g4 ,q1 }={1,2,3,4,5}

3 Y

(iql )

ζjq

l

+

l=1

!

.

The last equality obviously agree with (30). 3. Comparison of the theorem 2 with representations of multiple Ito stochastic integrals, based on Hermite polynomials Note, that rightness of formulas (26) – (31) can be verified by the fact, that if i1 = . . . = i6 = i = 1, . . . , m and ψ1 (s), . . . , ψ6 (s) ≡ ψ(s), then we can deduce the following equalities which are right w. p. 1 [1]: 1 J[ψ (1) ]T,t = δT,t , 1!
1 2 δ − ∆T,t , J[ψ (2) ]T,t = 2! T,t
1 3 J[ψ (3) ]T,t = δT,t − 3δT,t ∆T,t , 3!
1 4 2 (4) δT,t − 6δT,t ∆T,t + 3∆2T,t , J[ψ ]T,t = 4!
1 5 (5) 3 J[ψ ]T,t = δT,t − 10δT,t ∆T,t + 15δT,t ∆2T,t , 5!
1 6 (6) 4 2 J[ψ ]T,t = δT,t − 15δT,t ∆T,t + 45δT,t ∆2T,t − 15∆3T,t , 6!

14

D.F. KUZNETSOV

where δT,t =

ZT

ψ(s)dfs(i) ,

∆T,t =

t

ZT

ψ 2 (s)ds,

t

which can be independently obtained using the Ito formula and Hermite polynomials. When k = 1 everything is evident. Let’s examine the cases k = 2, 3. When k = 2 (we put p1 = p2 = p):   X p p X (i) (i) Cj1 j1 = Cj2 j1 ζj1 ζj2 − J[ψ (2) ]T,t = l.i.m. p→∞

= l.i.m. p→∞

X p jX 1 −1

(Cj2 j1 +

j1 =0

j1 ,j2 =0

(i) (i) Cj1 j2 ) ζj1 ζj2

+

p X

j1 =0

j1 =0 j2 =0

X p jX 1 −1

   2 (i) −1 = ζj1 Cj1 j1

  p 1 X 2  (i) 2 (i) (i) Cj1 Cj2 ζj1 ζj2 + = l.i.m. C −1 = ζj1 p→∞ 2 j =0 j1 j1 =0 j2 =0 1    X p p 1 X 2  (i) 2 1 (i) (i) C −1 = ζj1 Cj Cj ζ ζ + = l.i.m. p→∞ 2 j ,j =0 1 2 j1 j2 2 j =0 j1 1

1 2 j1 6=j2

(35)

 X 2  p p
1 1 X 2 1 2 (i) = l.i.m. − = Cj ζ C δ − ∆T,t . p→∞ 2 j =0 1 j1 2 j =0 j1 2! T,t 1

1

Let’s explain the last step in (35). For Ito stochastic integrals the following estimation [13] is right: q   ZT ZT q/2  2 M ξτ dfτ ≤ Kq M |ξτ | dτ ,

(36)

t

t

where q > 0 — is a fixed number; fτ — is a scalar standard Wiener process; ξτ ∈ M2 ([t, T ]); Kq — is a constant, depending only on q; ZT q/2  ZT 2 |ξτ | dτ < ∞ w. p. 1; M |ξτ |2 dτ < ∞. t

t

Since δT,t −

p X

(i) Cj1 ζj1

j1 =0

 ZT  p X Cj1 φj1 (s) dfs(i) , ψ(s) − = t

j1 =0

then using the estimation (36) to the right part of this expression and considering, that 2 ZT  p X Cj1 φj1 (s) ds → 0 ψ(s) − j1 =0

t

if p → ∞ we obtain (37)

ZT

ψ(s)dfs(i) = q -l.i.m. p→∞

t

p X

(i)

Cj1 ζj1 , q > 0.

j1 =0

Hence, if q = 4, then it is easy to conclude, that 2 X p (i) 2 = δT,t . Cj1 ζj1 l.i.m. p→∞

j1 =0

EXPANSION OF MULTIPLE ITO STOCHASTIC INTEGRALS

15

This equality was used in the last transition of the formula (35). If k = 3 (we put p1 = p2 = p3 = p): J[ψ (3) ]T,t = ! p p p p X X X X (i) (i) (i) (i) (i) (i) Cj1 j2 j1 ζj2 = Cj2 j2 j1 ζj1 − Cj3 j1 j1 ζj3 − Cj3 j2 j1 ζj1 ζj2 ζj3 − = l.i.m. p→∞

p X

= l.i.m. p→∞

= l.i.m. p→∞

+

p jX 1 −1 X

(i) (i) (i) Cj3 j2 j1 ζj1 ζj2 ζj3

j1 ,j2 ,j3 =0

p jX 1 −1 j2 −1 X X

j1 ,j2 =0

j1 ,j2 =0

j1 ,j3 =0

j1 ,j2 ,j3 =0

−

p X

(Cj3 j1 j1 + Cj1 j1 j3 +

(i) Cj1 j3 j1 ) ζj3

j1 ,j3 =0

Cj3 j2 j1 + Cj3 j1 j2 + Cj2 j1 j3 + Cj2 j3 j1 +Cj1 j2 j3 + Cj1 j3 j2

j1 =0 j2 =0 j3 =0

(Cj3 j1 j3 + Cj1 j3 j3 + Cj3 j3 j1 )

j1 =0 j3 =0

+

p X

j1 =0



(i) ζj3

2

(i) ζj1 +

p jX 1 −1 X

j1 =0 j3 =0

j1 ,j3 =0

= l.i.m. p→∞

=

(i) (i) (i)

ζj1 ζj2 ζj3 +

 2 (i) (i) ζj3 + (Cj3 j1 j1 + Cj1 j1 j3 + Cj1 j3 j1 ) ζj1

p  3 X (i) (i) (Cj3 j1 j1 + Cj1 j1 j3 + Cj1 j3 j1 ) ζj3 − Cj1 j1 j1 ζj1 p jX 1 −1 j2 −1 X X

!

!

!

=

(i) (i) (i)

Cj1 Cj2 Cj3 ζj1 ζj2 ζj3 +

j1 =0 j2 =0 j3 =0

p j1 −1 p j1 −1  2  2 1 X X 2 1 X X 2 (i) (i) (i) (i) + ζj1 + ζj3 + Cj3 Cj1 ζj3 Cj1 Cj3 ζj1 2 j =0 j =0 2 j =0 j =0 3 3 1 1 ! p p p   X X X 3 1 1 1 (i) (i) (i) (i) (i) + − Cj31 ζj1 Cj21 Cj3 ζj3 = l.i.m. Cj1 Cj2 Cj3 ζj1 ζj2 ζj3 + p→∞ 6 j =0 2 j ,j =0 6 j ,j ,j =0 1

1

3

1 2 3 j1 6=j2 ,j2 6=j3 ,j1 6=j3

p j1 −1 p j1 −1  2  2 1 X X 2 1 X X 2 (i) (i) (i) (i) ζj1 + ζj3 + Cj3 Cj1 ζj3 Cj1 Cj3 ζj1 2 j =0 j =0 2 j =0 j =0 3 3 1 1 ! p p p   X X 3 1 1 X 1 (i) (i) (i) (i) (i) + − Cj31 ζj1 Cj21 Cj3 ζj3 = l.i.m. Cj Cj Cj ζ ζ ζ − p→∞ 6 j =0 2 j ,j =0 6 j ,j ,j =0 1 2 3 j1 j2 j3 1 1 3 1 2 3 ! p j −1 p j p 1 1 −1  3  2  2 X X X X X 1 (i) (i) (i) (i) (i) 3 2 2 + Cj1 ζj1 ζj3 + Cj1 Cj3 ζj1 ζj1 + 3 Cj3 Cj1 ζj3 − 3 6 j =0 j =0 j =0 j =0 j =0

+

1

1

3

1

3

p j1 −1 p j1 −1  2  2 1 X X 2 1 X X 2 (i) (i) (i) (i) + ζj1 + ζj3 + Cj3 Cj1 ζj3 Cj1 Cj3 ζj1 2 j =0 j =0 2 j =0 j =0 3 3 1 1 ! !3 ! p p p p p   3 1 X 1 X 1 X 2 X 1 X 3 (i) (i) (i) (i) 2 = = l.i.m. − − Cj ζ ζ + Cj ζ C C Cj ζ C p→∞ 6 j =0 j1 j1 2 j ,j =0 j1 3 j3 6 j =0 1 j1 2 j =0 j1 j =0 3 j3 1

1

1

3


1 3 δT,t − 3δT,t ∆T,t . 3! The last step in (38) is arisen from the equality 3 X p (i) 3 = δT,t , Cj1 ζj1 l.i.m.

(38)

=

p→∞

j1 =0

which can be obtained easily when q = 8 (see (37)).

1

3

16

D.F. KUZNETSOV

In addition, we used the following correlations between the Fourier coefficients for the examined case: Cj1 j2 + Cj2 j1 = Cj1 Cj2 , 2Cj1 j1 = Cj21 , Cj1 j2 j3 + Cj1 j3 j2 + Cj2 j3 j1 + Cj2 j1 j3 + Cj3 j2 j1 + Cj3 j1 j2 = Cj1 Cj2 Cj3 , 2 (Cj1 j1 j3 + Cj1 j3 j1 + Cj3 j1 j1 ) = Cj21 Cj3 , 6Cj1 j1 j1 = Cj31 . 4. On usage of complete orthonormal discontinuous systems of functions in the theorem 1 Analyzing the proof of the theorem 1, we can ask a natural question: can we weaken the condition of continuity of functions φj (x); j = 1, 2, . . .? We will tell, that the function f (x) : [t, T ] → ℜ1 satisfies the condition (⋆), if it is continuous at the interval [t, T ] except may be for the finite number of points of the finite discontinuity, as well as it is continuous from the right at the interval [t, T ]. Afterwards, let’s suppose, that {φj (x)}∞ j=0 — is a complete orthonormal system of functions in the space L2 ([t, T ]), moreover φj (x), j < ∞ satisfies the condition (⋆). It is easy to see, that the continuity of function φj (x) was used substantially for proving of the theorem 1 in two places: lemma 3 and formula (13). It’s clear, that without damage to generality, partition {τj }N j=0 of the interval [t, T ] in lemma 3 and formula (13) can be taken so "small that among the points τj of this partition will be all points of jumps of functions ϕ1 (τ ) = φj1 (τ ), . . . , ϕk (τ ) = φjk (τ ); j1 , . . . , jk < ∞ and among the points (τj1 , . . . , τjk ); 0 ≤ j1 < . . . < jk ≤ N − 1 there will be all points of jumps of function Φ(t1 , . . . , tk ). Let’s demonstrate how to modify proofs of lemma 3 and formula (13) in the case when {φj (x)}∞ j=0 — is a complete orthonormal system of functions in the space L2 ([t, T ]), moreover φj (x), j < ∞ satisfies the condition (⋆). At first, appeal to lemma 3. Proving this lemma we got the following relations: 4  NX  NX −1 n 4 o −1 M J[∆ϕl ]τj+1 ,τj + J[∆ϕl ]τj+1 ,τj = M j=0

j=0

(39)

+6

N −1 X j=0

j−1 n n 2 o X 2 o M J[∆ϕl ]τj+1 ,τj M J[∆ϕl ]τq+1 ,τq , q=0

(40) τZj+1  τZj+1 2 n n 4 o 2 o 2 2 =3 (ϕl (τj ) − ϕl (s)) ds, M J[∆ϕl ]τj+1 ,τj (ϕl (τj ) − ϕl (s)) ds . = M J[∆ϕl ]τj+1 ,τj τj

τj

−1 Propose, that functions ϕl (s); l = 1, . . . , k satisfy the condition (⋆), and the partition {τj }N j=0 includes all points of jumps of functions ϕl (s); l = 1, . . . , k. It means, that, for the integral τZj+1

τj

(ϕl (τj ) − ϕl (s))2 ds

the subintegral function is continuous at the interval [τj , τj+1 ) and possibly it has finite discontinuity in the point τj+1 . Let µ ∈ (0, ∆τj ) is fixed, then, because of continuity which means uniform continuity of the functions ϕl (s); l = 1, . . . , k at the interval [τj , τj+1 − µ] we have: (41) τZj+1 τj+1 τZj+1 Z −µ 2 2 (ϕl (τj ) − ϕl (s)) ds + (ϕl (τj ) − ϕl (s))2 ds < ε2 (∆τj − µ) + M 2 µ. (ϕl (τj ) − ϕl (s)) ds = τj

τj

τj+1 −µ

EXPANSION OF MULTIPLE ITO STOCHASTIC INTEGRALS

17

Obtaining the inequality (41) we proposed, that ∆τj < δ; j = 0, 1, . . . , N − 1 (δ > 0 is exist for all ε > 0 and it doesn’t depend on s); |ϕl (τj ) − ϕl (s)| < ε if s ∈ [τj+1 − µ, τj+1 ] (because of uniform continuity of functions ϕl (s); l = 1, . . . , k); |ϕl (τj ) − ϕl (s)| < M , M — is a constant; potential point of discontinuity of function ϕl (s) is supposed in the point τj+1 . Performing the passage to the limit in the inequality (41) when µ → +0, we get τZj+1

(ϕl (τj ) − ϕl (s))2 ds ≤ ε2 ∆τj .

τj

Using this estimation for evaluation of the right part (39) we obtain (42) 4    NX  N j−1 N −1 −1 −1 X X X
∆τj ∆τq < 3ε4 δ(T − t) + (T − t)2 . (∆τj )2 + 6 J[∆ϕl ]τj+1 ,τj ≤ ε4 3 M j=0

j=0

j=0

q=0

This implies, that

4   NX −1 J[∆ϕl ]τj+1 ,τj → 0 M j=0

when N → ∞ and lemma 3 remains reasonable. Now, let’s present explanations concerning the rightness of the formula (13), when {φj (x)}∞ j=0 — is a full orthonormal system of functions in the space L2 ([t, T ]), moreover φj (x), j < ∞ satisfies the condition (⋆). Let’s examine the case k = 3 and representation (15). We can demonstrate, that in the studied case the first limit in the right part of (15) equals to zero (similarly we demonstrate, that the second limit in the right part of (15) equals to zero; proving of the second limit equality to zero in the right part of the formula (14) is the same as for the case of continuous functions φj (x); j = 0, 1, . . .). The second moment of prelimit expression of the first limit in the right part of (15) looks as follows: N −1 jX 2 −1 3 −1 jX X

τj2 +1 τj1 +1

Z

Z

j3 =0 j2 =0 j1 =0 τ j2

τj1

(Φ(t1 , t2 , τj3 ) − Φ(t1 , τj2 , τj3 ))2 dt1 dt2 ∆τj3 .

Further, for the fixed µ ∈ (0, ∆τj2 ) and ρ ∈ (0, ∆τj1 ) we have τj2 +1 τj1 +1

Z

Z

τj1

τj2

 τj2Z+1 −µ = +

τj2 +1

Z

τj2 +1 −µ

τj2

 τj1Z+1 −ρ + τj1

τj2 +1 −µ τj1 +1 −ρ

=

Z

τj2

Z

+

τj1

2

(Φ(t1 , t2 , τj3 ) − Φ(t1 , τj2 , τj3 )) dt1 dt2 = τj1 +1

Z

τj1 +1 −ρ



τj2 +1

τj2 +1 −µ τj1 +1

Z

τj2

Z

τj1 +1 −ρ

2

(Φ(t1 , t2 , τj3 ) − Φ(t1 , τj2 , τj3 )) dt1 dt2 =

+

Z

τj2 +1 −µ

τj2 +1

τj1 +1 −ρ

Z

τj1

+

Z

τj1 +1

Z

τj2 +1 −µ τj1 +1 −ρ

!

×

2

× (Φ(t1 , t2 , τj3 ) − Φ(t1 , τj2 , τj3 )) dt1 dt2 < (43)

< ε2 (∆τj2 − µ) (∆τj1 − ρ) + M 2 ρ (∆τj2 − µ) + M 2 µ (∆τj1 − ρ) + M 2 µρ,

where M — is a constant; ∆τj < δ; j = 0, 1, . . . , N − 1 (δ > 0 is exists for all ε > 0 and it doesn’t −1 depend on points (t1 , t2 , τj3 ), (t1 , τj2 , τj3 )); we also propose that, the partition {τj }N j=0 contains all points of discontinuity of the function Φ(t1 , t2 , t3 ) as points τj (for every variable).

18

D.F. KUZNETSOV

When obtaining of (43) we also suppose, that potential points of discontinuity of this function (for every variable) are in points τj1 +1 , τj2 +1 , τj3 +1 . Let’s explain in details how we obtained the inequality (43). Since the function Φ(t1 , t2 , t3 ) is continuous at the closed bounded set Q3 = {(t1 , t2 , t3 ) : t1 ∈ [τj1 , τj1 +1 − ρ], t2 ∈ [τj2 , τj2 +1 − µ], t3 ∈ [τj3 , τj3 +1 − ν], }, where ρ, µ, ν — are fixed small positive numbers (ν ∈ (0, ∆τj3 ), µ ∈ (0, ∆τj2 ), ρ ∈ (0, ∆τj1 )), then this function is also uniformly continous at this set and bounded at closed set D3 . Since the distance between points (t1 , t2 , τj3 ), (t1 , τj2 , τj3 ) ∈ Q3 is obviously less than δ (∆τj < δ; j = 0, 1, . . . , N −1), then |Φ(t1 , t2 , τj3 )−Φ(t1 , τj2 , τj3 )| < ε. This inequality was used during estimation of the first double integral in (43). Estimating of three remaining double integrals we used the feature of limitation of function Φ(t1 , t2 , t3 ) in form of inequality |Φ(t1 , t2 , τj3 ) − Φ(t1 , τj2 , τj3 )| < M. Performing the passage to the limit in the inequality (43) when µ, ρ → +0 we obtain the estimation τj2 +1 τj1 +1

Z

τj2

Z

τj1

2

(Φ(t1 , t2 , τj3 ) − Φ(t1 , τj2 , τj3 )) dt1 dt2 ≤ ε2 ∆τj2 ∆τj1 .

Usage of this estimation provides N −1 jX 3 −1 j2 −1 X X

τj2 +1 τj1 +1

Z

j3 =0 j2 =0 j1 =0 τ j2

≤ ε2

Z

τj1

2

(Φ(t1 , t2 , τj3 ) − Φ(t1 , τj2 , τj3 )) dt1 dt2 ∆τj3 ≤

N −1 jX 3 −1 j2 −1 X X

j3 =0 j2 =0 j1 =0

∆τj1 ∆τj2 ∆τj3 < ε2

(T − t)3 . 6

The last evaluation means, that in the considered case the first limit in the right part of (15) equals to zero (similarly we may demonstrate, that the second limit in the right part of (15) equals to zero). Consequently, formula (13) is reasonable when k = 3 in the analyzed case. Similarly, we perform argumentation for the case when k = 2 and k > 3. Consequently, in theorem 1 we can use complete orthonormal systems of functions {φj (x)}∞ j=0 in the space L2 ([t, T ]), for which φj (x), j < ∞ satisfies the condition (⋆). The example of such system of functions may serve as a complete orthonormal system of Haar functions in the space L2 ([t, T ]) :   1 x−t 1 φ0 (x) = √ , , φnj (x) = √ ϕnj T −t T −t T −t

where n = 0, 1, . . . ; j = 1, 2, . . . , 2n and functions ϕnj (x) has the following form:  n j−1 j−1 1  2 2 , x ∈ [ 2n , 2n + 2n+1 ) n j 1 ; ϕnj (x) = −2 2 , x ∈ [ j−1 2n + 2n+1 , 2n )   0 otherwise

n = 0, 1, . . . ; j = 1, 2, . . . , 2n (we choose the values of Haar functions in the points of discontinuity in order they will be continuous at the right). The other example of similar system of functions is a full orthonormal Rademacher-Walsh system of functions in the space L2 ([t, T ]): 1 φ0 (x) = √ , T −t     1 x−t x−t φm1 ...mk (x) = √ . . . ϕmk , ϕm1 T −t T −t T −t m

where 0 < m1 < . . . < mk ; m1 , . . . , mk = 1, 2, . . . ; k = 1, 2, . . . ; ϕm (x) = (−1)[2 m = 1, 2, . . . ; [y] — integer part of y.

x]

; x ∈ [0, 1];

EXPANSION OF MULTIPLE ITO STOCHASTIC INTEGRALS

19

5. Remarks about usage of complete orthonormal systems in the theorem 1 Note, that actually functions φj (s) of the complete orthonormal system of functions {φj (s)}∞ j=0 in the space L2 ([t, T ]) depend not only on s, but on t, T. For example, complete orthonormal systems of Legendre polynomials and trigonometric functions in the space L2 ([t, T ]) have the following form: r    T +t 2 2j + 1 s− , Pj φj (s, t, T ) = T −t 2 T −t

Pj (s) — Legendre polynomials;

  √

1 1 2πr(s−t) 2sin T −t φj (s, t, T ) = √ T −t √2cos 2πr(s−t) T −t

when j = 0 when j = 2r − 1; when j = 2r

r = 1, 2, . . . . Note, that the specified systems of functions will be used in the context of realizing of numerical methods for Ito stochastic differential equations for the sequences of time intervals: [T0 , T1 ], [T1 , T2 ], [T2 , T3 ], . . . , and spaces L2 ([T0 , T1 ]), L2 ([T1 , T2 ]), L2 ([T2 , T3 ]), . . . . We can explain, that the dependence of functions φj (s, t, T ) from t, T (hereinafter these constants will mean fixed moments of time) will not affect the main characteristics of independence of random variables. ZT (il ) ζ(jl )T,t = φjl (s, t, T )dws(il ) ; il 6= 0; l = 1, . . . , k. t

Indeed, for fixed t, T due to orthonormality of mentioned systems of functions, we have: o n (i ) (i ) M ζ(jll )T,t ζ(jrr )T,t = 1{il =ir 6=0} 1{jl =jr } ,

where

(i ) ζ(jll)T,t

=

ZT t

φjl (s, t, T )dws(il ) ; il 6= 0; l, r = 1, . . . , k.

On the other side random variables ZT2 ZT1 (il ) (il ) (il ) ζ(jl )T1 ,t1 = φjl (s, t1 , T1 )dws , ζ(jl )T2 ,t2 = φjl (s, t2 , T2 )dws(il ) t1

t2

are independent if [t1 , T1 ] ∩ [t2 , T2 ] = ∅ (the case T1 = t2 is possible) according to the property of Ito stochastic integrals. (i ) Therefore, two important characteristics of random variables ζ(jll)T,t which are the basic motive of their usage are stored. 6. Convergence in the mean of degree 2n of expansion of multiple Ito stochastic integrals from the theorem 1 Creating expansions of Ito stochastic integrals from theorem 1 we stored all information about these integrals, that is why it is natural to expect, that the mentioned expansions will be converged not only in the mean-square sense but in the stronger probability meanings. We will obtain the common evaluation which proves convergence in the mean of degree 2n, n ∈ N of approximations from the theorem 1. According to notations of the theorem 1:

20

D.F. KUZNETSOV

p1 ,...,pk RT,t

(44)

X

=

ZT

(t1 ,...,tk ) t

...

Zt2

(i )

(i )

Rp1 ...pk (t1 , . . . , tk )dft1 1 . . . dftkk ,

t

def

Rp1 ...pk (t1 , . . . , tk ) = K(t1 , . . . , tk ) −

p1 X

pk X

...

jk =0

j1 =0

Cjk ...j1

k Y

φjl (tl ).

l=1

For definiteness we will consider, that i1 , . . . , ik = 1, . . . , m (it is obviously quite enough for unified Taylor-Ito expansion [6] - [11] and we can see decoding of other notations used in this section at the text of proving of the theorem 1. Note, that proving of the theorem 1 we obtained, that Z p1 ,...,pk 2 M{(RT,t ) } ≤ Ck Rp21 ...pk (t1 , . . . , tk )dt1 . . . dtk , [t,T ]k

Ck < ∞ and Ck = k! for the case i1 , . . . , ik = 1, . . . , m. Assume, that (l−1) def ηtl ,t =

Ztl

...

t

Zt2

(i

(i )

)

l−1 ; l = 2, 3, . . . , k + 1, Rp1 ...pk (t1 , . . . , tk )dft1 1 . . . dftl−1

t

(k)

def

(k)

ηtk+1 ,t = ηT,t , (k) def ηT,t =

ZT

...

t

Zt2

(i )

(i )

Rp1 ...pk (t1 , . . . , tk )dft1 1 . . . dftkk .

t

Using Ito formula it is easy to demonstrate, that M

Zt

ξτ dfτ

t0

2n 

= n(2n − 1)

Zt

M

t0

 Zs t0

2n−2  ξs2 ds. ξu dfu

Using the Holder inequality in the right part under the sign of integration if p = n/(n − 1), q = n and using the increase of value 2n  Zt M ξτ dfτ t0

with the growth t, we get: 2n (n−1)/n Zt 2n   Zt Zt ≤ n(2n − 1) M (M{ξs2n })1/n ds. M ξτ dfτ ξτ dfτ t0

t0

t0

Raising to power n the obtained inequality and dividing it on 2n n−1  Zt M ξτ dfτ t0

we get the following estimation (45)

M

Zt t0

ξτ dfτ

2n 

n

≤ (n(2n − 1))

Zt t0

(M{ξs2n })1/n ds

n

.

EXPANSION OF MULTIPLE ITO STOCHASTIC INTEGRALS

21

Using estimation (45) we have (k)

M{(ηT,t )2n } ≤ (n(2n − 1))n ≤ (n(2n − 1))n

ZT 

(n(2n − 1))n

t

ZT t

(k−1)

(M{(ηtk ,t )2n })1/n dtk

Ztk

(k−2)

(M{(ηtk−1 ,t )2n })1/n dtk−1

t

n

≤

n 1/n

dtk

n

=

ZT Ztk n (k−2) 2n 1/n 2n = (n(2n − 1)) ≤ ... (M{(ηtk−1 ,t ) }) dtk−1 dtk t

n(k−1)

. . . ≤ (n(2n − 1))

t

n(k−1)

= (n(2n − 1))

t

ZT Ztk t

(2n − 1)!!

n(k−1)

≤ (n(2n − 1))

n Zt3 (1) 2n 1/n . . . (M{(ηt2 ,t ) }) dt3 . . . dtk−1 dtk = t

ZT

...

t

(2n − 1)!!

Zt2

Rp21 ...pk (t1 , . . . , tk )dt1 . . . dtk

t

 Z

Rp21 ...pk (t1 , . . . , tk )dt1

. . . dtk

[t,T ]k

n

n

≤

.

The next to last step was obtained using the formula Zt2 n (1) M{(ηt2 ,t )2n } = (2n − 1)!! Rp21 ...pk (t1 , . . . , tk )dt1 , t

which follows from gaussianity of (1) ηt2 ,t

=

Zt2

(i )

Rp1 ...pk (t1 , . . . , tk )dft1 1 .

t

Similarly we estimate each summand in the right part of (44). Then, from (44) using Minkowski inequality we finally get n 1/2n 2n    Z p1 ,...,pk 2n M{(RT,t ) } ≤ k! (n(2n − 1))n(k−1) (2n − 1)!! = Rp21 ...pk (t1 , . . . , tk )dt1 . . . dtk [t,T ]k

= (k!)2n (n(2n − 1))n(k−1) (2n − 1)!!

(46)

 Z

Rp21 ...pk (t1 , . . . , tk )dt1 . . . dtk

[t,T ]k

We have: Z (47)

Rp21 ...pk (t1 , . . . , tk )dt1

. . . dtk =

Z

2

K (t1 , . . . , tk )dt1 . . . dtk −

[t,T ]k

[t,T ]k

p1 X

...

n

.

pk X

Cj2k ...j1 .

j1 =0

jk =0

p1 X

pk X

Let’s substitute (47) into (46): p1 ,...,pk 2n M{(RT,t ) }≤

(48)

≤ (k!)2n (n(2n − 1))n(k−1) (2n − 1)!!

 Z

[t,T ]k

K 2 (t1 , . . . , tk )dt1 . . . dtk −

j1 =0

...

jk =0

Cj2k ...j1

n

.

22

D.F. KUZNETSOV

The inequality (46) (or (48)) means, that approximations of multiple Ito stochastic integrals, obtained using the theorem 1, converge in the mean of degree 2n, n ∈ N, as according to Parseval equality Z Rp21 ...pk (t1 , . . . , tk )dt1 . . . dtk → 0 if p1 , . . . , pk → ∞.

[t,T ]k

References [1] Kloeden P.E., Platen E. Numerical Solution of Stochastic Differential Equations. Springer, Berlin, 1995. 632 pp. [2] Milstein G.N. Numerical Integration of Stochastic Differential Equations. Ural University Press, Sverdlovsk, 1988. 225 pp. [3] Milstein G.N., Tretyakov M.V. Stochastic Numerics for Mathematical Physics. Springer, Berlin, 2004. 616 pp. [4] Kulchitskiy O.Yu., Kuznetsov D.F. Approximation of multiple Ito stochastic integrals. (In Russian). VINITI. 1679-V94 (1994), 38 pp. [5] Kuznetsov D.F. A method of expansion and approximation of repeated stochastic Stratonovich integrals based on multiple Fourier series on full orthonormal systems. (In Russian). Electronic Journal Differential Equations and Control Processes. 1997, no. 1. 18-77. Available at: http://www.math.spbu.ru/diffjournal/pdf/j002.pdf [6] Kuznetsov D.F. Numerical Integration of Stochastic Differential Equations. 2 (In Russian). Polytechnical University Publishing House: St.-Petersburg, 2006, 764 pp. (DOI: 10.18720/SPBPU/2/s17-227). Available at: http://www.sde-kuznetsov.spb.ru/downloads/kuz_2006.pdf [7] Kuznetsov D.F. Stochastic Differential Equations: Theory and Practice of Numerical Solution. With MatLab programs. 4th Ed. (In Russian). Polytechnical University Publishing House: St.-Petersburg, 2010, XXX+786 pp. (DOI: 10.18720/SPBPU/2/s17-231). Available at: http://www.sde-kuznetsov.spb.ru/downloads/4_ed_kuz.pdf [8] Dmitriy F. Kuznetsov. Strong Approximation of Multiple Ito and Stratonovich Stochastic Integrals: Multiple Fourier Series Approach. 2nd Ed. (In English). Polytechnical University Publishing House, St.-Petersburg, 2011, 284 pp. (DOI: 10.18720/SPBPU/2/s17-233). Available at: http://www.sde-kuznetsov.spb.ru/downloads/kuz_2011_2_ed.pdf [9] Dmitriy F. Kuznetsov. Multiple Ito and Stratonovich Stochastic Integrals: Approximations, Properties, Formulas. (In English). Polytechnical University Publishing House, St.-Petersburg, 2013, 382 pp. (DOI: 10.18720/SPBPU/2/s17-234). Available at: http://www.sde-kuznetsov.spb.ru/downloads/kuz_2013.pdf [10] Dmitriy F. Kuznetsov. Multiple Ito and Stratonovich Stochastic Integrals: Fourier-Legendre and Trigonometric Expansions, Approximations, Formulas (In English). Electronic Journal Differential Equations and Control Processes, no. 1, 2017, 385 (A.1 - A.385) pp. (DOI: 10.18720/SPBPU/2/z17-3). Available at: http://www.math.spbu.ru/diffjournal/pdf/kuznetsov_book2.pdf [11] Kuznetsov D.F. Stochastic Differential Equations: Theory and Practice of Numerical Solution. With MatLab Programs. 5th Ed. (In Russian). Electronic Journal "Differential Equations and Control Processes, no. 2, 2017, 1000 (A.1 - A.1000) pp. (DOI: 10.18720/SPBPU/2/z17-4). Available at: http://www.math.spbu.ru/diffjournal/pdf/kuznetsov_book3.pdf [12] Skhorokhod A.V. Stochastc Processes with Independent Augments. (In Russian). Moscow, Nauka Publ., 1964. 280 pp. [13] Gichman I.I., Skorochod A.V. Stochastic Differential Equations and its Applications. Naukova Dumka, Kiev, 1982. 612 pp. Dmitriy Feliksovich Kuznetsov Peter the Great Saint-Petersburg Polytechnic University, Polytechnicheskaya ul., 29, 195251, Saint-Petersburg, Russia E-mail address: [email protected]