Rosenblatt过程的逼近及其相关分析
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摘要
本文主要研究Rosenblatt过程及其更一般的Hermite过程的鞅差逼近.所谓指数为H∈(1/2,1)的k-阶Hermite过程是指由如下积分所确定的过程：
Zkh(t)其中W为一个标准布朗运动,而核KH由如下定义：Hermite过程具有以下性质：
(1)对任意的c>0,(Zkh(ct))和(cHZkh(t))有同分布,知Z是H阶自相似过程；
(2)对h>0,联合分布(Zkh(t+h)—Zkh(t),t∈[O,T])是独立的,知它有稳定增量；
(3)由结合Kolmogorov连续性质,知ZHk有δ (4)由知过程Z有长相依性；(当H>1/2时,fBM和Rosenblatt过程有此性质)
(5)协方差函数为：
当κ=1时,它是众所周知的分数布朗运动；κ=2时,Hermite过程被称为Rosenblatt过程.值得注意的是这类过程既不是Gaussian过程也不是Markov过程或半鞅除非H=1/2,即它是Brownian运动,但是它却具有与是Gaussian过程的分数布朗运动完全相同的相依结构!
首先,我们考虑Rosenblatt过程的鞅差逼近,我们证明过程列在n→∞时,弱收敛于Rosenblatt过程ZH的,这里{ξ(n),n≥1}是满足适当条件的鞅差序列.
其次,对于满足适当条件的鞅差序列我们构造了随机过程列的递推累加和
其中,Q(n)(t,i_1/n,...,i_k/n)为：我们证明过程列{Zn,n=1,2,...}在n→∞时,弱收敛于Hermite过程ZH.
In this article, we study the martingale difference approximation of Rosenblatt process and Hermite process. The so-called Hermite process of order k with index H∈(1/2,1) is the process given by the integral
Zkh(t)
where W is a standard Brownian motion, and the kernel KH has the expression:
Hermite process admits the following properties:
(1) the process Z is H-selfsimilar:for any c> 0, (ZkH(ct)) and (cHZkH(t)) have the same distribution;
(2) the process has the stationarity of increments:for any h>0, the joint distribu-tion(ZkH(t+h)-ZkH(t),t∈[O,T]) is independent;
(3) the mean square of the increment is given by as a consequence, it follows will little extra effort from Kolmogorov's continuity criterion that the Hermite process ZkH has Holder continuous paths of orderδ< H: 6
(4) it exhibits long-range dependence in the sense that (This property is identical to that of fBm since the processes share the same covariance structure, and the property is well-known for f Bm with H>1/2.)
(5) the covariance function is:
For k=1, it is well known Brownian motion; for k=2, the Hermite process is known as the Rosenblatt process.It should be noted that this type of process is neither Gaussian process is not Markov process or semi-martingale unless H=1/2, that it is a Brownian motion, but it is a Gaussian process with fractional Brownian motion in exactly the same dependence structure!
Firstly, we consider the martingale difference approximation of the Rosenblatt process, We prove that the sequence of the process converges weakly to the Rosenblatt process ZH, as n tends to infinity, where{ξ(n),n≥1} is a sequence of martingale-differences satisfying.appropriate conditions.
Secondly, for the sequence of martingale-differences satisfying appropriate condi-tions, we construct the sums of product'of a sequence of the stochastic process where, Q(n) (t,i_1/n,...,i_k/n)is an approximation of Q(t, u, v): and we prove that the processes{Zn,n= 1,2,...} converges weakly to the Hermite process ZH, as n tends to infinity.
Zkh(t)其中W为一个标准布朗运动,而核KH由如下定义：Hermite过程具有以下性质：
(1)对任意的c>0,(Zkh(ct))和(cHZkh(t))有同分布,知Z是H阶自相似过程；
(2)对h>0,联合分布(Zkh(t+h)—Zkh(t),t∈[O,T])是独立的,知它有稳定增量；
(3)由结合Kolmogorov连续性质,知ZHk有δ
(5)协方差函数为：
当κ=1时,它是众所周知的分数布朗运动；κ=2时,Hermite过程被称为Rosenblatt过程.值得注意的是这类过程既不是Gaussian过程也不是Markov过程或半鞅除非H=1/2,即它是Brownian运动,但是它却具有与是Gaussian过程的分数布朗运动完全相同的相依结构!
首先,我们考虑Rosenblatt过程的鞅差逼近,我们证明过程列在n→∞时,弱收敛于Rosenblatt过程ZH的,这里{ξ(n),n≥1}是满足适当条件的鞅差序列.
其次,对于满足适当条件的鞅差序列我们构造了随机过程列的递推累加和
其中,Q(n)(t,i_1/n,...,i_k/n)为：我们证明过程列{Zn,n=1,2,...}在n→∞时,弱收敛于Hermite过程ZH.
In this article, we study the martingale difference approximation of Rosenblatt process and Hermite process. The so-called Hermite process of order k with index H∈(1/2,1) is the process given by the integral
Zkh(t)
where W is a standard Brownian motion, and the kernel KH has the expression:
Hermite process admits the following properties:
(1) the process Z is H-selfsimilar:for any c> 0, (ZkH(ct)) and (cHZkH(t)) have the same distribution;
(2) the process has the stationarity of increments:for any h>0, the joint distribu-tion(ZkH(t+h)-ZkH(t),t∈[O,T]) is independent;
(3) the mean square of the increment is given by as a consequence, it follows will little extra effort from Kolmogorov's continuity criterion that the Hermite process ZkH has Holder continuous paths of orderδ< H: 6
(4) it exhibits long-range dependence in the sense that (This property is identical to that of fBm since the processes share the same covariance structure, and the property is well-known for f Bm with H>1/2.)
(5) the covariance function is:
For k=1, it is well known Brownian motion; for k=2, the Hermite process is known as the Rosenblatt process.It should be noted that this type of process is neither Gaussian process is not Markov process or semi-martingale unless H=1/2, that it is a Brownian motion, but it is a Gaussian process with fractional Brownian motion in exactly the same dependence structure!
Firstly, we consider the martingale difference approximation of the Rosenblatt process, We prove that the sequence of the process converges weakly to the Rosenblatt process ZH, as n tends to infinity, where{ξ(n),n≥1} is a sequence of martingale-differences satisfying.appropriate conditions.
Secondly, for the sequence of martingale-differences satisfying appropriate condi-tions, we construct the sums of product'of a sequence of the stochastic process where, Q(n) (t,i_1/n,...,i_k/n)is an approximation of Q(t, u, v): and we prove that the processes{Zn,n= 1,2,...} converges weakly to the Hermite process ZH, as n tends to infinity.
引文
[1]P. Abry, V. Pipiras, Wavelet-based synthesis of the Rosenblatt process, To appear in Signal Processing,2005.
[2]J.M.P. Albin, A note on the Rosenblatt distributions, Statist. Probab. Lett.,40(1) (1998),83-91.
[3]J.M.P. Albin, On extremal theory for self similar processes, Anal. of Probab., 26(2) (1998),743-793.
[4]E. Alos, O. Mazet and D. Nualart, Stochastic calculus with respect to Gaussian processes. Annals of Probability.,29 (2001),766-801.
[5]E. Alos, D. Nualart, Stochastic integration with respect to the fractional Brownian motion, Stochastics and Stochastics Reports.,75(3) (2003),129-152.
[6]X. Bardina, M. Jolis, Weak convergence to the multiple Stratonovich integrals, Stochastic Process. Appl.,90(2) (2000),277-300.
[7]J. Beran, Statistics for Long-Memory Processes, Chapman and Hall, (1994).
[8]K. Bertin, S. Torres and C. A. Tudor, Maximum likelihood estimators and random walks in long memory models, proprint, (2008).
[9]P. Billingsley, Convergence of Probability Measures, Wiley, New York 1968.
[10]Yu.V. Borovskikh and V.S. Korolyuk, Martingale Approximation, VSP Utrecht, The Netherlands 1997.
[11]P. Carmona, L. Coutin and G. Montseny:Stochastic integration with respect to fractional Brownian motion. Ann. Institut Henri Poincare.,39(1) (2003),27-68.
[12]P. Cheridito, H. Kawaguchi and M. Maejima, Fractional Ornstein-Uhlenbeck pro-cesses. Electronic Journal of Probability.,8 (2003),1-14.
[13]A. Chronopoulou, C.A. Tudor, Self-similarity parameter estimation and reproduc-tion property for non-Gaussian Hermite processes, ESAIM Probability and Statis-tics,12 (2008),230-257.
[14]A. Chronopoulou, C.A. Tudor, Variations and Hurst index estimation for a Rosen-blatt process using longer fiters, Electronic Journal of Statistics,2008.
[15]J.F. Coeurjolly, Estimating the parameters of a fractional Brownian motion by discrte variations of its sample paths, Statistical Inference for Stochastic Processes, 4 (2001),199-227.
[16]R.Cont, Long range dependence in financial markets, http://www.cmap.polytechnique.fr/rama/papers/FE05.pdf, 2006.
[17]W. Dai, C. C. Heyde:Ito formula with respect to fractional Brownian motion and its application. Journal of Appl. Math. and Stoch. An.,9 (1996),439-448.
[18]A. Dasgupta, Fractional Brownian motion:its properties and applications to stochastic integration, University of North Carolina:Ph.D. Thesis (1998).
[19]L. Decreusefond and A.S. Ustunel, Stochastic analysis of the fractional Brownian motion. Potential Analysis.,10 (1998),177-214.
[20]R.L. Dobrushin and P. Major, Non-central limit theorems for non-linear function-als of Gaussian felds, Z. Wahrsch. Gebiete,50 (1979),27-52.
[21]T. Duncan, Y. Z. Hu and B. Pasik-Duncan, Stochastic Calculus for Fractional Brownian Motion I.Theory. SI AM J. Control Optim.,38(2) (2000),582-612.
[22]P. Embrechts and M. Maejima, Self-similar processes, Princeton University Press 2002.
[23]P. Hall and C.C. Heyde, Martingale Limit Theorem and Application, Academic Press 1980.
[24]Y.Z. Hu and P.A. Meyer, Sur les inte grales multiples de Stratonovich,Séminaire de Probabilitiés XXII, Lecture Notes in Mathematics., (1988),72-81.
[25]Y.Z. Hu, D. Nualart, Parameter estimation for fractional Ornstein-Uhlenbeck pro-cesses, Probability.
[26]K. Itó, Multiple Wiener Integral, J. Math. Soc. Japan.,3 (1951),157-169.
[27]J. Jacod, A.N. Shiryaev, Limit theorems for stochastic processes. Second edition., Springer,2003.
[28]M. Jolis, M. Sanz, Integrator properties of the Skorohod integral, Stochastics and Stochastics Reports.,41(3) (1992),163-176.
[29]I. Kruk, F. Russo and C.A. Tudor, Wiener integrals, Malliavin calculus and co-variance structure measure, Journal of Func. Anal.,2007.
[30]J. Lamperti, Semi-stable stochastic processes, Trans. Amer. Math. Soc,104 (1962),62-78.
[31]N.N. Leonenko, V.V. Ahn, Rate of convergence to the Rosenblatt distribution for additive functionals of stochastic processes with long-range dependence. Journal of Appl. Mathematics and Stoch. Anal,14(1) (2001),27-46.
[32]S. J. Lin, Stochastic analysis of fractional Brownian motions, Stoch. and Stoch. Reports,55 (1995),121-140.
[33]R. S. Liptser and A. N. Shiryayev, On necessary and sufficient conditions in the funcitonal central limit theorem for semimartingales, Theory Probab. Appl.,26 (1981),130-135.
[34]A. Lo, Long memory in stock market prices, Econometrica,59 (1991),1279-1313.
[35]M. Maejima and C.A. Tudor, Wiener integrals with respect to the Hermite process and a Non-Central Limit Theorem, Stoch. Anal. Appl.,25 (2007),1043-1056.
[36]B. Mandelbrot, The variation of certain speculative prices, Journal of Business, XXXVI (2004),392-417.
[37]B. Mandelbrot, M. Van Ness, Fractional Brownian motions, fractional noises and applications, SIAM Rev.10 (1968),422-437.
[38]A. Nieminem, Fractional Brownian motion and Martingale differences, Statist. Probab. Lett.,70 (2004),1-10.
[39]I. Norros,On the use of the fractional Brownian motion in the theory of connec-tionless networks, IEEE J. Sel. Ar. Commun.,13, (1995),953-962.
[40]I. Nourdin, D.Nualart and C.A. Tudor, Central and non-central limit theorems for weighted ower variations of fractional Brownian motion. Preprint.
[41]I. Nourdin, C.A. Tudor, Some linear fractional stochastic equations, Stoch.,78(2) (2006),61-65.
[42]D. Nualart, Malliavin Calculus and Related Topics, Springer 2006.
[43]V. Pipiras, Wavelet type expansion of the Rosenblatt process, The Journal of Fourier Anal. and Appl.,10(6) (2004),599-634.
[44]Z. Qian, T. Lyons, System control and rough paths, Clarendon Press, Oxford,2002.
[45]C. Rogers, Arbitrage with fractional Brownian motion, Math. Finance,7 (1997), 95-105.
[46]M. Rosenblatt, Independence and dependence, Proc.4 th Berkeley Sympos. Math. Statist. Probab., (1961),411-443.
[47]F. Russo, P. Vallois, Forward backward and symmetric stochastic integration, Prob. Theory Rel. Fields,97 (1993),403-421.
[48]F. Russo, P. Vallois, Elements of stochastic calculus via regularization, Preprint.
[49]D. M. Salopek, Tolerance to arbitrage, Stoch. Proc. Appl.,76, (1998),217-230.
[50]G. Samorodnitsky and M. Taqqu, Stable Non-Gaussian random variables, Chap-man and Hall, London, (1994).
[51]A. N. Shiryaev, On arbitrage and replication for fractal models, Centre for Math. Physics and Stoch., (1998).
[52]A.N. Shiryaev, Probability, Springer, New York,1984.
[53]J.L. Sole and F. Utzet, Stratonovich integral and trace, Stochastics and Stochastics Reports,29 (2) (1990),203-220.
[54]T.Sottinen, Fractional Brownian motion, random walks and binary market models. Finance Stochastics.,5 (2001),343-355.
[55]M. Taqqu, A representation for self-similar processes, Stochastic Process. Appl.,7 (1978),55-64.
[56]M. Taqqu, Convergence of integrated processes of arbitrary Hermite rank, Z.Wahrsch. Gebiete,50 (1979),53-83.
[57]M. Taqqu, Weak convergence to the fractional Brownian motion and to the Rosen-blatt, Z.Wahrsch. Gebiete,31 (1975),287-302.
[58]H. Teicher, Distribution and Moment Convergence of Martingales, Probab. Th. Rel. Fields.,79 (1988),303-316.
[59]S. Torres and C.A. Tudor, Donsker type theorem for the Rosenblatt process and a binary market model, Stoch. Anal. Appl.,27 (2009),555-573.
[60]C.A. Tudor, Analysis of the Rosenblatt process, ESAIM Proba. and Statist.,12 (2008),230-257.
[61]C.A. Tudor, F. Viens, Variations and estimators through Malliavin calculus, An-nal. Probab.,2009.
[62]K. Yoshihara, weak convergence to some processes defined by multiple wiener integrals, Yokohama Math. Journal Vol.,38 (1990).
[63]W. Willinger, M. Taqqu and V. Teverovsky, Long range dependence and stock returns, Finance and Stoch.,3 (1999),1-13.
[64]M. Zaehle, Integration with respect to fractal functions and stochastic calculus, Prob. Theory Rel. Fields,111 (1998),333-374.
[2]J.M.P. Albin, A note on the Rosenblatt distributions, Statist. Probab. Lett.,40(1) (1998),83-91.
[3]J.M.P. Albin, On extremal theory for self similar processes, Anal. of Probab., 26(2) (1998),743-793.
[4]E. Alos, O. Mazet and D. Nualart, Stochastic calculus with respect to Gaussian processes. Annals of Probability.,29 (2001),766-801.
[5]E. Alos, D. Nualart, Stochastic integration with respect to the fractional Brownian motion, Stochastics and Stochastics Reports.,75(3) (2003),129-152.
[6]X. Bardina, M. Jolis, Weak convergence to the multiple Stratonovich integrals, Stochastic Process. Appl.,90(2) (2000),277-300.
[7]J. Beran, Statistics for Long-Memory Processes, Chapman and Hall, (1994).
[8]K. Bertin, S. Torres and C. A. Tudor, Maximum likelihood estimators and random walks in long memory models, proprint, (2008).
[9]P. Billingsley, Convergence of Probability Measures, Wiley, New York 1968.
[10]Yu.V. Borovskikh and V.S. Korolyuk, Martingale Approximation, VSP Utrecht, The Netherlands 1997.
[11]P. Carmona, L. Coutin and G. Montseny:Stochastic integration with respect to fractional Brownian motion. Ann. Institut Henri Poincare.,39(1) (2003),27-68.
[12]P. Cheridito, H. Kawaguchi and M. Maejima, Fractional Ornstein-Uhlenbeck pro-cesses. Electronic Journal of Probability.,8 (2003),1-14.
[13]A. Chronopoulou, C.A. Tudor, Self-similarity parameter estimation and reproduc-tion property for non-Gaussian Hermite processes, ESAIM Probability and Statis-tics,12 (2008),230-257.
[14]A. Chronopoulou, C.A. Tudor, Variations and Hurst index estimation for a Rosen-blatt process using longer fiters, Electronic Journal of Statistics,2008.
[15]J.F. Coeurjolly, Estimating the parameters of a fractional Brownian motion by discrte variations of its sample paths, Statistical Inference for Stochastic Processes, 4 (2001),199-227.
[16]R.Cont, Long range dependence in financial markets, http://www.cmap.polytechnique.fr/rama/papers/FE05.pdf, 2006.
[17]W. Dai, C. C. Heyde:Ito formula with respect to fractional Brownian motion and its application. Journal of Appl. Math. and Stoch. An.,9 (1996),439-448.
[18]A. Dasgupta, Fractional Brownian motion:its properties and applications to stochastic integration, University of North Carolina:Ph.D. Thesis (1998).
[19]L. Decreusefond and A.S. Ustunel, Stochastic analysis of the fractional Brownian motion. Potential Analysis.,10 (1998),177-214.
[20]R.L. Dobrushin and P. Major, Non-central limit theorems for non-linear function-als of Gaussian felds, Z. Wahrsch. Gebiete,50 (1979),27-52.
[21]T. Duncan, Y. Z. Hu and B. Pasik-Duncan, Stochastic Calculus for Fractional Brownian Motion I.Theory. SI AM J. Control Optim.,38(2) (2000),582-612.
[22]P. Embrechts and M. Maejima, Self-similar processes, Princeton University Press 2002.
[23]P. Hall and C.C. Heyde, Martingale Limit Theorem and Application, Academic Press 1980.
[24]Y.Z. Hu and P.A. Meyer, Sur les inte grales multiples de Stratonovich,Séminaire de Probabilitiés XXII, Lecture Notes in Mathematics., (1988),72-81.
[25]Y.Z. Hu, D. Nualart, Parameter estimation for fractional Ornstein-Uhlenbeck pro-cesses, Probability.
[26]K. Itó, Multiple Wiener Integral, J. Math. Soc. Japan.,3 (1951),157-169.
[27]J. Jacod, A.N. Shiryaev, Limit theorems for stochastic processes. Second edition., Springer,2003.
[28]M. Jolis, M. Sanz, Integrator properties of the Skorohod integral, Stochastics and Stochastics Reports.,41(3) (1992),163-176.
[29]I. Kruk, F. Russo and C.A. Tudor, Wiener integrals, Malliavin calculus and co-variance structure measure, Journal of Func. Anal.,2007.
[30]J. Lamperti, Semi-stable stochastic processes, Trans. Amer. Math. Soc,104 (1962),62-78.
[31]N.N. Leonenko, V.V. Ahn, Rate of convergence to the Rosenblatt distribution for additive functionals of stochastic processes with long-range dependence. Journal of Appl. Mathematics and Stoch. Anal,14(1) (2001),27-46.
[32]S. J. Lin, Stochastic analysis of fractional Brownian motions, Stoch. and Stoch. Reports,55 (1995),121-140.
[33]R. S. Liptser and A. N. Shiryayev, On necessary and sufficient conditions in the funcitonal central limit theorem for semimartingales, Theory Probab. Appl.,26 (1981),130-135.
[34]A. Lo, Long memory in stock market prices, Econometrica,59 (1991),1279-1313.
[35]M. Maejima and C.A. Tudor, Wiener integrals with respect to the Hermite process and a Non-Central Limit Theorem, Stoch. Anal. Appl.,25 (2007),1043-1056.
[36]B. Mandelbrot, The variation of certain speculative prices, Journal of Business, XXXVI (2004),392-417.
[37]B. Mandelbrot, M. Van Ness, Fractional Brownian motions, fractional noises and applications, SIAM Rev.10 (1968),422-437.
[38]A. Nieminem, Fractional Brownian motion and Martingale differences, Statist. Probab. Lett.,70 (2004),1-10.
[39]I. Norros,On the use of the fractional Brownian motion in the theory of connec-tionless networks, IEEE J. Sel. Ar. Commun.,13, (1995),953-962.
[40]I. Nourdin, D.Nualart and C.A. Tudor, Central and non-central limit theorems for weighted ower variations of fractional Brownian motion. Preprint.
[41]I. Nourdin, C.A. Tudor, Some linear fractional stochastic equations, Stoch.,78(2) (2006),61-65.
[42]D. Nualart, Malliavin Calculus and Related Topics, Springer 2006.
[43]V. Pipiras, Wavelet type expansion of the Rosenblatt process, The Journal of Fourier Anal. and Appl.,10(6) (2004),599-634.
[44]Z. Qian, T. Lyons, System control and rough paths, Clarendon Press, Oxford,2002.
[45]C. Rogers, Arbitrage with fractional Brownian motion, Math. Finance,7 (1997), 95-105.
[46]M. Rosenblatt, Independence and dependence, Proc.4 th Berkeley Sympos. Math. Statist. Probab., (1961),411-443.
[47]F. Russo, P. Vallois, Forward backward and symmetric stochastic integration, Prob. Theory Rel. Fields,97 (1993),403-421.
[48]F. Russo, P. Vallois, Elements of stochastic calculus via regularization, Preprint.
[49]D. M. Salopek, Tolerance to arbitrage, Stoch. Proc. Appl.,76, (1998),217-230.
[50]G. Samorodnitsky and M. Taqqu, Stable Non-Gaussian random variables, Chap-man and Hall, London, (1994).
[51]A. N. Shiryaev, On arbitrage and replication for fractal models, Centre for Math. Physics and Stoch., (1998).
[52]A.N. Shiryaev, Probability, Springer, New York,1984.
[53]J.L. Sole and F. Utzet, Stratonovich integral and trace, Stochastics and Stochastics Reports,29 (2) (1990),203-220.
[54]T.Sottinen, Fractional Brownian motion, random walks and binary market models. Finance Stochastics.,5 (2001),343-355.
[55]M. Taqqu, A representation for self-similar processes, Stochastic Process. Appl.,7 (1978),55-64.
[56]M. Taqqu, Convergence of integrated processes of arbitrary Hermite rank, Z.Wahrsch. Gebiete,50 (1979),53-83.
[57]M. Taqqu, Weak convergence to the fractional Brownian motion and to the Rosen-blatt, Z.Wahrsch. Gebiete,31 (1975),287-302.
[58]H. Teicher, Distribution and Moment Convergence of Martingales, Probab. Th. Rel. Fields.,79 (1988),303-316.
[59]S. Torres and C.A. Tudor, Donsker type theorem for the Rosenblatt process and a binary market model, Stoch. Anal. Appl.,27 (2009),555-573.
[60]C.A. Tudor, Analysis of the Rosenblatt process, ESAIM Proba. and Statist.,12 (2008),230-257.
[61]C.A. Tudor, F. Viens, Variations and estimators through Malliavin calculus, An-nal. Probab.,2009.
[62]K. Yoshihara, weak convergence to some processes defined by multiple wiener integrals, Yokohama Math. Journal Vol.,38 (1990).
[63]W. Willinger, M. Taqqu and V. Teverovsky, Long range dependence and stock returns, Finance and Stoch.,3 (1999),1-13.
[64]M. Zaehle, Integration with respect to fractal functions and stochastic calculus, Prob. Theory Rel. Fields,111 (1998),333-374.