New Simple Method of Expansion of Iterated Ito Stochastic integrals of Multiplicity 2 Based on Expansion of the Brownian Motion Using Legendre Polynomials and Trigonometric Functions
aa r X i v : . [ m a t h . P R ] O c t NEW SIMPLE METHOD FOR OBTAINMENT AN EXPANSION OF DOUBLESTOCHASTIC ITO INTEGRALS, BASED ON THE EXPANSION OFBROWNIAN MOTION USING LEGENDRE POLYNOMIALS ANDTRIGONOMETRIC FUNCTIONS
DMITRIY F. KUZNETSOV
Abstract.
The atricle is devoted to the new simple method for obtainment an expansionof double stochastic Ito integrals, based on the expansion of Brownian motion (standardWiener process) using complete orthonormalized systems of functions in the space L ([ t, T ]) . The cases of Legendre polynomials and trigonometric functions are considered in details.We obtained a new representation of the Levy stochastic area, based on the Legendrepolynomials. This representation also has been derived with using the method on expansionof iterated stochastic Ito integrals, based on generalized multple Fourier series. The mention-ed new representation of the Levy stochastic area has more simple form in comparison withthe classical trigonometric representation of the Levy stochastic area. Convergence in themean of degree n ( n ∈ N ) as well as with probability 1 of the Levy stochastic area is proven.The results of the article can be applied to numerical solution of Ito stochastic differentialequations as well as to numerical approximation of mild solution for non-commutative semi-linear stochastic partial differential equations by Milstein type method. Introduction
Let (Ω , F , P ) be a complete probability space, let { F t , t ∈ [0 , T ] } be a nondecreasing right-continousfamily of σ -subfields of F , and let w t be a standard m -dimensional Wiener stochastic process, whichis F t -measurable for any t ∈ [0 , T ] . We assume that the components w ( i ) t ( i = 1 , . . . , m ) of this processare independent. Consider an Ito stochastic differential equation in the integral form(1) x t = x + t Z a ( x τ , τ ) dτ + t Z B ( x τ , τ ) d w τ , x = x (0 , ω ) , ω ∈ Ω . Here x t is some n -dimensional stochastic process satisfying to equation (1). The nonrandom functions a : ℜ n × [0 , T ] → ℜ n , B : ℜ n × [0 , T ] → ℜ n × m guarantee the existence and uniqueness up to stochasticequivalence of a solution of equation (1) [1]. The second integral on the right-hand side of (1) isinterpreted as an Ito stochastic integral. Let x be an n -dimensional random variable, which is F -measurable and M {| x | } < ∞ ( M denotes a mathematical expectation). We assume that x and w t − w are independent when t > . One of the effective approaches to numerical integration of Ito stochastic differential equations is anapproach based on the Taylor–Ito expansion [2]-[4]. The most important feature of such expansion is apresence in them of the so-called iterated stochastic Ito integrals, which play the key role for solving
Mathematics Subject Classification: 60H05, 60H10, 42B05.Keywords: Iterated stochastic Ito integral, Generalized multiple Fourier series, Multiple Fourier–Legendre series, Levy stochastic area, Mean square-convergence, Milstein method, Ito stochasticdifferential equation, Approximation, Expansion. the problem of numerical integration of Ito stochastic differential equations and has the followingform(2) J [ ψ ( k ) ] T,t = T Z t ψ k ( t k ) . . . t Z t ψ ( t ) d w ( i ) t . . . d w ( i k ) t k ( i , . . . , i k = 0 , , . . . , m ) , where every ψ l ( τ ) ( l = 1 , . . . , k ) is a continuous non-random function on [ t, T ] , w ( i ) τ ( i = 1 , . . . , m )are independent standard Wiener processes, and w (0) τ = τ .In this article we pay special attention to the case k = 2 , i , i = 1 , . . . , m , ψ ( τ ) , ψ ( τ ) ≡ . Thiscase corresponds to the so-called Milstein method [3], [4] of numerical integration of Ito stochasticdifferential equations of form (1), which has the order of strong convergence 1.0 under the specificconditions [3], [4].The Milstein method has the following form [3], [4] y p +1 = y p + m X i =1 B i I ( i ) τ p +1 ,τ p + ∆ a + m X i ,i =1 G i B i ˆ I ( i i ) τ p +1 ,τ p , where ∆ = T /N ( N > is a constant (for simplicity) step of integration, τ p = p ∆ ( p = 0 , , . . . , N ) , G i = n X j =1 B ji ( x , t ) ∂∂ x j ( i = 1 , . . . , m ) ,B i is an i th column of the matrix function B and B ij is an ij th element of the matrix function B , a i is an i th element of the vector function a , and x i is an i th element of the column x , the columns B i , a , G i B i are calculated in the point ( y p , p ) ,I ( i ) τ p +1 ,τ p = τ p +1 Z τ p d w ( i ) τ , ˆ I ( i i ) τ p +1 ,τ p is an approximation of the following double (iterated) stochastic Ito integral I ( i i ) τ p +1 ,τ p = τ p +1 Z τ p s Z τ p d w ( i ) τ d w ( i ) s . The Levy stochastic area A ( i i ) T,t is defined as follows [5] A ( i i ) T,t = 12 (cid:16) I ( i i ) T,t − I ( i i ) T,t (cid:17) . It is clear that(3) I ( i i ) T,t = 12 I ( i ) T,t I ( i ) T,t + A ( i i ) T,t w. p. 1 , where i = i . EW SIMPLE METHOD FOR OBTAINMENT AN EXPANSION OF DOUBLE STOCHASTIC ITO INTEGRALS 3
From (3) it follows that the problem of numerical simulation of double (iterated) stochastic Itointegral I ( i i ) T,t is equivalent to the problem of numerical simulation of the Levy stochastic area.There are some methods for representation of the Levy stochastic area (see, for example, [2]–[4]).In this article we consider another representation of the Levy stochastic area, which is based on theLegendre polynomials and simpler than its existing analogues.2.
Approach to Expansion of Iterated stochastic Ito integrals, Based onGeneralized Multiple Fourier Series
Consider an iterated stochastic Ito integrals (2). Define the following function on a hypercube [ t, T ] k K ( t , . . . , t k ) = ψ ( t ) . . . ψ k ( t k ) , t < . . . < t k , otherwise , t , . . . , t k ∈ [ t, T ] , k ≥ , and K ( t ) ≡ ψ ( t ) , t ∈ [ t, T ] . Suppose that { φ j ( x ) } ∞ j =0 is a complete orthonormal system of functions in the space L ([ t, T ]) .The function K ( t , . . . , t k ) is sectionally continuous in the hypercube [ t, T ] k . At this situation it iswell known that the generalized multiple Fourier series of K ( t , . . . , t k ) ∈ L ([ t, T ] k ) is converging to K ( t , . . . , t k ) in the hypercube [ t, T ] k in the mean-square sense, i.e.(4) lim p ,...,p k →∞ (cid:13)(cid:13)(cid:13)(cid:13) K ( t , . . . , t k ) − p X j =0 . . . p k X j k =0 C j k ...j k Y l =1 φ j l ( t l ) (cid:13)(cid:13)(cid:13)(cid:13) L ([ t,T ] k ) = 0 , where(5) C j k ...j = Z [ t,T ] k K ( t , . . . , t k ) k Y l =1 φ j l ( t l ) dt . . . dt k is the Fourier coefficient and k f k L ([ t,T ] k ) = Z [ t,T ] k f ( t , . . . , t k ) dt . . . dt k / . Consider the partition { τ j } Nj =0 of [ t, T ] such that(6) t = τ < . . . < τ N = T, ∆ N = max ≤ j ≤ N − ∆ τ j → if N → ∞ , ∆ τ j = τ j +1 − τ j . Theorem 1 [6]-[25], [30]-[39], [41]-[43].
Suppose that every ψ l ( τ ) ( l = 1 , . . . , k ) is a continuous non-random function on [ t, T ] and { φ j ( x ) } ∞ j =0 is a complete orthonormal system of continous functionsin L ([ t, T ]) . Then
D.F. KUZNETSOV J [ ψ ( k ) ] T,t = l.i.m. p ,...,p k →∞ p X j =0 . . . p k X j k =0 C j k ...j k Y l =1 ζ ( i l ) j l − (7) − l.i.m. N →∞ X ( l ,...,l k ) ∈ G k φ j ( τ l )∆ w ( i ) τ l . . . φ j k ( τ l k )∆ w ( i k ) τ lk ! , where G k = H k \ L k , H k = { ( l , . . . , l k ) : l , . . . , l k = 0 , , . . . , N − } , L k = { ( l , . . . , l k ) : l , . . . , l k = 0 , , . . . , N − l g = l r ( g = r ); g, r = 1 , . . . , k } , l . i . m . is a limit in the mean-square sense, i , . . . , i k = 0 , , . . . , m, C j k ...j is the Fourier coefficient (5), every (8) ζ ( i ) j = T Z t φ j ( s ) d w ( i ) s is a standard Gaussian random variable for various i or j ( if i = 0 ), ∆ w ( i ) τ j = w ( i ) τ j +1 − w ( i ) τ j ( i =0 , , . . . , m ) , { τ j } Nj =0 is a partition of [ t, T ] , which satisfies the condition (6) . In order to evaluate the significance of Theorem 1 for practice we will demonstrate its transformedparticular cases for k = 1 , . . . , [6]-[17], [32](9) J [ ψ (1) ] T,t = l.i.m. p →∞ p X j =0 C j ζ ( i ) j , (10) J [ ψ (2) ] T,t = l.i.m. p ,p →∞ p X j =0 p X j =0 C j j (cid:18) ζ ( i ) j ζ ( i ) j − { i = i =0 } { j = j } (cid:19) ,J [ ψ (3) ] T,t = l.i.m. p ,...,p →∞ p X j =0 p X j =0 p X j =0 C j j j (cid:18) ζ ( i ) j ζ ( i ) j ζ ( i ) j − (11) − { i = i =0 } { j = j } ζ ( i ) j − { i = i =0 } { j = j } ζ ( i ) j − { i = i =0 } { j = j } ζ ( i ) j (cid:19) ,J [ ψ (4) ] T,t = l.i.m. p ,...,p →∞ p X j =0 . . . p X j =0 C j ...j (cid:18) Y l =1 ζ ( i l ) j l −− { i = i =0 } { j = j } ζ ( i ) j ζ ( i ) j − { i = i =0 } { j = j } ζ ( i ) j ζ ( i ) j −− { i = i =0 } { j = j } ζ ( i ) j ζ ( i ) j − { i = i =0 } { j = j } ζ ( i ) j ζ ( i ) j −− { i = i =0 } { j = j } ζ ( i ) j ζ ( i ) j − { i = i =0 } { j = j } ζ ( i ) j ζ ( i ) j + EW SIMPLE METHOD FOR OBTAINMENT AN EXPANSION OF DOUBLE STOCHASTIC ITO INTEGRALS 5 + { i = i =0 } { j = j } { i = i =0 } { j = j } + { i = i =0 } { j = j } { i = i =0 } { j = j } + (12) + { i = i =0 } { j = j } { i = i =0 } { j = j } (cid:19) ,J [ ψ (5) ] T,t = l.i.m. p ,...,p →∞ p X j =0 . . . p X j =0 C j ...j Y l =1 ζ ( i l ) j l −− { i = i =0 } { j = j } ζ ( i ) j ζ ( i ) j ζ ( i ) j − { i = i =0 } { j = j } ζ ( i ) j ζ ( i ) j ζ ( i ) j −− { i = i =0 } { j = j } ζ ( i ) j ζ ( i ) j ζ ( i ) j − { i = i =0 } { j = j } ζ ( i ) j ζ ( i ) j ζ ( i ) j −− { i = i =0 } { j = j } ζ ( i ) j ζ ( i ) j ζ ( i ) j − { i = i =0 } { j = j } ζ ( i ) j ζ ( i ) j ζ ( i ) j −− { i = i =0 } { j = j } ζ ( i ) j ζ ( i ) j ζ ( i ) j − { i = i =0 } { j = j } ζ ( i ) j ζ ( i ) j ζ ( i ) j −− { i = i =0 } { j = j } ζ ( i ) j ζ ( i ) j ζ ( i ) j − { i = i =0 } { j = j } ζ ( i ) j ζ ( i ) j ζ ( i ) j ++ { i = i =0 } { j = j } { i = i =0 } { j = j } ζ ( i ) j + { i = i =0 } { j = j } { i = i =0 } { j = j } ζ ( i ) j ++ { i = i =0 } { j = j } { i = i =0 } { j = j } ζ ( i ) j + { i = i =0 } { j = j } { i = i =0 } { j = j } ζ ( i ) j ++ { i = i =0 } { j = j } { i = i =0 } { j = j } ζ ( i ) j + { i = i =0 } { j = j } { i = i =0 } { j = j } ζ ( i ) j ++ { i = i =0 } { j = j } { i = i =0 } { j = j } ζ ( i ) j + { i = i =0 } { j = j } { i = i =0 } { j = j } ζ ( i ) j ++ { i = i =0 } { j = j } { i = i =0 } { j = j } ζ ( i ) j + { i = i =0 } { j = j } { i = i =0 } { j = j } ζ ( i ) j ++ { i = i =0 } { j = j } { i = i =0 } { j = j } ζ ( i ) j + { i = i =0 } { j = j } { i = i =0 } { j = j } ζ ( i ) j ++ { i = i =0 } { j = j } { i = i =0 } { j = j } ζ ( i ) j + { i = i =0 } { j = j } { i = i =0 } { j = j } ζ ( i ) j + (13) + { i = i =0 } { j = j } { i = i =0 } { j = j } ζ ( i ) j ! , where A is the indicator of the set A .It was shown in [12]-[17], [32] that Theorem 1 is valid for convergence in the mean of degree n ( n ∈ N ). Moreover, the complete orthonormal in L ([ t, T ]) systems of Haar and Rademacher–Walsh functions also can be applied in Theorem 1 [6]-[17]. The generalization of Theorem 1 forcomplete orthonormal with weigth r ( x ) ≥ system of functions in the space L ([ t, T ]) can be foundin [10], [42]. Recently, Theorem 1 has been applied to expansion and mean-square approximation ofiterated stochastic Ito integrals with respect to the infinite-dmensional Q -Wiener process [24], [25].These results can be directly applied to construction of high-order strong numerical methods fornon-commutative semilinear stochastic partial differential equations [24], [25].Note that we obtain the following useful possibilities of the approach, based on Theorem 1.1. There is an obvious formula (see (5)) for calculation of expansion coefficients of iterated stochasticIto integral (2) with any fixed multiplicity k .2. We have possibilities for exact calculation of the mean-square error of approximation of iteratedstochastic Ito integral (2) [8]-[10], [18], [33].3. Since the used multiple Fourier series is a generalized in the sense that it is built using variouscomplete orthonormal systems of functions in the space L ([ t, T ]) , then we have new possibilities forapproximation — we may use not only trigonometric functions as in [2]-[4] but Legendre polynomials.4. As it turned out [6]-[25], [30]-[39], [41]-[43] it is more convenient to work with Legendre poly-nomials for building of approximations of iterated stochastic Ito integrals (2). Approximations basedon the Legendre polynomials essentially simpler than their analogues based on the trigonometric D.F. KUZNETSOV functions. Another advantages of the application of Legendre polynomials in the framework of thementioned question are considered in [22], [30].5. The approach to expansion of iterated stochastic Ito integrals, based on the Karhunen–Loeveexpansion of the Wiener process as well as the approach from [45] (we will discuss this work inSect. 5) lead to iterated application of the operation of limit transition (the operation of limittransition is implemented only once in Theorem 1 (see above)) starting from second multiplicity(in the general case) and third multiplicity (for the case ψ ( s ) , ψ ( s ) , ψ ( s ) ≡ i , i , i = 1 , . . . , m )of iterated stochastic Ito integrals. Multiple series from Theorem 1 (the operation of limit transitionis implemented only once) are more convenient for approximation than the iterated ones (iteratedapplication of operation of the limit transition), since partial sums of multiple series converge for anypossible case of convergence to infinity of their upper limits of summation (let us denote them as p , . . . , p k ). For example, for more simple and convenient for practice case when p = . . . = p k = p →∞ . For iterated series it is obviously not the case. However, in [2] the authors unreasonably use thecondition p = p = p = p → ∞ within the application of the mentioned approach, based on theKarhunen–Loeve expansion of the Wiener process [3].Note that rightness of formulas (9)–(13) can be verified by the fact that if i = . . . = i = i =1 , . . . , m and ψ ( s ) , . . . , ψ ( s ) ≡ ψ ( s ) in (9)–(13), then we can obtain from (9)–(13) the followingequalities which are right w. p. 1 J [ ψ (1) ] T,t = 11! δ T,t ,J [ ψ (2) ] T,t = 12! (cid:0) δ T,t − ∆ T,t (cid:1) ,J [ ψ (3) ] T,t = 13! (cid:0) δ T,t − δ T,t ∆ T,t (cid:1) ,J [ ψ (4) ] T,t = 14! (cid:0) δ T,t − δ T,t ∆ T,t + 3∆ T,t (cid:1) ,J [ ψ (5) ] T,t = 15! (cid:0) δ T,t − δ T,t ∆ T,t + 15 δ T,t ∆ T,t (cid:1) , where δ T,t = T Z t ψ ( s ) d w ( i ) s , ∆ T,t = T Z t ψ ( s ) ds, which can be independently obtained using the Ito formula and Hermite polynomials.The cases k = 2 , and p = p = p = p are considered in [7]-[17].3. New Representation of the Levy Stochastic Area, Based on the LegendrePolynomials
Let us consider (10) for the case i = i , ψ ( s ) , ψ ( s ) ≡ . At that we suppose that { φ j ( x ) } ∞ j =0 isa complete orthonormal system of Legendre polynomials in the space L ([ t, T ]) . Then(14) I ( i i ) T,t = T − t ζ ( i )0 ζ ( i )0 + ∞ X i =1 √ i − (cid:16) ζ ( i ) i − ζ ( i ) i − ζ ( i ) i ζ ( i ) i − (cid:17)! , where EW SIMPLE METHOD FOR OBTAINMENT AN EXPANSION OF DOUBLE STOCHASTIC ITO INTEGRALS 7 (15) I ( i i ) T,t = T Z t s Z t d w ( i ) τ d w ( i ) s ( i , i = 1 , . . . , m ) ,ζ ( i ) j are independent standard Gaussian random variables (for various i or j ), which have the followingform ζ ( i ) j = T Z t φ j ( s ) d w ( i ) s , where(16) φ i ( s ) = r i + 1 T − t P i (cid:18)(cid:18) s − t − T − t (cid:19) T − t (cid:19) , i = 0 , , , . . . , and P i ( x ) ( i = 0 , , , . . . ) is the Legendre polynomial.From (14) we obtain T − t ∞ X i =1 √ i − (cid:16) ζ ( i ) i − ζ ( i ) i − ζ ( i ) i ζ ( i ) i − (cid:17) = 12 (cid:16) I ( i i ) T,t − I ( i i ) T,t (cid:17) . Then, the new representation of the Levy stochastic area, which is based on the Legendre polynomials,has the following form(17) A ( i i ) T,t = T − t ∞ X i =1 √ i − (cid:16) ζ ( i ) i − ζ ( i ) i − ζ ( i ) i ζ ( i ) i − (cid:17) . The Classical Representation of the Levy Stochastic Area
Let us consider (10) for the case i = i , ψ ( s ) , ψ ( s ) ≡ . At that we suppose that { φ j ( x ) } ∞ j =0 isa complete orthonormal system of trigonometric functions in L ([ t, T ]) . Then I ( i i ) T,t = 12 ( T − t ) ζ ( i )0 ζ ( i )0 + 1 π ∞ X r =1 r (cid:18) ζ ( i )2 r ζ ( i )2 r − − ζ ( i )2 r − ζ ( i )2 r + (18) + √ (cid:16) ζ ( i )2 r − ζ ( i )0 − ζ ( i )0 ζ ( i )2 r − (cid:17)(cid:19)! , where we use the same notations as in (14), but φ j ( s ) has the following form D.F. KUZNETSOV (19) φ j ( s ) = 1 √ T − t , if j = 0 √ πr ( s − t ) / ( T − t )) , if j = 2 r − √ πr ( s − t ) / ( T − t )) , if j = 2 r , r = 1 , , . . . From (18) we obtain T − t π ∞ X r =1 r (cid:16) ζ ( i )2 r ζ ( i )2 r − − ζ ( i )2 r − ζ ( i )2 r + √ (cid:16) ζ ( i )2 r − ζ ( i )0 − ζ ( i )0 ζ ( i )2 r − (cid:17)(cid:17) == 12 (cid:16) I ( i i ) T,t − I ( i i ) T,t (cid:17) . Then, the representation of the Levy stochastic area, which is based on the trigonometric functions,has the following form(20) ˆ A ( i i ) T,t = T − t π ∞ X r =1 r (cid:16) ζ ( i )2 r ζ ( i )2 r − − ζ ( i )2 r − ζ ( i )2 r + √ (cid:16) ζ ( i )2 r − ζ ( i )0 − ζ ( i )0 ζ ( i )2 r − (cid:17)(cid:17) . Note that Milstein G.N. proposed in [3] the method of expansion of iterated stochastic Ito integralsof second multiplicity based on trigonometric Fourier expansion of the Brownian bridge process(version of the so-called Karunen–Loeve expansion) w t − t ∆ w ∆ , t ∈ [0 , ∆] , ∆ > , where w t is a standard vector Wiener process with independent components w ( i ) t , i = 1 , . . . , m. The trigonometric Fourier expansion of the Brownian bridge process has the form(21) w ( i ) t − t ∆ w ( i )∆ = 12 a i, + ∞ X r =1 (cid:18) a i,r cos 2 πrt ∆ + b i,r sin 2 πrt ∆ (cid:19) , where a i,r = 2∆ ∆ Z (cid:16) w ( i ) s − s ∆ w ( i )∆ (cid:17) cos 2 πrs ∆ ds,b i,r = 2∆ ∆ Z (cid:16) w ( i ) s − s ∆ w ( i )∆ (cid:17) sin 2 πrs ∆ ds,r = 0 , , . . . , i = 1 , . . . , m. It is easy to demonstrate [3] that the random variables a i,r , b i,r are Gaussian ones and they satisfythe following relations M { a i,r b i,r } = M { a i,r b i,k } = 0 , EW SIMPLE METHOD FOR OBTAINMENT AN EXPANSION OF DOUBLE STOCHASTIC ITO INTEGRALS 9 M { a i,r a i,k } = M { b i,r b i,k } = 0 , M { a i ,r a i ,r } = M { b i ,r b i ,r } = 0 , M (cid:8) a i,r (cid:9) = M (cid:8) b i,r (cid:9) = ∆2 π r , where i, i , i = 1 , . . . , m, r = k, i = i . According to (21) we have(22) w ( i ) t = w ( i )∆ t ∆ + 12 a i, + ∞ X r =1 (cid:18) a i,r cos 2 πrt ∆ + b i,r sin 2 πrt ∆ (cid:19) , where the series converges in the mean-square sense.Expansion (18) has been obtained in [3] using (22).5. New Simple Method for Obtainment of Representations of the Levy StochasticArea
It is well known that the idea of representing of the Wiener process as a functional series withrandom coefficients that are independent standard Gaussian random variables, with using of acomplete orthonormal system of trigonometric functions in L ([0 , T ]) goes back to the works of Wiener[46] (1924) and Levy [47] (1951). The specified series was used in [46] and [47] for construction of theBrownian motion process (Wiener process). A little later, Ito and McKean in [48] (1965) used for thispurpose the complete orthonormal system of Haar functions in L ([0 , T ]) .Let w τ , τ ∈ [0 , ˆ T ] be an m -dimestional Wiener process with independent components w ( i ) τ ( i =1 , . . . , m ) . We have w ( i ) s − w ( i ) t = s Z t d w ( i ) τ = T Z t { τ
Consider the Fourier expansion of { τ
Let φ j ( τ ) ( j = 0 , , . . . ) be a complete orthonormal system of functions in the space L ([ t, T ]) . Let
EW SIMPLE METHOD FOR OBTAINMENT AN EXPANSION OF DOUBLE STOCHASTIC ITO INTEGRALS 11 (26) T Z t (cid:16) w ( i ) s − w ( i ) t (cid:17) m d w ( i ) s = m X j =0 T Z t φ j ( τ ) d w ( i ) τ T Z t s Z t φ j ( τ ) dτ d w ( i ) τ be an approximation of iterated stochastic Ito integral T Z t s Z t d w ( i ) τ d w ( i ) s ( i = i ) , where i , i = 1 , . . . , m . Then T Z t s Z t d w ( i ) τ d w ( i ) s = l.i.m. m →∞ T Z t (cid:16) w ( i ) s − w ( i ) t (cid:17) m d w ( i ) s , where i , i = 1 , . . . , m . Proof.
Using the standard properties of Ito stochastic integral as well as (25) and the property oforthonormality of functions φ j ( τ ) ( j = 0 , , . . . ) at the interval [ t, T ] , we obtain M T Z t s Z t d w ( i ) τ d w ( i ) s − T Z t (cid:16) w ( i ) s − w ( i ) t (cid:17) m d w ( i ) s == T Z t M w ( i ) s − w ( i ) t − (cid:16) w ( i ) s − w ( i ) t (cid:17) m ! ds == T Z t T Z t { τ
If for the sequence of random variables ξ q and for some α > the number series ∞ X q =1 M {| ξ q | α } converges, then the sequence ξ q converges to zero w. p. . From (38) and (41) ( n = 2) we obtain M (cid:26)(cid:16) I ( i i ) T,t − I ( i i ) qT,t (cid:17) (cid:27) = M (cid:26)(cid:16) A ( i i ) T,t − A ( i i ) qT,t (cid:17) (cid:27) ≤ Kq , where constant K does not depend on q. Since the series ∞ X q =1 Kq converges, then according to Lemma 1 we obtain that A ( i i ) T,t − A ( i i ) qT,t → if q → ∞ w. p. 1. Then A ( i i ) qT,t → A ( i i ) T,t if q → ∞ w. p. 1. References
EW SIMPLE METHOD FOR OBTAINMENT AN EXPANSION OF DOUBLE STOCHASTIC ITO INTEGRALS 17
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Dmitriy Feliksovich KuznetsovPeter the Great Saint-Petersburg Polytechnic University,Polytechnicheskaya ul., 29,195251, Saint-Petersburg, Russia
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