- Email: [email protected]
Nonhomogeneous fractional Poisson processes
Chaos, Solitons and Fractals 31 (2007) 236–241 www.elsevier.com/locate/chaos
Nonhomogeneous fractional Poisson processes a,*
, Shi-Ying Zhang a, Shen Fan
Xiao-Tian Wang b
q
b
a School of Management, Tianjin University, Tianjin 300072, PR China Computer and Information School, Zhejiang Wanli University, Ningbo 315100, PR China
Accepted 19 September 2005
Abstract In this paper, we propose a class of non-Gaussian stationary increment processes, named nonhomogeneous fracðjÞ tional Poisson processes W H ðtÞ, which permit the study of the effects of long-range dependance in a large number ðjÞ of fields including quantum physics and finance. The processes W H ðtÞ are self-similar in a wide sense, exhibit more fatter tail than Gaussian processes, and converge to the Gaussian processes in distribution in some cases. In addition, we ðjÞ also show that the intensity function k(t) strongly influences the existence of the highest finite moment of W H ðtÞ and the ðjÞ behaviour of the tail probability of W H ðtÞ. 2005 Elsevier Ltd. All rights reserved.
1. Introduction Recently stochastic models appropriate for long-range dependent phenomena have been given a great deal of attention and numerous theoretical results and successful applications have been reported. Studies have found that long-range dependence (LRD) is an important issue in modelling observed data in a large number of fields including economics [2], signal processing [26], network traffic [13], geophysical phenomena [14,29], fluid mechanics [9], and physiological signals [11]. Fractional Brownian motion (fBm) is a Gaussian process, which is fundamental and popular in the modelling and simulation of the phenomenon of LRD. This process was originally defined by Kolmogonov within a Hillbert space framework. The commonly adopted version of fBm which is defined for all times was first introduced by Mandelbrot and van Ness [16], that is, for 0 < H < 1, the fractional Brownian motion process BH(t) is defined by Z 0 Z t 1 H12 H12 H 12 dBðsÞ þ ðt sÞ ðsÞ ðt sÞ dBðsÞ ; t 2 R; ð1:1Þ BH ðtÞ :¼ CðH þ 12Þ 1 0 Rt R0 where B(t) is a standard Brownian motion process, and for t < 0, the notation 0 should be interpreted as t . Notice that if H ¼ 12, we get B12 ¼ BðtÞ; q *
t 2 R.
Supported by NSFC(70471050). Corresponding author. E-mail address: [email protected] (X.-T. Wang).
0960-0779/$ - see front matter 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.chaos.2005.09.063
X.-T. Wang et al. / Chaos, Solitons and Fractals 31 (2007) 236–241
237
The basic properties which characterize the fBm are the Gaussian distribution, the self-similarity and the stationary of increment processes. However, for many practical applications the properties above turn out to be too restrictive because (i) the fBm is self-similar in a strict sense while random variables describing many physical systems are self-similar in a wide sense; (ii) the distribution for a variety of complex systems are often characterized by fat tails while the fBm has the Gaussian distribution and hence does not exhibit fat tail phenomenon (see [25,27]). To describe the above phenomena, in this paper, we propose a class of stationary increment processes, named nonðjÞ ðjÞ homogeneous fractional Poisson processes W H ðtÞ (nfPp), as a suitable alternative to fBm. The process W H ðtÞ (H 6¼ 12) exhibits long-range dependence, has more fatter tail than Gaussian processes, and is self-similar in a wide sense. In addiðjÞ tion, the W H ðtÞ has the same spectral density as the fBm. ðjÞ The straight impulse for the studies of the W H ðtÞ came from the authorsÕ interest in the following two areas: (i) In finance, it is widely accepted that financial data exhibit the fat tails and long-memory volatility persistence and that financial market charts are neither Gaussian nor self-similar in a strict sense (see [1,15,23]). (ii) In quantum physics, it has been recognized that fractals play an important role and that quantum physics could be a result of the fractal nature of space–time in micro-physics as discussed extensively by the Canadian physicist G. Ord and the French physicist L. Nottale as well as the Egyptian physicist M.S. El Naschie (see, e.g., [4– 8,17–20]). In particular, Sidharth did a fundamental work on many aspects of E-infinity theory, and Iovane obtained some important results by applying E-infinity theory to cosmological models of the universe (see [21,24]). ðjÞ
Remark. The process W H ðtÞ is different from the ÔFractional Poisson processÕ, which is defined by Jumarie through the Liouville–Riemann fractional derivative of order a 2 R (see Definition 4.1. in [10]).
2. Nonhomogeneous fractional Poisson processes In this section, we will give the definition and the fundamental properties of the nonhomogeneous fractional Poisson process. We work on a probability space ðX; F; P Þ. Assume that B(t) a standard Brownian motion, and N(t) a nonhomogeneous Poisson process with intensity function k(t) > 0, are defined on the probability space ðX; F; P Þ. In addition, we assume that B(t) is independent of N(t), and B(0) = N(0) = 0. Definition 2.1. A random process Y(t), t P 0 is said to be self-similar with parameter H > 0 (a) in a strict sense, if for any a > 0 d
Y ðatÞ ¼ aH Y ðtÞ;
t P 0;
(b) in a wide sense, if E[Y(t)] = 0 qðt; sÞ ¼ E½Y ðtÞY ðsÞ < 1 and
qðat; asÞ ¼ a2H qðt; sÞ.
BH(t) with Hurst exponent H 2 (0, 1) is a continuous Gaussian process with the following properties: (i) E[BH(t)] = 0 for all t 2 R; (ii) E½BH ðtÞBH ðsÞ ¼ 12 V 2H ðjsj2H þ jtj2H js tj2H Þ for all s; t 2 R, where V 2H ¼ E½BH ð1Þ2 is a constant. If V 2H ¼ 1, then BH(t) is called standard fractional Brownian motion. Clearly (a)–(b) are equivalent for fBm. ðjÞ
Definition 2.2. Let H 2 (0, 1). We call the following random process W H ðtÞ (j = 1, 2, 3), nonhomogeneous fractional Poisson process.
238
X.-T. Wang et al. / Chaos, Solitons and Fractals 31 (2007) 236–241 ðjÞ
For t > 0, W H ðtÞ is defined by 0 1 Z 0 ðt sÞH12 ðsÞH12 Z t H 12 1 ðt sÞ ð1Þ @ pffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffi dqðsÞA; dqðsÞ þ W H ðtÞ ¼ CðH þ 12Þ kðsÞ kðsÞ 1 0 ! 1 Z 0 Z t 1 ðt sÞH2 ð2Þ H12 H12 pffiffiffiffiffiffiffiffi dqðsÞ dBðsÞ þ ðt sÞ ðsÞ W H ðtÞ ¼ CðH þ 12Þ 1 kðsÞ 0
ð2:1Þ ð2:2Þ
and
! 1 1 Z 0 Z t 1 ððt sÞH 2 ðsÞH2 Þ H12 pffiffiffiffiffiffiffiffi ¼ W dqðsÞ þ ðt sÞ dBðsÞ ; CðH þ 12Þ 1 kðsÞ 0 Rt Rt R0 where qðtÞ ¼ N ðtÞ 0 kðsÞ ds, and for all t < 0, 0 should be interpreted as t . ð3Þ H ðtÞ
Remark. E[q(t)] = 0, E½qðtÞ2 ¼
Rt 0
ð2:3Þ
ðjÞ
kðsÞ ds, and for H 2 (0, 1), W H ðtÞ is a ‘‘fractal shot noise process’’. ðjÞ
ðjÞ
By definition, we know that W H ¼ ðW H ðtÞÞtP0 has the following properties: ðjÞ
ðjÞ
(1) W H ð0Þ ¼ 0, and E½W H ðtÞ ¼ 0 for all t P 0. ðjÞ
(2) The covariance function of W H ðtÞ is 1 2 ðjÞ ðjÞ E½W H ðtÞW H ðsÞ ¼ V H ðjsj2H þ jtj2H js tj2H Þ; 2 2 R0 2 H 12 H 12 1 1 where V H ¼ ðCðHþ1ÞÞ2 2H þ 1 ð1 sÞ ðsÞ ds .
ð2:4Þ
2
ðjÞ
(3) W H ðtÞ is self-similar in a wide sense.In fact, for any given a > 0, from Eq. (2.4), we have ðjÞ
ðjÞ
ðjÞ
ðjÞ
E½W H ðasÞW H ðatÞ ¼ a2H E½W H ðsÞW H ðtÞ
ð2:5Þ
and ðjÞ
2
ðjÞ
E½W H ðtÞ W H ðsÞ2 ¼ V H jt sj2H . (4) W
ðjÞ H ðtÞ
ð2:6Þ
is a wide-sense stationary increment process.In fact, for any b > 0, ðjÞ
ðjÞ
ðjÞ
ðjÞ
F ðs tÞ ¼ E½ðW H ðt þ bÞ W H ðtÞÞðW H ðs þ bÞ W H ðsÞÞ 1 2 ¼ V H jðs tÞ þ bj2H þ jðs tÞ bj2H 2js tj2H . 2
ð2:7Þ
ðjÞ
(5) If H 6¼ 12, W H ðtÞ is not a Gaussian process. ð3Þ
ð1Þ
ð2Þ
(6) If H ¼ 12, then W H ðtÞ ¼ BðtÞ and W H ðtÞ ¼ W H ðtÞ ¼
Rt 0
p1ffiffiffiffiffiffi dqðsÞ. kðsÞ
ðjÞ
(7) From Eq. (2.6), we know that W H ðtÞ is continuous in the sense of ‘‘the 2nd mean’’ (q.m.). On the other hand, from Eq. (2.6) and [3], we know that ð3Þ • for H P 12, W H ðtÞ has continuous paths; ð1Þ
ð2Þ
• for H > 12, W H ðtÞ and W H ðtÞ have continuous paths. ðjÞ (8) If H > 12, from Eq. (2.4) we know that the process W H ðtÞ exhibits long-range dependence. ðjÞ (9) The spectral representation of W H ðtÞ. For any b > 0, from the formula of the spectral representation of a wide-sense-stationary process, we have Z 1 Z 1 ðjÞ ðjÞ ðjÞ ðjÞ cos kt dU b ðkÞ þ sin kt dV b ðkÞ ðt P 0Þ; W H ðt þ bÞ W H ðtÞ ¼ 0 ðjÞ Ub
ðjÞ b
0
and V are uncorrelated to each other and are centered second order processes with orthogonal where increments. ðjÞ (10) The spectrum of W H ðtÞ. By definition, the Wigner–Vile spectrum (WVS) of a nonstationary process X(t) with covariance function qX(t, s) is given by Z 1 s s qX t þ ; t eixs ds. S X ðt; xÞ ¼ 2 2 1
X.-T. Wang et al. / Chaos, Solitons and Fractals 31 (2007) 236–241
239
ðjÞ
If we apply this definition to W H ðtÞ, we get S W ðjÞ ðt; xÞ ¼ ð1 212H cos 2xtÞ H
C jxj2Hþ1
;
2
pH V H and H 6¼ 12. where C ¼ Cð12HÞ cos pH
ðjÞ
3. A simple theorem of the convergence of W H ðtÞ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðjÞ W H ðtÞ ðjÞ , as t ! +1. Here r ðtÞ ¼ VarðW H ðtÞÞ. In this section, our interest is in the asymptotic distribution of H r ðtÞ H a ðjÞ s ; s 2 ½0; þ1Þ W ðtÞ . Then, for a > 1, rHH ðtÞ converges in distribution to a normal N(0, 1) limit, Theorem 3.1. Let kðsÞ ¼ k; s 2 ð1; 0Þ as t ! +1, where k > 0 is a constant. Proof. See Appendix A.
h ðjÞ
Remark. From Theorem 3.1, we know that W H ðtÞ asymptotically converges to the fractional Brownian motion BH(t) in distribution. ðjÞ
4. The existence of the moments of W H ðtÞ In this section, we show that the intensity function k(t) strongly influences the existence of the highest finite moment ðjÞ ðjÞ of W H ðtÞ and the behaviour of the tail probability of W H ðtÞ (j = 1, 2, 3). Assume that 1 Z t Z 0 ðt sÞH12 ðsÞH12 ðt sÞH 2 pffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffi dqðsÞ; Y 2 ðtÞ ¼ dqðsÞ. Y 1 ðtÞ ¼ kðsÞ kðsÞ 0 1 From [22], we know that the characteristic function of Yj(t) (j = 1, 2) has the following form: ( ! ) H 1 Z t pffiffiffiffiffiffiffiffi Z t 2 ðtsÞ iw pffiffiffiffi H12 kðsÞ /Y 1 ðwÞ ¼ exp wi kðsÞ e 1 ds kðsÞðt sÞ ds þ 0
and /Y 2 ðwÞ ¼ exp
(Z
0
"
ð4:1Þ
0
!# ) H 1 H 1 2 ðsÞ 2 ðtsÞ pffiffiffiffiffiffiffiffi pffiffiffiffi iw H 12 H12 kðsÞ wi þ kðsÞ e kðsÞ ðt sÞ ðsÞ 1 ds .
ð4:2Þ
1
To simplify the calculation in Eqs. (4.1) and (4.2), in the following we assume that H > 12. If k(s) = k > 0 is a constant, from Eqs. (4.1) and (4.2) we know that ðjÞ
for any given k > 0, EjW H ðtÞjk < þ1 ðjÞ
and for large x > 0, P ðjW H ðtÞj > xÞ <
ðjÞ
EjW H ðtÞjk xk
.
On the other hand, if k(t) = ta and a 2 (1, 2), from Eq. (4.1) we know that ð2Þ
EjW H ðtÞj3 < þ1, ð2Þ
4 EjW H ðtÞj ¼ þ1 and ð2Þ 3 EjW ðtÞj ð2Þ P jW H ðtÞj > x < Hx3 , ð2Þ
which exhibits that W H ðtÞ has a fat-tailed marginal distribution. Remark. Define rt as the return of a given stock in a given time interval Dt. In econometrics, a large number of empirical studies have reported that the tails of the stock return distributions can be well described by a power-law decay, i.e.
240
X.-T. Wang et al. / Chaos, Solitons and Fractals 31 (2007) 236–241
P ðjrt j > xÞ xb with b 2 (2.5, 4).
5. Conclusion ðjÞ
In this paper, we give the definition of the nonhomogeneous fractional Poisson process W H ðtÞ, obtain the fundamenðjÞ ðjÞ tal properties and the asymptotic distribution of W H ðtÞ in some cases. We believe W H ðtÞ do provide useful models for a host of natural phenomena including quantum physics and finance.
Appendix A Proof of Theorem 3.1. In the following, we assume that H 6¼ 12 ; a > 1; kðsÞ ¼ Let 1 1 Z t Z t ðt sÞH2 ðt sÞH 2 pffiffiffiffiffiffiffiffi dqðsÞ; Y1 ðtÞ ¼ pffiffiffiffiffiffiffiffi dN ðsÞ. Y 1 ðtÞ ¼ kðsÞ kðsÞ 0 0
sa ; k;
sP0 ¼ min and b s < 0.
2 4H jaj ; j12H j
.
Since t2H r1 2 ðtÞ ¼ VarðY1 ðtÞÞ ¼ ! 0; 2H and for some d 2 ð0; bÞ Rt
ðH 1Þð2þdÞ 2 1þd ðkðsÞÞ 2
kðsÞðtsÞ
0
ðr1 ðtÞÞ2þd
ds
d
¼
ð2H Þð1þ2Þ
as t ! þ1
R1 0
1
ð1 sÞðH 2Þð2þdÞ s 2 ds t
ad
ð1þaÞd 2
from Theorem 3 and Corollary 4 of [12], we know that process, i.e.
! 0;
Y 1 ðtÞ r1 ðtÞ
as t ! þ1;
converges in distribution to the zero-mean real Gaussian
Y 1 ðtÞ is asymptotically N ð0; 1Þ as t ! þ1; r1 ðtÞ where r21 ðtÞ ¼ VarðY 1 ðtÞÞ. On the other hand, since k(s) = k > 0 is a constant for s < 0, from the proof of the Theorem ðjÞ W ðtÞ 3.1 of [28], we know that rHH ðtÞ converges in distribution to a normal N(0, 1) limit, i.e. ðjÞ
W H ðtÞ is asymptotically N ð0; 1Þ. rH ðtÞ
References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14]
Bollerslev T, Mikkelsen HO. J Econ 1996;73:151. Anh VV, Heyde CC. J Stat Plan Infer 1990;80(1/2):1. Ash RB, Gardner MF. Topics in stochastic process. New York: Academic Press; 1975. El Naschie MS. Chaos, Solitons & Fractals 2005;25:509. El Naschie MS. Chaos, Solitons & Fractals 2004;19:209. El Naschie MS. Chaos, Solitons & Fractals 1992;1(5):485. El Naschie MS. Int J Nonlinear Sci Numer Simul 2004;5(3):191. El Naschie MS. Chaos, Solitons & Fractals 2005;23:711. Hurst HE, Black RP, Sinaika YM. Proc IEEE 1991:3557. Jumarie G. Chaos, Solitons & Fractals 2001;12:2577. Kaalda J. Fractals 1993;1(2):191. Lane JA. J Appl Prob 1984;21:287. Leland WE et al. IEEE/ACM Trans Network 1994;2:1. Lo T et al. Proc IEEE 1993;140:243.
X.-T. Wang et al. / Chaos, Solitons and Fractals 31 (2007) 236–241 [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29]
Mandelbrot BB. Scient Am 1999;4:71. Mandelbrot BB, van Ness JW. SIAM Rev 1968;10(4):422. El Naschie MS. Chaos, Solitons & Fractals 2003;18:401. Nottale L. Chaos, Solitons & Fractals 1996;7(6):877. Ord GN. J Phys A: Math Gen 1983;16:1869. Ord GN. Chaos, Solitons & Fractals 2003;17:609. Sidharth BG. Chaos, Solitons & Fractals 2003;18(1):197. Papoulis A. Probability, random variables, and stochastic process. McGraw-Hill Book Company; 1965. Plerou V et al. Phys Rev E 2000;62(3):3023. Iovane G. Chaos, Solitons & Fractals 2005;23(3):691. Stanley HE et al. Physica A 2001;299:1. Ve´hel JL. Fractals 1995;3(4):755. Voss R. Phys Rev Lett 1992;68:3805. Wang X-T, Wen Z-X. Chaos, Solitons & Fractals 2003;18:169. Lapenna V et al. Chaos, Solitons & Fractals 2004;21:491.
241
Nonhomogeneous fractional Poisson processes a,*
, Shi-Ying Zhang a, Shen Fan
Xiao-Tian Wang b
q
b
a School of Management, Tianjin University, Tianjin 300072, PR China Computer and Information School, Zhejiang Wanli University, Ningbo 315100, PR China
Accepted 19 September 2005
Abstract In this paper, we propose a class of non-Gaussian stationary increment processes, named nonhomogeneous fracðjÞ tional Poisson processes W H ðtÞ, which permit the study of the effects of long-range dependance in a large number ðjÞ of fields including quantum physics and finance. The processes W H ðtÞ are self-similar in a wide sense, exhibit more fatter tail than Gaussian processes, and converge to the Gaussian processes in distribution in some cases. In addition, we ðjÞ also show that the intensity function k(t) strongly influences the existence of the highest finite moment of W H ðtÞ and the ðjÞ behaviour of the tail probability of W H ðtÞ. 2005 Elsevier Ltd. All rights reserved.
1. Introduction Recently stochastic models appropriate for long-range dependent phenomena have been given a great deal of attention and numerous theoretical results and successful applications have been reported. Studies have found that long-range dependence (LRD) is an important issue in modelling observed data in a large number of fields including economics [2], signal processing [26], network traffic [13], geophysical phenomena [14,29], fluid mechanics [9], and physiological signals [11]. Fractional Brownian motion (fBm) is a Gaussian process, which is fundamental and popular in the modelling and simulation of the phenomenon of LRD. This process was originally defined by Kolmogonov within a Hillbert space framework. The commonly adopted version of fBm which is defined for all times was first introduced by Mandelbrot and van Ness [16], that is, for 0 < H < 1, the fractional Brownian motion process BH(t) is defined by Z 0 Z t 1 H12 H12 H 12 dBðsÞ þ ðt sÞ ðsÞ ðt sÞ dBðsÞ ; t 2 R; ð1:1Þ BH ðtÞ :¼ CðH þ 12Þ 1 0 Rt R0 where B(t) is a standard Brownian motion process, and for t < 0, the notation 0 should be interpreted as t . Notice that if H ¼ 12, we get B12 ¼ BðtÞ; q *
t 2 R.
Supported by NSFC(70471050). Corresponding author. E-mail address: [email protected] (X.-T. Wang).
0960-0779/$ - see front matter 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.chaos.2005.09.063
X.-T. Wang et al. / Chaos, Solitons and Fractals 31 (2007) 236–241
237
The basic properties which characterize the fBm are the Gaussian distribution, the self-similarity and the stationary of increment processes. However, for many practical applications the properties above turn out to be too restrictive because (i) the fBm is self-similar in a strict sense while random variables describing many physical systems are self-similar in a wide sense; (ii) the distribution for a variety of complex systems are often characterized by fat tails while the fBm has the Gaussian distribution and hence does not exhibit fat tail phenomenon (see [25,27]). To describe the above phenomena, in this paper, we propose a class of stationary increment processes, named nonðjÞ ðjÞ homogeneous fractional Poisson processes W H ðtÞ (nfPp), as a suitable alternative to fBm. The process W H ðtÞ (H 6¼ 12) exhibits long-range dependence, has more fatter tail than Gaussian processes, and is self-similar in a wide sense. In addiðjÞ tion, the W H ðtÞ has the same spectral density as the fBm. ðjÞ The straight impulse for the studies of the W H ðtÞ came from the authorsÕ interest in the following two areas: (i) In finance, it is widely accepted that financial data exhibit the fat tails and long-memory volatility persistence and that financial market charts are neither Gaussian nor self-similar in a strict sense (see [1,15,23]). (ii) In quantum physics, it has been recognized that fractals play an important role and that quantum physics could be a result of the fractal nature of space–time in micro-physics as discussed extensively by the Canadian physicist G. Ord and the French physicist L. Nottale as well as the Egyptian physicist M.S. El Naschie (see, e.g., [4– 8,17–20]). In particular, Sidharth did a fundamental work on many aspects of E-infinity theory, and Iovane obtained some important results by applying E-infinity theory to cosmological models of the universe (see [21,24]). ðjÞ
Remark. The process W H ðtÞ is different from the ÔFractional Poisson processÕ, which is defined by Jumarie through the Liouville–Riemann fractional derivative of order a 2 R (see Definition 4.1. in [10]).
2. Nonhomogeneous fractional Poisson processes In this section, we will give the definition and the fundamental properties of the nonhomogeneous fractional Poisson process. We work on a probability space ðX; F; P Þ. Assume that B(t) a standard Brownian motion, and N(t) a nonhomogeneous Poisson process with intensity function k(t) > 0, are defined on the probability space ðX; F; P Þ. In addition, we assume that B(t) is independent of N(t), and B(0) = N(0) = 0. Definition 2.1. A random process Y(t), t P 0 is said to be self-similar with parameter H > 0 (a) in a strict sense, if for any a > 0 d
Y ðatÞ ¼ aH Y ðtÞ;
t P 0;
(b) in a wide sense, if E[Y(t)] = 0 qðt; sÞ ¼ E½Y ðtÞY ðsÞ < 1 and
qðat; asÞ ¼ a2H qðt; sÞ.
BH(t) with Hurst exponent H 2 (0, 1) is a continuous Gaussian process with the following properties: (i) E[BH(t)] = 0 for all t 2 R; (ii) E½BH ðtÞBH ðsÞ ¼ 12 V 2H ðjsj2H þ jtj2H js tj2H Þ for all s; t 2 R, where V 2H ¼ E½BH ð1Þ2 is a constant. If V 2H ¼ 1, then BH(t) is called standard fractional Brownian motion. Clearly (a)–(b) are equivalent for fBm. ðjÞ
Definition 2.2. Let H 2 (0, 1). We call the following random process W H ðtÞ (j = 1, 2, 3), nonhomogeneous fractional Poisson process.
238
X.-T. Wang et al. / Chaos, Solitons and Fractals 31 (2007) 236–241 ðjÞ
For t > 0, W H ðtÞ is defined by 0 1 Z 0 ðt sÞH12 ðsÞH12 Z t H 12 1 ðt sÞ ð1Þ @ pffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffi dqðsÞA; dqðsÞ þ W H ðtÞ ¼ CðH þ 12Þ kðsÞ kðsÞ 1 0 ! 1 Z 0 Z t 1 ðt sÞH2 ð2Þ H12 H12 pffiffiffiffiffiffiffiffi dqðsÞ dBðsÞ þ ðt sÞ ðsÞ W H ðtÞ ¼ CðH þ 12Þ 1 kðsÞ 0
ð2:1Þ ð2:2Þ
and
! 1 1 Z 0 Z t 1 ððt sÞH 2 ðsÞH2 Þ H12 pffiffiffiffiffiffiffiffi ¼ W dqðsÞ þ ðt sÞ dBðsÞ ; CðH þ 12Þ 1 kðsÞ 0 Rt Rt R0 where qðtÞ ¼ N ðtÞ 0 kðsÞ ds, and for all t < 0, 0 should be interpreted as t . ð3Þ H ðtÞ
Remark. E[q(t)] = 0, E½qðtÞ2 ¼
Rt 0
ð2:3Þ
ðjÞ
kðsÞ ds, and for H 2 (0, 1), W H ðtÞ is a ‘‘fractal shot noise process’’. ðjÞ
ðjÞ
By definition, we know that W H ¼ ðW H ðtÞÞtP0 has the following properties: ðjÞ
ðjÞ
(1) W H ð0Þ ¼ 0, and E½W H ðtÞ ¼ 0 for all t P 0. ðjÞ
(2) The covariance function of W H ðtÞ is 1 2 ðjÞ ðjÞ E½W H ðtÞW H ðsÞ ¼ V H ðjsj2H þ jtj2H js tj2H Þ; 2 2 R0 2 H 12 H 12 1 1 where V H ¼ ðCðHþ1ÞÞ2 2H þ 1 ð1 sÞ ðsÞ ds .
ð2:4Þ
2
ðjÞ
(3) W H ðtÞ is self-similar in a wide sense.In fact, for any given a > 0, from Eq. (2.4), we have ðjÞ
ðjÞ
ðjÞ
ðjÞ
E½W H ðasÞW H ðatÞ ¼ a2H E½W H ðsÞW H ðtÞ
ð2:5Þ
and ðjÞ
2
ðjÞ
E½W H ðtÞ W H ðsÞ2 ¼ V H jt sj2H . (4) W
ðjÞ H ðtÞ
ð2:6Þ
is a wide-sense stationary increment process.In fact, for any b > 0, ðjÞ
ðjÞ
ðjÞ
ðjÞ
F ðs tÞ ¼ E½ðW H ðt þ bÞ W H ðtÞÞðW H ðs þ bÞ W H ðsÞÞ 1 2 ¼ V H jðs tÞ þ bj2H þ jðs tÞ bj2H 2js tj2H . 2
ð2:7Þ
ðjÞ
(5) If H 6¼ 12, W H ðtÞ is not a Gaussian process. ð3Þ
ð1Þ
ð2Þ
(6) If H ¼ 12, then W H ðtÞ ¼ BðtÞ and W H ðtÞ ¼ W H ðtÞ ¼
Rt 0
p1ffiffiffiffiffiffi dqðsÞ. kðsÞ
ðjÞ
(7) From Eq. (2.6), we know that W H ðtÞ is continuous in the sense of ‘‘the 2nd mean’’ (q.m.). On the other hand, from Eq. (2.6) and [3], we know that ð3Þ • for H P 12, W H ðtÞ has continuous paths; ð1Þ
ð2Þ
• for H > 12, W H ðtÞ and W H ðtÞ have continuous paths. ðjÞ (8) If H > 12, from Eq. (2.4) we know that the process W H ðtÞ exhibits long-range dependence. ðjÞ (9) The spectral representation of W H ðtÞ. For any b > 0, from the formula of the spectral representation of a wide-sense-stationary process, we have Z 1 Z 1 ðjÞ ðjÞ ðjÞ ðjÞ cos kt dU b ðkÞ þ sin kt dV b ðkÞ ðt P 0Þ; W H ðt þ bÞ W H ðtÞ ¼ 0 ðjÞ Ub
ðjÞ b
0
and V are uncorrelated to each other and are centered second order processes with orthogonal where increments. ðjÞ (10) The spectrum of W H ðtÞ. By definition, the Wigner–Vile spectrum (WVS) of a nonstationary process X(t) with covariance function qX(t, s) is given by Z 1 s s qX t þ ; t eixs ds. S X ðt; xÞ ¼ 2 2 1
X.-T. Wang et al. / Chaos, Solitons and Fractals 31 (2007) 236–241
239
ðjÞ
If we apply this definition to W H ðtÞ, we get S W ðjÞ ðt; xÞ ¼ ð1 212H cos 2xtÞ H
C jxj2Hþ1
;
2
pH V H and H 6¼ 12. where C ¼ Cð12HÞ cos pH
ðjÞ
3. A simple theorem of the convergence of W H ðtÞ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðjÞ W H ðtÞ ðjÞ , as t ! +1. Here r ðtÞ ¼ VarðW H ðtÞÞ. In this section, our interest is in the asymptotic distribution of H r ðtÞ H a ðjÞ s ; s 2 ½0; þ1Þ W ðtÞ . Then, for a > 1, rHH ðtÞ converges in distribution to a normal N(0, 1) limit, Theorem 3.1. Let kðsÞ ¼ k; s 2 ð1; 0Þ as t ! +1, where k > 0 is a constant. Proof. See Appendix A.
h ðjÞ
Remark. From Theorem 3.1, we know that W H ðtÞ asymptotically converges to the fractional Brownian motion BH(t) in distribution. ðjÞ
4. The existence of the moments of W H ðtÞ In this section, we show that the intensity function k(t) strongly influences the existence of the highest finite moment ðjÞ ðjÞ of W H ðtÞ and the behaviour of the tail probability of W H ðtÞ (j = 1, 2, 3). Assume that 1 Z t Z 0 ðt sÞH12 ðsÞH12 ðt sÞH 2 pffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffi dqðsÞ; Y 2 ðtÞ ¼ dqðsÞ. Y 1 ðtÞ ¼ kðsÞ kðsÞ 0 1 From [22], we know that the characteristic function of Yj(t) (j = 1, 2) has the following form: ( ! ) H 1 Z t pffiffiffiffiffiffiffiffi Z t 2 ðtsÞ iw pffiffiffiffi H12 kðsÞ /Y 1 ðwÞ ¼ exp wi kðsÞ e 1 ds kðsÞðt sÞ ds þ 0
and /Y 2 ðwÞ ¼ exp
(Z
0
"
ð4:1Þ
0
!# ) H 1 H 1 2 ðsÞ 2 ðtsÞ pffiffiffiffiffiffiffiffi pffiffiffiffi iw H 12 H12 kðsÞ wi þ kðsÞ e kðsÞ ðt sÞ ðsÞ 1 ds .
ð4:2Þ
1
To simplify the calculation in Eqs. (4.1) and (4.2), in the following we assume that H > 12. If k(s) = k > 0 is a constant, from Eqs. (4.1) and (4.2) we know that ðjÞ
for any given k > 0, EjW H ðtÞjk < þ1 ðjÞ
and for large x > 0, P ðjW H ðtÞj > xÞ <
ðjÞ
EjW H ðtÞjk xk
.
On the other hand, if k(t) = ta and a 2 (1, 2), from Eq. (4.1) we know that ð2Þ
EjW H ðtÞj3 < þ1, ð2Þ
4 EjW H ðtÞj ¼ þ1 and ð2Þ 3 EjW ðtÞj ð2Þ P jW H ðtÞj > x < Hx3 , ð2Þ
which exhibits that W H ðtÞ has a fat-tailed marginal distribution. Remark. Define rt as the return of a given stock in a given time interval Dt. In econometrics, a large number of empirical studies have reported that the tails of the stock return distributions can be well described by a power-law decay, i.e.
240
X.-T. Wang et al. / Chaos, Solitons and Fractals 31 (2007) 236–241
P ðjrt j > xÞ xb with b 2 (2.5, 4).
5. Conclusion ðjÞ
In this paper, we give the definition of the nonhomogeneous fractional Poisson process W H ðtÞ, obtain the fundamenðjÞ ðjÞ tal properties and the asymptotic distribution of W H ðtÞ in some cases. We believe W H ðtÞ do provide useful models for a host of natural phenomena including quantum physics and finance.
Appendix A Proof of Theorem 3.1. In the following, we assume that H 6¼ 12 ; a > 1; kðsÞ ¼ Let 1 1 Z t Z t ðt sÞH2 ðt sÞH 2 pffiffiffiffiffiffiffiffi dqðsÞ; Y1 ðtÞ ¼ pffiffiffiffiffiffiffiffi dN ðsÞ. Y 1 ðtÞ ¼ kðsÞ kðsÞ 0 0
sa ; k;
sP0 ¼ min and b s < 0.
2 4H jaj ; j12H j
.
Since t2H r1 2 ðtÞ ¼ VarðY1 ðtÞÞ ¼ ! 0; 2H and for some d 2 ð0; bÞ Rt
ðH 1Þð2þdÞ 2 1þd ðkðsÞÞ 2
kðsÞðtsÞ
0
ðr1 ðtÞÞ2þd
ds
d
¼
ð2H Þð1þ2Þ
as t ! þ1
R1 0
1
ð1 sÞðH 2Þð2þdÞ s 2 ds t
ad
ð1þaÞd 2
from Theorem 3 and Corollary 4 of [12], we know that process, i.e.
! 0;
Y 1 ðtÞ r1 ðtÞ
as t ! þ1;
converges in distribution to the zero-mean real Gaussian
Y 1 ðtÞ is asymptotically N ð0; 1Þ as t ! þ1; r1 ðtÞ where r21 ðtÞ ¼ VarðY 1 ðtÞÞ. On the other hand, since k(s) = k > 0 is a constant for s < 0, from the proof of the Theorem ðjÞ W ðtÞ 3.1 of [28], we know that rHH ðtÞ converges in distribution to a normal N(0, 1) limit, i.e. ðjÞ
W H ðtÞ is asymptotically N ð0; 1Þ. rH ðtÞ
References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14]
Bollerslev T, Mikkelsen HO. J Econ 1996;73:151. Anh VV, Heyde CC. J Stat Plan Infer 1990;80(1/2):1. Ash RB, Gardner MF. Topics in stochastic process. New York: Academic Press; 1975. El Naschie MS. Chaos, Solitons & Fractals 2005;25:509. El Naschie MS. Chaos, Solitons & Fractals 2004;19:209. El Naschie MS. Chaos, Solitons & Fractals 1992;1(5):485. El Naschie MS. Int J Nonlinear Sci Numer Simul 2004;5(3):191. El Naschie MS. Chaos, Solitons & Fractals 2005;23:711. Hurst HE, Black RP, Sinaika YM. Proc IEEE 1991:3557. Jumarie G. Chaos, Solitons & Fractals 2001;12:2577. Kaalda J. Fractals 1993;1(2):191. Lane JA. J Appl Prob 1984;21:287. Leland WE et al. IEEE/ACM Trans Network 1994;2:1. Lo T et al. Proc IEEE 1993;140:243.
X.-T. Wang et al. / Chaos, Solitons and Fractals 31 (2007) 236–241 [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29]
Mandelbrot BB. Scient Am 1999;4:71. Mandelbrot BB, van Ness JW. SIAM Rev 1968;10(4):422. El Naschie MS. Chaos, Solitons & Fractals 2003;18:401. Nottale L. Chaos, Solitons & Fractals 1996;7(6):877. Ord GN. J Phys A: Math Gen 1983;16:1869. Ord GN. Chaos, Solitons & Fractals 2003;17:609. Sidharth BG. Chaos, Solitons & Fractals 2003;18(1):197. Papoulis A. Probability, random variables, and stochastic process. McGraw-Hill Book Company; 1965. Plerou V et al. Phys Rev E 2000;62(3):3023. Iovane G. Chaos, Solitons & Fractals 2005;23(3):691. Stanley HE et al. Physica A 2001;299:1. Ve´hel JL. Fractals 1995;3(4):755. Voss R. Phys Rev Lett 1992;68:3805. Wang X-T, Wen Z-X. Chaos, Solitons & Fractals 2003;18:169. Lapenna V et al. Chaos, Solitons & Fractals 2004;21:491.
241