arXiv:1801.01079v2 [math.PR] 5 Jun 2018
EXACT CALCULATION OF MEAN-SQUARE ERROR OF APPROXIMATION OF MULTIPLE ITO STOCHASTIC INTEGRALS FOR THE METHOD, BASED ON THE MULTIPLE FOURIER SERIES DMITRIY F. KUZNETSOV
Abstract. The article is devoted to the obtainment of exact and approximate expressions for mean-square error of approximation of multiple Ito stochastic integrals from the stochastic Taylor-Ito expansion for the method, based on generalized multiple Fourier series. As a result, we won’t need to consider redundant terms of expansions of multiple Ito stochastic integrals, that may complicate numerical methods for Ito stochastic differential equations. The results of the article may be applied to numerical integration of Ito stochastic differential equations.
1. Introduction In this article we develop the method of expansion and mean-square approximation of multiple Ito stochastic integrals, based on generalized multiple Fourier series, converging in the mean, which was proposed by author [1] - [9]. Hereinafter, this method referred to as the method of multiple Fourier series. The question about how estimate or even calculate exactly mean-square error of approximation of multiple Ito stochastic integrals for the method of multiple Fourier series composes the subject of the article. From the one side the mentioned question is essentially difficult in the case of multi-dimensional Wiener process, because of we need to take into account different combinations of components of multi-dimensional Wiener process. From the other side the effective solution of the mentioned problem allows to constructing more economic numerical methods for Ito stochastic differential equations. In the section 1 we give the formulation of the main theorem (theorem 1), which is represents the base of the method of multiple Fourier series [1] and introduce denotations. Section 2 is devoted to the formulation and proof of the theorem (theoren 2) which allows to calculate exacly the mean-square error of approximation of multiple Ito stochastic integrals of arbitrary multiplicity k for the method of multiple Fourier series. In [8] we consider the scheme of the proof, but here we give the complete proof of the theorem 2. The particular cases (k = 1, . . . , 5) of the theorem from the section 2 are considered in details in the section 3. In the secton 4 we prove the effective estimate for the mean-square error of approximation of multiple Ito stochastic integrals of arbitrary multiplicity k for the method of multiple Fourier series (theorem 3). The results of the article can be useful for numerical integration of Ito stochastic differential equations in accordance with the strong (mean-square) criterion [10] - [12]. 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.
Mathematics Subject Classification: 60H05, 60H10, 42B05. Keywords: Multiple Ito stochastic integral, Multiple Fourier series, Mean-Square Error, Approximation, Expansion. 1
2
D.F. KUZNETSOV
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 p we can write down: Y X X p p k n n X X (i1 ...ik ) (il ) (k) n Cjk ...j1 ζjl − Sj1 ,...,jk = + ... + J[ψ ]T,t = jk =0
j1 =p+1
j1 =0
jk =p+1
l=1
p+1,n = J[ψ (k) ]pT,t + ξ[ψ (k) ]T,t .
(17)
(i)
Not difficult to understand (see (9) – (11)) that due to independence of random variables ζj (i ...i ) Sj11,...,jkk
different j (i 6= 0) and due to special structure of random variables are correct: Y k (i ) (i ...i ) (18) M ζjl l − Sj11,...,jkk = 0,
the following relations
l=1
(19)
M
Y k
l=1
where
(i )
(i ...i )
ζjl l − Sj11,...,jkk
Y k l=1
(i )
(i ...i )
ζj ′ l − Sj ′ 1,...,jk′ l
1
k
= 0,
(j1 , . . . , jk ) ∈ Kp , (j1′ , . . . , jk′ ) ∈ Kn \Kp
and
Kn = {(j1 , . . . , jk ) : 0 ≤ j1 , . . . , jk ≤ n} , Kp = {(j1 , . . . , jk ) : 0 ≤ j1 , . . . , jk ≤ p} . From (18) and (19) we obtain: o n p+1,n = 0. M J[ψ (k) ]pT,t ξ[ψ (k) ]T,t Due to (16), (13) and (17) we can write down:
p+1,n ξ[ψ (k) ]T,t = J[ψ (k) ]nT,t − J[ψ (k) ]pT,t , def
p+1,n l.i.m. ξ[ψ (k) ]T,t = J[ψ (k) ]T,t − J[ψ (k) ]pT,t = ξ[ψ (k) ]p+1 T,t . n→∞
We have: n o (k) p 0 ≤ M ξ[ψ (k) ]p+1 ]T,t = T,t J[ψ
n o p+1,n (k) p+1,n = M ξ[ψ (k) ]p+1 ]T,t + ξ[ψ (k) ]T,t J[ψ (k) ]pT,t = T,t − ξ[ψ
n o n o p+1,n (k) p+1,n ≤ M ξ[ψ (k) ]p+1 ]T,t J[ψ (k) ]pT,t + M ξ[ψ (k) ]T,t J[ψ (k) ]pT,t = T,t − ξ[ψ
≤
s
M
n o = M J[ψ (k) ]T,t − J[ψ (k) ]nT,t J[ψ (k) ]pT,t ≤
J[ψ (k) ]
T,t
−
J[ψ (k) ]nT,t
for
s 2 2 p (k) ≤ M J[ψ ]T,t
6
D.F. KUZNETSOV
≤ ×
s
(20)
M
≤K
s
M
2 × J[ψ (k) ]T,t − J[ψ (k) ]nT,t
J[ψ (k) ]pT,t
s
M
−
J[ψ (k) ]
T,t
2
+
r
M
n
J[ψ (k) ]
T,t
2 o
!
≤
2 J[ψ (k) ]T,t − J[ψ (k) ]nT,t → 0 if n → ∞,
where K is a constant. From (20) it follows: or M
n
n o (k) p M ξ[ψ (k) ]p+1 J[ψ ] T,t T,t = 0
o J[ψ (k) ]T,t − J[ψ (k) ]pT,t J[ψ (k) ]pT,t = 0.
The last equality means that n o 2 (k) (k) p (k) p (21) M J[ψ ]T,t J[ψ ]T,t = M J[ψ ]T,t .
Taking into account (21) we obtain 2 2 + =M J[ψ (k) ]T,t M J[ψ (k) ]T,t − J[ψ (k) ]pT,t
2 o 2 n (k) (k) (k) p (k) p − J[ψ ]T,t − 2M J[ψ ]T,t J[ψ ]T,t = M +M J[ψ ]T,t n o −M J[ψ (k) ]T,t J[ψ (k) ]pT,t =
(22)
=
Z
[t,T ]k
o n K 2 (t1 , . . . , tk )dt1 . . . dtk − M J[ψ (k) ]T,t J[ψ (k) ]pT,t .
Let’s calculate the value
n o M J[ψ (k) ]T,t J[ψ (k) ]pT,t .
(i ...i )
Note that the value Sj11,...,jkk consists of finite sum of constants and multiple Itˆo stochastic integrals with multiplicities less, than k (see (9) – (11)). Then from standard moment proterties of stochastic integrals [13] we obtain o n (i ...i ) M J[ψ (k) ]T,t Sj11,...,jkk = 0. In result we have: Y p p k n o X X (i ) (i ...i ) = ζjl l − Sj11,...,jkk Cjk ...j1 M J[ψ (k) ]T,t ... M J[ψ (k) ]T,t J[ψ (k) ]pT,t = j1 =0
(23)
=
p X
j1 =0
...
p X
jk =0
jk =0
l=1
k Y (il ) (k) . ζjl Cjk ...j1 M J[ψ ]T,t l=1
Let’s sustitute (23) into (22). Then 2 (k) (k) p = M J[ψ ]T,t − J[ψ ]T,t
EXACT CALCULATION OF MEAN-SQUARE ERROR OF APPROXIMATION
(24)
Z
=
K 2 (t1 , . . . , tk )dt1 . . . dtk −
[t,T ]k
Let’s consider the value
p X
...
p X
jk =0
j1 =0
7
k Y (i ) Cjk ...j1 M J[ψ (k) ]T,t ζjl l . l=1
k Y (il ) (k) . M J[ψ ]T,t ζjl l=1
Using the Ito formula we get: k Y
(i ) ζjl l
l=1
(25)
=
ZT
X
=
ZT
(i ) φjk (tk )dftkk
t
φjk (tk ) . . .
(j1 ,...,jk ) t (i ...i ) Gj11,...,jkk
where less, than k;
...
ZT
(i )
φj1 (t1 )dft1 1 =
t
Zt2
(i )
(i )
(i ...i )
φj1 (t1 )dft1 1 . . . dftkk + Gj11,...,jkk w. p. 1,
t
consists of finite sum of constants and multiple Itˆo stochastic integrals with multiplicities X
(j1 ,...,jk )
means the sum according to all possible derangements (j1 , . . . , jk ), at the same time if jr changed places with jq in the derangement (j1 , . . . , jk ), then ir changes places with iq in the derangement (i1 , . . . , ik ); another denotations see in the theorem 1. As we noted above o n (i ...i ) (26) M J[ψ (k) ]T,t Gj11,...,jkk = 0. For example, for the case k = 3 we have: ZT
(i )
φj3 (t3 )dft3 3
φj3 (t3 )
t
+
ZT
+
φj2 (t2 )
t
Zt2
φj3 (t3 )
t
Zt3
ZT
φj1 (t3 )
=
t
φj2 (t3 )
t
t
Zt2
φj2 (t3 )
t
Zt2
φj3 (t1 )dft1 3 dft2 1 dft3 2 +
t
Zt3
φj1 (t2 )
t
ZT
Zt2
ZT
φj1 (t3 )
Zt3
φj3 (t2 )
Zt2
φj2 (t1 )dft1 2 dft2 3 dft3 1 +
t
φj1 (t2 )
φj2 (t1 )dft1 2 dft2 1 dft3 3 +
t
Zt2
Zt3
φj2 (t2 )
(i ) (i ) (i ) φj1 (t1 )dft1 1 dft2 2 dft3 3
(i )
t
ZT
φj1 (t3 )
t ZT t
(i )
(i )
(i )
t
φj3 (t3 )
+1{i2 =i3 6=0}
(i )
(i )
φj3 (t1 )dft1 3 dft2 2 dft3 1 +
Zt3
+1{i1 =i2 6=0}
(i )
φj1 (t1 )dft1 1 = Zt3
+
t
ZT t
ZT
ZT
+1{i1 =i3 6=0}
(i )
φj2 (t2 )dft2 2
t
t
Zt3
ZT
ZT
t
t
+ 1{i1 =i3 6=0}
t
ZT
Zt3
+ 1{i2 =i3 6=0}
ZT
(i ) φj1 (t2 )φj2 (t2 )dt2 dft3 3
t
(i ) φj2 (t2 )φj3 (t2 )dt2 dft3 1
t
φj3 (t3 )φj1 (t3 )
Zt3 t
(i )
φj2 (t1 )dft1 2 dt3 + 1{i1 =i2 6=0}
t ZT t
(i )
(i )
(i )
(i )
(i )
(i )
(i )
(i )
(i )
φj3 (t2 )
φj1 (t1 )dft1 1 dft2 3 dft3 2 +
t
t
t
φj2 (t3 )
Zt3
(i )
φj1 (t2 )φj3 (t2 )dt2 dft3 2 +
t
φj3 (t3 )φj2 (t3 )
φj1 (t3 )φj2 (t3 )
Zt3
t Zt3 t
(i )
φj1 (t1 )dft1 1 dt3 +
(i )
φj3 (t1 )dft1 3 dt3 =
8
D.F. KUZNETSOV
=
X
ZT
φj3 (t3 )
(j1 ,j2 ,j3 ) t
Zt3
φj2 (t2 )
t
Zt2
(i )
(i )
(i )
(i i i )
φj1 (t1 )dft1 1 dft2 2 dft3 3 + Gj11,j22 ,j33 w. p. 1.
t
From (2), (25) and (26) we obtain: Z 2 (k) (k) p M J[ψ ]T,t − J[ψ ]T,t =
K 2 (t1 , . . . , tk )dt1 . . . dtk −
[t,T ]k
−
p X
j1 =0
...
p X
jk =0
Cjk ...j1 M J[ψ (k) ]T,t
ZT
X
φjk (tk ) . . .
(j1 ,...,jk ) t
The theorem 1 is proven.
Zt2
(i )
(i )
φj1 (t1 )dft1 1 . . . dftkk
t
.
3. Exact calculation of mean-square errors of approximation for the cases k = 1, . . . , 5 Let’s denote Z
def
M{(J[ψ (k) ]T,t − J[ψ (k) ]pT,t )2 } = Ekp ,
def
K 2 (t1 , . . . , tk )dt1 . . . dtk = Ik .
[t,T ]k
3.1. The case k = 1. In this case from the theorem 1 we obtain: p X Cj21 . E1p = I1 − j1 =0
3.2. The case k = 2. In this case from the theorem 1 we obtain: (I). i1 6= i2 : (27)
E2p
(II). i1 = i2 : E2p = I2 −
= I2 −
p X
j1 ,j2 =0
p X
Cj22 j1 ,
j1 ,j2 =0
Cj22 j1 −
p X
Cj2 j1 Cj1 j2 .
j1 ,j2 =0
Example 1. Let’s consider the following double Ito stochastic integral from the stochastic TaylorIto expansion [10] - [12]: (28)
(i1 i2 ) I(00)T,t
=
ZT Zt2 t
(i )
(i )
dft1 1 dft2 2 ,
t
where i1 , i2 = 1, . . . , m. Approximation, based on the expansion (9) for the integral (28) (case of Legendre polynomials) has the following form [1]: p T − t (i1 ) (i2 ) X T −t 1 (i1 i2 )p (i1 ) (i2 ) (i1 ) (i2 ) √ (29) I(00)T,t = − 1{i1 =i2 } ζ0 ζ0 + ζi−1 ζi − ζi ζi−1 . 2 2 2 4i − 1 i=1
EXACT CALCULATION OF MEAN-SQUARE ERROR OF APPROXIMATION
9
Using (27) we obtain [1]: p 2 (T − t)2 1 X 1 (i1 i2 )p (i1 i2 ) (i1 6= i2 ). − = − I(00)T,t (30) M I(00)T,t 2 2 i=1 4i2 − 1 3.3. The case k = 3. In this case from the theorem 1 we obtain: (I). i1 6= i2 , i1 6= i3 , i2 6= i3 : p X Cj23 j2 j1 , E3p = I3 − j3 ,j2 ,j1 =0
(II). i1 = i2 = i3 :
E3p = I3 − (III).1. i1 = i2 6= i3 : E3p = I3 −
(31)
E3p = I3 −
Cj23 j2 j1 −
p X
Cj23 j2 j1 −
p X
Cj23 j2 j1 −
j3 ,j2 ,j1 =0
X
(j1 ,j2 ,j3 )
p X
j3 ,j2 ,j1 =0
(III).3. i1 = i3 6= i2 :
Cj3 j2 j1
j1 ,j2 ,j3 =0
j3 ,j2 ,j1 =0
(III).2. i1 6= i2 = i3 : E3p = I3 −
p X
Cj3 j2 j1 ,
p X
Cj3 j1 j2 Cj3 j2 j1 ,
p X
Cj2 j3 j1 Cj3 j2 j1 ,
p X
Cj3 j2 j1 Cj1 j2 j3 .
j3 ,j2 ,j1 =0
j3 ,j2 ,j1 =0
j3 ,j2 ,j1 =0
Example 2. Let’s consider the following triple Ito stochastic integral from the stochastic Taylor-Ito expansion [10] - [12]: (i1 i2 i3 ) I(000)T,t
(32)
=
ZT Zt3 Zt2 t
t
(i )
(i )
(i )
dft1 1 dft2 2 dft3 3 ,
t
where i1 , i2 , i3 = 1, . . . , m. Approximation, based on the expansion (10) for the integral (32) (case of Legendre polynomials; p1 = p2 = p3 = p) has the following form [9]: (i i i )p
1 2 3 = I(000)T,t
p X
j1 ,j2 ,j3 =0
(33)
(i )
(i ) (i ) (i )
Cj3 j2 j1 ζj11 ζj22 ζj33 − 1{i1 =i2 } 1{j1 =j2 } ζj33 −
(i ) −1{i2 =i3 } 1{j2 =j3 } ζj11
−
(i ) 1{i1 =i3 } 1{j1 =j3 } ζj22
!
,
where (34)
p (2j1 + 1)(2j2 + 1)(2j3 + 1) Cj3 j2 j1 = (T − t)3/2 C¯j3 j2 j1 , 8 Zy Zz Z1 ¯ Cj3 j2 j1 = Pj3 (z) Pj2 (y) Pj1 (x)dxdydz, −1
−1
−1
where Pi (x) is the Legendre polynomial (i = 0, 1, 2, . . .).
10
D.F. KUZNETSOV
For example, using (31) we obtain: p p 2 (T − t)3 X X (i1 i2 i3 )p (i1 i2 i3 ) 2 − Ckji − = Ckij Ckji (i1 = i2 6= i3 ). M I(000)T,t − I(000)T,t 6 i,j,k=0
i,j,k=0
As mentioned in [9], the exact values of coefficients C¯kji when i, j, k = 0, 1, . . . , p can be calculated using DERIVE (computer packs of symbol transformations). In [9] we can find tables of FourierLegendre coefficients for approximations of multiple Ito stochastic integrals of multiplicity 1-5. For the case i1 = i2 = i3 it is comfortable to use the following formula [10] - [12]: 3 1 (i1 i1 i1 ) (i1 ) (i1 ) 3/2 w. p. 1. I(000)T,t = (T − t) ζ0 − 3ζ0 6 3.4. The case k = 4. In this case from the theorem 1 we obtain: (I). i1 , . . . , i4 — are pairwise different: E4p = I4 −
p X
Cj24 ...j1 ,
j1 ,...,j4 =0
(II). i1 = i2 = i3 = i4 : E4p
= I4 −
Cj4 ...j1
j1 ,...,j4 =0
(III).1. i1 = i2 6= i3 , i4 ; i3 6= i4 : E4p
p X
= I4 −
(III).2. i1 = i3 6= i2 , i4 ; i2 6= i4 : E4p = I4 − (III).3. i1 = i4 6= i2 , i3 ; i2 6= i3 : E4p = I4 − (III).4. i2 = i3 6= i1 , i4 ; i1 6= i4 : E4p = I4 − (III).5. i2 = i4 6= i1 , i3 ; i1 6= i3 : E4p = I4 − (III).6. i3 = i4 6= i1 , i2 ; i1 6= i2 : E4p = I4 −
X
(j1 ,...,j4 )
p X
Cj4 ...j1
p X
Cj4 ...j1
p X
Cj4 ...j1
p X
Cj4 ...j1
p X
Cj4 ...j1
p X
Cj4 ...j1
j1 ,...,j4 =0
Cj4 ...j1 ,
X
Cj4 ...j1 ,
X
Cj4 ...j1 ,
X
Cj4 ...j1 ,
X
Cj4 ...j1 ,
X
Cj4 ...j1 ,
(j1 ,j3 )
j1 ,...,j4 =0
(j1 ,j4 )
j1 ,...,j4 =0
(j2 ,j3 )
j1 ,...,j4 =0
j1 ,...,j4 =0
X
(j1 ,j2 )
j1 ,...,j4 =0
Cj4 ...j1 ,
(j2 ,j4 )
(j3 ,j4 )
EXACT CALCULATION OF MEAN-SQUARE ERROR OF APPROXIMATION
(IV).1. i1 = i2 = i3 6= i4 : E4p = I4 − (IV).2. i2 = i3 = i4 6= i1 : E4p
= I4 −
(IV).3. i1 = i2 = i4 6= i3 : E4p
= I4 −
(IV).4. i1 = i3 = i4 6= i2 : E4p = I4 − (V).1. i1 = i2 6= i3 = i4 : E4p = I4 − (V).2. i1 = i3 6= i2 = i4 : E4p
= I4 −
(V).3. i1 = i4 6= i2 = i3 : E4p = I4 −
p X
Cj4 ...j1
j1 ,...,j4 =0
p X
Cj4 ...j1 ,
X
Cj4 ...j1 ,
X
Cj4 ...j1 ,
X
Cj4 ...j1 ,
(j1 ,j2 ,j3 )
Cj4 ...j1
j1 ,...,j4 =0
(j2 ,j3 ,j4 )
p X
Cj4 ...j1
p X
Cj4 ...j1
j1 ,...,j4 =0
(j1 ,j2 ,j4 )
j1 ,...,j4 =0
p X
X
(j1 ,j3 ,j4 )
Cj4 ...j1
j1 ,...,j4 =0
Cj4 ...j1
X X
, Cj4 ...j1
X X
Cj4 ...j1
(j1 ,j2 )
p X
Cj4 ...j1
p X
Cj4 ...j1
j1 ,...,j4 =0
(j1 ,j3 )
j1 ,...,j4 =0
,
X X
(j1 ,j4 )
(j3 ,j4 )
(j2 ,j4 )
(j2 ,j3 )
.
3.5. The case k = 5. In this case from the theorem 1 we obtain: (I). i1 , . . . , i5 — are pairwise different: E5p = I5 −
p X
j1 ,...,j5 =0
(II). i1 = i2 = i3 = i4 = i5 : E5p = I5 −
p X
Cj25 ...j1 ,
Cj5 ...j1
j1 ,...,j5 =0
X
(j1 ,...,j5 )
Cj5 ...j1 ,
(III).1. i1 = i2 6= i3 , i4 , i5 (i3 , i4 , i5 — are pairwise different): X p X p Cj5 ...j1 , Cj5 ...j1 E5 = I5 − j1 ,...,j5 =0
(j1 ,j2 )
(III).2. i1 = i3 6= i2 , i4 , i5 (i2 , i4 , i5 — are pairwise different): X p X Cj5 ...j1 , Cj5 ...j1 E5p = I5 − j1 ,...,j5 =0
(j1 ,j3 )
11
12
D.F. KUZNETSOV
(III).3. i1 = i4 6= i2 , i3 , i5 (i2 , i3 , i5 — are pairwise different): X p X p Cj5 ...j1 , Cj5 ...j1 E5 = I5 − j1 ,...,j5 =0
(j1 ,j4 )
(III).4. i1 = i5 6= i2 , i3 , i4 (i2 , i3 , i4 — are pairwise different): X p X p Cj5 ...j1 , Cj5 ...j1 E5 = I5 − j1 ,...,j5 =0
(j1 ,j5 )
(III).5. i2 = i3 6= i1 , i4 , i5 (i1 , i4 , i5 — are pairwise different): X p X p Cj5 ...j1 , Cj5 ...j1 E5 = I5 − j1 ,...,j5 =0
(j2 ,j3 )
(III).6. i2 = i4 6= i1 , i3 , i5 (i1 , i3 , i5 — are pairwise different): X p X p Cj5 ...j1 , Cj5 ...j1 E5 = I5 − j1 ,...,j5 =0
(j2 ,j4 )
(III).7. i2 = i5 6= i1 , i3 , i4 (i1 , i3 , i4 — are pairwise different): X p X p Cj5 ...j1 , Cj5 ...j1 E5 = I5 − j1 ,...,j5 =0
(j2 ,j5 )
(III).8. i3 = i4 6= i1 , i2 , i5 (i1 , i2 , i5 — are pairwise different): X p X p Cj5 ...j1 , Cj5 ...j1 E5 = I5 − j1 ,...,j5 =0
(j3 ,j4 )
(III).9. i3 = i5 6= i1 , i2 , i4 (i1 , i2 , i4 — are pairwise different): X p X p Cj5 ...j1 , Cj5 ...j1 E5 = I5 − j1 ,...,j5 =0
(j3 ,j5 )
(III).10. i4 = i5 6= i1 , i2 , i3 (i1 , i2 , i3 — are pairwise different): X p X p Cj5 ...j1 , Cj5 ...j1 E5 = I5 − j1 ,...,j5 =0
(IV).1. i1 = i2 = i3 6= i4 , i5 (i4 6= i5 ): E5p
= I5 −
p X
Cj5 ...j1
p X
Cj5 ...j1
j1 ,...,j5 =0
X
Cj5 ...j1 ,
X
Cj5 ...j1 ,
X
Cj5 ...j1 ,
(j1 ,j2 ,j3 )
j1 ,...,j5 =0
(IV).3. i1 = i2 = i5 6= i3 , i4 (i3 6= i4 ): E5p = I5 −
Cj5 ...j1
j1 ,...,j5 =0
(IV).2. i1 = i2 = i4 6= i3 , i5 (i3 6= i5 ): E5p = I5 −
p X
(j4 ,j5 )
(j1 ,j2 ,j4 )
(j1 ,j2 ,j5 )
EXACT CALCULATION OF MEAN-SQUARE ERROR OF APPROXIMATION
(IV).4. i2 = i3 = i4 6= i1 , i5 (i1 6= i5 ): E5p
= I5 −
p X
(IV).5. i2 = i3 = i5 6= i1 , i4 (i1 6= i4 ): E5p
= I5 −
p X
= I5 −
= I5 −
= I5 −
= I5 −
= I5 −
(V).1. i1 = i2 = i3 = i4 6= i5 : E5p
= I5 −
(V).2. i1 = i2 = i3 = i5 6= i4 : E5p
= I5 −
(V).3. i1 = i2 = i4 = i5 6= i3 : E5p = I5 − (V).4. i1 = i3 = i4 = i5 6= i2 : E5p = I5 −
p X
Cj5 ...j1
Cj5 ...j1
Cj5 ...j1
j1 ,...,j5 =0
Cj5 ...j1 ,
X
Cj5 ...j1 ,
X
Cj5 ...j1 ,
X
Cj5 ...j1 ,
(j1 ,j4 ,j5 )
X
Cj5 ...j1 ,
X
Cj5 ...j1 ,
X
Cj5 ...j1 ,
X
Cj5 ...j1 ,
(j1 ,j2 ,j3 ,j4 )
p X
Cj5 ...j1
p X
Cj5 ...j1
p X
Cj5 ...j1
j1 ,...,j5 =0
(j1 ,j2 ,j3 ,j5 )
j1 ,...,j5 =0
j1 ,...,j5 =0
X
(j1 ,j3 ,j4 )
j1 ,...,j5 =0
p X
Cj5 ...j1 ,
(j1 ,j3 ,j5 )
j1 ,...,j5 =0
(IV).10. i1 = i4 = i5 6= i2 , i3 (i2 6= i3 ): E5p
Cj5 ...j1
p X
X
(j3 ,j4 ,j5 )
j1 ,...,j5 =0
(IV).9. i1 = i3 = i4 6= i2 , i5 (i2 6= i5 ): E5p
Cj5 ...j1
p X
Cj5 ...j1 ,
(j2 ,j4 ,j5 )
j1 ,...,j5 =0
(IV).8. i1 = i3 = i5 6= i2 , i4 (i2 6= i4 ): E5p
Cj5 ...j1
p X
X
(j2 ,j3 ,j5 )
j1 ,...,j5 =0
(IV).7. i3 = i4 = i5 6= i1 , i2 (i1 6= i2 ): E5p
Cj5 ...j1
p X
Cj5 ...j1 ,
(j2 ,j3 ,j4 )
j1 ,...,j5 =0
(IV).6. i2 = i4 = i5 6= i1 , i3 (i1 6= i3 ): E5p
Cj5 ...j1
j1 ,...,j5 =0
X
(j1 ,j2 ,j4 ,j5 )
(j1 ,j3 ,j4 ,j5 )
13
14
D.F. KUZNETSOV
(V).5. i2 = i3 = i4 = i5 6= i1 : E5p
= I5 −
(VI).1. i5 6= i1 = i2 6= i3 = i4 6= i5 : E5p
= I5 −
= I5 −
= I5 −
= I5 −
= I5 −
= I5 −
= I5 −
= I5 −
p X
Cj5 ...j1
p X
p X
p X
, Cj5 ...j1
X X
, Cj5 ...j1
X X
, Cj5 ...j1
X X
, Cj5 ...j1
X X
, Cj5 ...j1
X X
, Cj5 ...j1
X X
, Cj5 ...j1
X X
, Cj5 ...j1
X X
Cj5 ...j1
,
X X
Cj5 ...j1
,
(j1 ,j2 )
Cj5 ...j1
(j1 ,j5 )
Cj5 ...j1
(j2 ,j5 )
Cj5 ...j1
(j2 ,j5 )
p X
Cj5 ...j1
p X
Cj5 ...j1
(j1 ,j2 )
j1 ,...,j5 =0
Cj5 ...j1 ,
X X
(j1 ,j4 )
j1 ,...,j5 =0
(VI).9. i3 6= i2 = i4 6= i1 = i5 6= i3 : E5p = I5 −
Cj5 ...j1
j1 ,...,j5 =0
(VI).8. i3 6= i1 = i2 6= i4 = i5 6= i3 : E5p
p X
X
(j2 ,j3 ,j4 ,j5 )
(j1 ,j3 )
j1 ,...,j5 =0
(VI).7. i3 6= i2 = i5 6= i1 = i4 6= i3 : E5p
Cj5 ...j1
j1 ,...,j5 =0
VI).6. i4 6= i2 = i5 6= i1 = i3 6= i4 : E5p
p X
(j1 ,j2 )
j1 ,...,j5 =0
(VI).5. i4 6= i1 = i5 6= i2 = i3 6= i4 : E5p
Cj5 ...j1
j1 ,...,j5 =0
(VI).4. i4 6= i1 = i2 6= i3 = i5 6= i4 : E5p
p X
j1 ,...,j5 =0
(VI).3. i5 6= i1 = i4 6= i2 = i3 6= i5 : E5p
Cj5 ...j1
j1 ,...,j5 =0
j1 ,...,j5 =0
(VI).2. i5 6= i1 = i3 6= i2 = i4 6= i5 : E5p
p X
(j2 ,j4 )
(j3 ,j4 )
(j2 ,j4 )
(j2 ,j3 )
(j3 ,j5 )
(j2 ,j3 )
(j1 ,j3 )
(j1 ,j4 )
(j4 ,j5 )
(j1 ,j5 )
(VI).10. i2 6= i1 = i4 6= i3 = i5 6= i2 : E5p = I5 −
p X
j1 ,...,j5 =0
Cj5 ...j1
(j1 ,j4 )
(j3 ,j5 )
EXACT CALCULATION OF MEAN-SQUARE ERROR OF APPROXIMATION
(VI).11. i2 6= i1 = i3 6= i4 = i5 6= i2 : E5p
= I5 −
p X
Cj5 ...j1
j1 ,...,j5 =0
X X
, Cj5 ...j1
X X
, Cj5 ...j1
X X
, Cj5 ...j1
X X
, Cj5 ...j1
X X
, Cj5 ...j1
(j1 ,j3 )
(j4 ,j5 )
(VI).12. i2 6= i1 = i5 6= i3 = i4 6= i2 : E5p
= I5 −
p X
Cj5 ...j1
j1 ,...,j5 =0
(j1 ,j5 )
(j3 ,j4 )
(VI).13. i1 6= i2 = i3 6= i4 = i5 6= i1 : E5p
= I5 −
p X
Cj5 ...j1
j1 ,...,j5 =0
(j2 ,j3 )
(j4 ,j5 )
(VI).14. i1 6= i2 = i4 6= i3 = i5 6= i1 : E5p
= I5 −
p X
Cj5 ...j1
j1 ,...,j5 =0
(j2 ,j4 )
(j3 ,j5 )
(VI).15. i1 6= i2 = i5 6= i3 = i4 6= i1 : E5p
= I5 −
(VII).1. i1 = i2 = i3 6= i4 = i5 : E5p
= I5 −
(VII).2. i1 = i2 = i4 6= i3 = i5 : E5p
= I5 −
(VII).3. i1 = i2 = i5 6= i3 = i4 : Ep = I − (VII).4. i2 = i3 = i4 6= i1 = i5 : E5p
= I5 −
(VII).5. i2 = i3 = i5 6= i1 = i4 : E5p = I5 − (VII).6. i2 = i4 = i5 6= i1 = i3 : E5p = I5 −
p X
Cj5 ...j1
j1 ,...,j5 =0
p X
(j2 ,j5 )
X X
, Cj5 ...j1
X X
, Cj5 ...j1
X X
, Cj5 ...j1
X X
, Cj5 ...j1
X X
Cj5 ...j1
,
X X
Cj5 ...j1
,
Cj5 ...j1
j1 ,...,j5 =0
p X
(j4 ,j5 )
Cj5 ...j1
j1 ,...,j5 =0
p X
(j3 ,j5 )
Cj5 ...j1
j1 ,...,j5 =0
(j3 ,j4 )
p X
Cj5 ...j1
p X
Cj5 ...j1
p X
Cj5 ...j1
j1 ,...,j5 =0
(j1 ,j5 )
j1 ,...,j5 =0
j1 ,...,j5 =0
(j3 ,j4 )
(j1 ,j4 )
(j1 ,j3 )
(j1 ,j2 ,j3 )
(j1 ,j2 ,j4 )
(j1 ,j2 ,j5 )
(j2 ,j3 ,j4 )
(j2 ,j3 ,j5 )
(j2 ,j4 ,j5 )
15
16
D.F. KUZNETSOV
(VII).7. i3 = i4 = i5 6= i1 = i2 : E5p = I5 −
= I5 −
Cj5 ...j1
Cj5 ...j1
,
X X
Cj5 ...j1
,
X X
, Cj5 ...j1
X X
Cj5 ...j1
(j3 ,j4 ,j5 )
(j1 ,j3 ,j5 )
(j2 ,j4 )
p X
Cj5 ...j1
p X
Cj5 ...j1
j1 ,...,j5 =0
(VII).10. i1 = i4 = i5 6= i2 = i3 : E5p = I5 −
p X
X X (j1 ,j2 )
j1 ,...,j5 =0
(VII).9. i1 = i3 = i4 6= i2 = i5 : E5p
Cj5 ...j1
j1 ,...,j5 =0
(VI).8. i1 = i3 = i5 6= i2 = i4 : E5p = I5 −
p X
(j1 ,j3 ,j4 )
(j2 ,j5 )
j1 ,...,j5 =0
(j1 ,j4 ,j5 )
(j2 ,j3 )
.
4. Estimate for Mean-Square Error of Approximation of Multple Ito Stochastic Integrals Theorem 3 (see [5], [6]). Suppose that every ψl (τ ) (l = 1, . . . , k) is a continuous on [t, T ] function and {φj (x)}∞ j=0 is a complete orthonormal system of continuous functions in L2 ([t, T ]). Then the estimate 2 1 ,...,pk ≤ M J[ψ (k) ]T,t − J[ψ (k) ]pT,t (35)
≤ k!
Z
2
K (t1 , . . . , tk )dt1 . . . dtk −
[t,T ]k
p1 X
...
j1 =0
pk X
jk =0
Cj2k ...j1
!
,
is valid for the following cases: 1. i1 , . . . , ik = 1, . . . , m and T − t < ∞; 2. i1 , . . . , ik = 0, 1, . . . , m and T − t < 1, 1 ,...,pk where J[ψ (k) ]T,t is an integral (1), J[ψ (k) ]pT,t is a prelimit expression in the right part of (6); another denotations see in the theorem 1. Proof. Prooving the theorem 1 (see [1], [6], [7]) we obtained w. p. 1 the following representation: p1 ,...,pk 1 ,...,pk J[ψ (k) ]T,t = J[ψ (k) ]pT,t + RT,t ,
where p1 ,...,pk RT,t =
(36)
=
X
ZT
(t1 ,...,tk ) t
...
Zt2 t
K(t1 , . . . , tk ) −
p1 X
j1 =0
...
pk X
jk =0
Cjk ...j1
k Y l=1
(i ) (i ) φjl (tl ) dwt11 . . . dwtkk , (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
EXACT CALCULATION OF MEAN-SQUARE ERROR OF APPROXIMATION
17
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 [13], that Ito stochastic integral exists in the mean-square sence for any ξ ∈ M2 ([0, T ]). Let’s consider standard moment properties of stochastic integrals [13] if ξτ ∈ M2 ([t0 , t]): 2 Z t n Zt o 2 M ξτ dfτ = M |ξτ | dτ,
(37)
t0
t0
2 Zt Zt n o 2 M ξτ dτ ≤ (t − t0 ) M |ξτ | dτ.
(38)
t0
t0
In the case of any fixed k and numbers i1 , . . . , ik = 1, . . . , m the integrals in the right part of (36) p1 ,...,pk will be depend in the stochastic sense. Let’s estimate the second moment of RT,t . From (36), (37) and inequality (a1 + a2 + . . . + am )2 ≤ m a21 + a22 + . . . + a2m
(39)
we obtain the following estimate for the case i1 , . . . , ik = 1, . . . , m (T − t < ∞) : M
≤ k!
X
ZT
...
(t1 ,...,tk ) t
= k!
Zt2 t
Z
(40)
= k!
Z
p1 ,...,pk RT,t
K(t1 , . . . , tk ) −
K(t1 , . . . , tk ) −
[t,T ]k
p1 X
p1 X
...
...
≤
pk X
Cjk ...j1
jk =0
j1 =0
j1 =0
2
pk X
k Y
φjl (tl )
l=1
Cjk ...j1
jk =0
k Y
!2
φjl (tl )
l=1
K 2 (t1 , . . . , tk )dt1 . . . dtk −
[t,T ]k
!2
p1 X
j1 =0
...
pk X
dt1 . . . dtk
!
=
dt1 . . . dtk =
!
Cj2k ...j1 .
jk =0
For the case of any fixed k and numbers i1 , . . . , ik = 0, 1, . . . , m from (36) – (39) we obtain: M
≤ Ck
X
ZT
(t1 ,...,tk ) t
...
Zt2 t
p1 ,...,pk RT,t
K(t1 , . . . , tk ) −
p1 X
j1 =0
2
...
≤
pk X
jk =0
Cjk ...j1
k Y l=1
!2
φjl (tl )
dt1 . . . dtk =
18
D.F. KUZNETSOV
= Ck
Z
K(t1 , . . . , tk ) −
[t,T ]k
= Ck
Z
p1 X
j1 =0
...
pk X
Cjk ...j1
jk =0
k Y
!2
φjl (tl )
l=1
2
K (t1 , . . . , tk )dt1 . . . dtk −
[t,T ]k
p1 X
j1 =0
...
pk X
dt1 . . . dtk =
Cj2k ...j1
jk =0
!
,
where Ck – is a constant. Not difficult to see that constant Ck depend on k (k – is a multiplicity of multiple Ito stochastic integral) and T − t (T − t – is an interval of integration of multiple Ito stochastic integral). Moreover Ck has the following form Ck = k! · max {(T − t)α1 , (T − t)α2 , . . . , (T − t)αk! } , where α1 , α2 , . . . , αk! = 0, 1, . . . , k − 1. However, as we noted before, it is obvious, that the interval T − t of integration of multiple Ito stochastic integrals is a step of numerical procedures for Ito stochastic differential equations which is a small value. For example T − t < 1. Then Ck ≤ k!. It means, that for the case any fixed k and numbers i1 , . . . , ik = 0, 1, . . . , m (T − t < 1) we can write down the (35). The theorem 3 is proven. (i1 i2 i3 ) of the Example 3. Let’s consider the estimate (35) for the triple Ito stochastic integral I(000)T,t form (32): p 2 X (T − t)3 (i1 i2 i3 )p (i1 i2 i3 ) 2 M ≤6 − I(000)T,t I(000)T,t Ckji (i1 , i2 , i3 = 1, . . . , m), − 6 i,j,k=0
where Ckji has the form (34). References [1] 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 [2] 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 [3] 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 [4] 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 [5] 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 [6] 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 [7] Dmitriy F. Kuznetsov. Expansion of Multiple Ito Stochastic Integrals of Arbitrary Multiplicity, Based on Generalized Multiple Fourier Series, Converging in the Mean. arXiv:1712.09746 [math.PR]. 2017, 22 pp. [8] Dmitriy F. Kuznetsov. Application of the Fourier Method to the Mean-Square Approximation of Multiple Ito and Stratonovich Stochastic Integrals. arXiv:1712.08991 [math.PR]. 2017, 15 pp.
EXACT CALCULATION OF MEAN-SQUARE ERROR OF APPROXIMATION
19
[9] Dmitriy F. Kuznetsov. Mean-Square Approximation of Multiple Ito and Stratonovich Stochastic Integrals from the Taylor-Ito and Taylor-Stratonovich Expansions, Using Legendre Polynomials. arXiv:1801.00231 [math.PR]. 2017, 26 pp. [10] Kloeden P.E., Platen E. Numerical Solution of Stochastic Differential Equations. Springer, Berlin, 1995. 632 pp. [11] Milstein G.N. Numerical Integration of Stochastic Differential Equations. Ural University Press, Sverdlovsk, 1988. 225 pp. [12] Milstein G.N., Tretyakov M.V. Stochastic Numerics for Mathematical Physics. Springer, Berlin, 2004. 616 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] 
