The Study of a Class of the Fractional Brownian Motion Base on Wavelet. 1 Introduction. 2 Basic definition

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1 ISSN (print, (online International Journal of Nonlinear Science Vol.(0 No.,pp.90-9 The Study of a Cla of the Fractional Brownian Motion Bae on Wavelet Xuewen Xia Hunan Intitute of Engineering,Xiangtan,0,China Hunan Normal Univerity,Changha,008,China (eceived 7 January 00, accepted 8 October 00 Abtract: In thi paper, we ue the wavelet tranform to the Fractional Brownian Motion by Haar wavelet,we obtain ome tatitical propertie about the tochatic procee and it denity degree and wavelet expre. Keyword: Fractional Brownian Motion; wavelet analyi;wavelet tranform; haar wavelet;denity degree. Introduction The Fractional Brownian Motion i a cla of important tochatic procee,about the procee,there are ome tudy work([-].in thi paper,we tudy it ue wavelet method. With the rapid development of computerized cientific intrument come a wide variety of intereting problem for data analyi and ignal proceing.in field ranging from Extragalactic Atronomy to Molecular Spectrocopy to Medical Imaging to computer viion,one mut recover a ignal,curve,image,pectrum,or denity from incomplete,indirect,and noiy data.wavelet have contributed to thi already intenely developed and rapidly advancing field. Wavelet analyi i a remarkable tool for analyzing function of one or everal variable that appear in mathematic or in ignal and image proceing. With hindight the wavelet tranform can be viewed a divere a mathematic, phyic and electrical engineering.the baic idea i to ue a family of building block to repreent the object at hand in an efficient and inightful way, the building block come in different ize, and are uitable for decribing feature with a reolution commenurate with their ize. Some peron have tudied wavelet problem of tochatic proce or tochatic ytem (ee[6]-[].in thi paper,we tudy a cla of random procee uing wavelet analyi methodand tudy it energy,the energy expre by denity degree. Baic definition Definition Fractional Brownian motion: Gau procee X t (t 0 be called φ-fractional Brownian motion (0 < φ <, if X 0 = 0, EX t = 0, and E(X X t = (φ + t φ t φ ( When φ =, we have E(XtX = ( + t t ( Definition X t i tochatic procee, then it wavelet alternative i W (, x = x t X t ψ( dt ( where ψ i continuou wavelet Correponding author. addre: xxw67@6.com Copyright c World Academic Pre, World Academic Union IJNS.0.06./7

2 X. Xia: The Study of a Cla of the Fractional Brownian Motion Bae on Wavelet 9 Definition ψ(x i ψ(x =, 0 x <, x < ( Call ψ(x a haar wavelet. 0, other Statitic property and energy Ue(, we have We have E[W (, x] = 0. Then relation function of W (, x i W (, x = [ ] x x(tdt x(tdt (τ = E[W (, xw (, x + τ] = x u x + τ v E[x(ux(v]ψ( ψ( = { E[x(ux(v]duduv +τ x+τ + +τ dudv +τ x S+τ +τ +τ } Ue (, we have = I + I + I + I u > v, then we have E[x(ux(v] = (u + v u v I = = = I = [ +τ x+τ +τ (u + v (u v dudv +τ u dudv+ +τ v dudv ] (u v dudv +τ = { [x (x ] + [(x + τ (x + τ ] [( τ ( τ ] = = +τ +τ +τ +τ +τ (u + v (u v dudv u dudv +τ +τ v x +τ dudv + (u v dudv +τ = 6 [x (x ] 6 [x + τ (x + τ ] + [( τ + ( τ ( τ ] IJNS homepage:

3 9 International Journal of Nonlinear Science, Vol.(0, No., pp x+τ I = = +τ x+τ +τ (u + v (u v dudv [ 6 (x (x ] [(x + τ (x + τ ] I = = = [ = [ ( +τ +τ +τ +τ +τ +τ (u + v (u v dudv (x (x u dudv + + ( ] ( ( τ ( τ ( τ +τ +τ v dudv (x + τ (x + τ +τ +τ (u v dudv ] ( Then, (τ = (I + I + I + I = (x + τ (x + τ [ ( τ + ( τ ] 6 [ (x + τ (x + τ ] + [ ( τ ( τ + ( τ ] [ (x + τ (x + τ ] [ ( τ + ( τ + ( τ ] + [ (x + τ (x + τ ] [ ( τ + ( τ + ( τ ] (τ = 8 (x + τ (x + τ [ ( τ ( τ 6 [ (x + τ (x + τ ] + [ ( τ + ( τ ( τ ] [ (x + τ (x + τ ] [ ( τ ( τ ( τ ] + [ (x + τ (x + τ ] [ ( τ ( τ ( τ ] (0 = 8 x (x ( + 6 [ (x (x ] + [ ( ] [ x (x ] [ ( ( ] + [ (x (x ] [ ( ( ] Ue above, We can obtain ( (0, then we can obtain the zero denity of W(,x (0 π (0 and the avergy denity ( (0. π (0 IJNS for contribution: editor@nonlinearcience.org.uk

4 X. Xia: The Study of a Cla of the Fractional Brownian Motion Bae on Wavelet 9 Wavelet repreentation real function φ i tandard orthogonal element of multireolution analyi {V j }j Z (ee [], then exit h k l, have φ(t = k φ(t k ψ(t = k ( k h k φ(t k Then wavelet expre of X(t in mean quare i Where X(t = J Cn J φ( J t n + j d j nψ( j t n j J K C j n = j d j n = j n Z X(tφ( j t ndt X(tψ( j t ndt Then have E [ d j nd k ] m = j+k E [X(tX(]ψ( j t nψ( k mddt We can obtain the value on above. We may alo can analye Cn j a above. Now we conider function ψ(t that exit compact upport et on [ k, k ], k, k 0, and exit enough large M, have tm ψ(tdt = 0, 0 m M, then φ exit compact upport et on [ k, k ] atify k + k = k + k, k, k 0. b(j, k =< X(t, ψ jk >, a(j, k =< X(t, φ jk > J i a contant, then are a tandard orthonormal bai of pace L (, then have X(t = J { } { } J φ( J x k, k Z j ψ( j t k, k Z j J K Z a(j, Kφ( J t K + j J Therefore, the elf-correlation function of b(j, m b (j, K; m, n = E [b(j, mb(k, n] = j+k F ( j K, t = j b(j, Kψ( j t K K Z E [X(tX(] ψ( j t mψ( K ndtd ψ( j K tψ(d If ψ(t have (M -order waning moment, then F ( j K, t have (M order waning moment. In actual, t m ψ( j K tψ(ddt = ( j K t m ψ(tψ(ddt = Cm( n j K m n ( t n ψ(tψ(ddt = 0, n m < M Therefore we have IJNS homepage:

5 9 International Journal of Nonlinear Science, Vol.(0, No., pp Theorem X(t i olution proce of ytem (, ψ(t have compact upported et on [ K, K ], K, K > 0, and ψ(t have (M -order waning moment, and ψ(t i tandard orthonormal wavelet function. Then tochatic proce b(j, m are tationary proce. eference [] Bernt Qkendal.Stochatic differential equation.springer-verlag.00 [] Paul Malliavin.Stochatic analyi.springer-verlag.00 [] Pairick Flandrin.Wavelet analyi and ynthei of Fractional Brownian Motion.EEE Tran.,On Information Theory,8((99: [] Y.Meyer. In Ondelette et operatear(m. Hermann 990 [] Adam Bobrowki.Functional analyi for probability and tochatic procee.cambridge Univerity Pre.007 [6] Cambanc. Wavelet Approximation of Determinitic and andom Signal[J]. IEEE Tran.on Information Theory, 0((99:0-09. [7] Flandrin. Wavelet Analyi and Synthei of Fractional Brownian Motion[J]. IEEE Tran.on Information Theory, 8((99: [8] Krim. Multire olution Analyi of a cla ntationary Procee[J]. IEEE Tran.on Information Theory, ((99: [9] Haobo en. Wavelet etimation for jump on a heteroedatic regreion model. Acta Mathematica Scientia, ((00: [0] Xuewen Xia. Wavelet analyi of the tochatic ytem with coular tationary noie. Engineering Science,(00: -6. [] Xuewen Xia.Wavelet denity degree of continuou parameter A model.international Journal Nonlinear Science,7((008:7-. [] Xuewen Xia.Wavelet analyi of Browain motion.world Journal of Modelling and Simulation,007(:06-. [] Xuewen Xia, Ting Dai.Wavelet denity degree of a cla of Wiener procee. International Journal of Nonlinear Science,7((009:7-. [] A.Hendi.New exact travelling wave olution for ome nonlinear evolution equation. International Journal of Nonlinear Science,7((009:9-67. IJNS for contribution: editor@nonlinearcience.org.uk