On Fractional Stable Processes And Fields: White Noise Approach

In the theoretical and applied research fields such as economics, finance, telecom, hydrologics and nuclear physics, etc., for modelling the system we require the underlying stochastic processes own the following key statistical properties: long-range dependence, self-similarity and high variability. Since the general fractional stable processes have these properties, its statistical properties and related stochastic analysis have evoked great research interests. Especially, the study related to fractional Brownian motion has obtained many interesting results, where the application of white noise analysis methods is much more enlightenment.White noise analysis theory was initiated by T. Hida in 1975. In essence, it is an infinite dimensional Schwartz distribution theory, which has a deepgoing physical background and has been getting much attention in recent years. Firstly, Hida took the functionals of Brownian motion as the white noise functionals which laid the foundation of white noise analysis. Then, through the hard work of many experts, white noise analysis has been refined as an efficient mathematical theory gradually; its novel and distinctive ideas and methods could be taken as inspirations in many fields.Let S(R) and S^*(R) be the Schwartz space of rapidly decreasing smooth functions and the space of tempered distributions on R respectively. Denote by < · , · > the dual pairing on S^*(R)×S(R). In 1995, Albeverio S. and Wu J. had put forward a method to construct relativity quantum fields as S^*(R)-valued stochastic processes (random fields), by convolution of a generalized white noise stochastic process (random field) and a linear continuous operator G on S(R).Inspired by the above ideas, we consider the following fractional integral operatorswhere the kernel Moreover, we try to construct fractional Brownian motions and general fractional stable processes@ by repalcing linear continuous operator Q with /_. However, the operator I is not continuous from : 0,is a fractional Brownian motion with Hurst constant H(H = (3 + |). Moreover,where Wt(oS) = {u, l[o,t]} is the standard Brownian motion.The treatment of looking fractional Brownian motions as functionals of the standard Gaussian white noises, makes " the fractional white noise calculus " follow directly from the classical white noise calculus. And as examples, we prove that the fractional Girsanov formula, the Ito type integrals and the fractional Black-Scholes formula are easy consequences of their classical counterparts. Moreover,by replacing I with Riesz poly-potential (Riesz potential ), we could construct naturally anisotropic fractional (isotropic fractional) Brownian fields; and prove its stationariness of increments, self-similarity, Holder continuity of its trajectories.Conclusion 2 As for the study of stable non-Gaussian case, the above ideas could be generalized. Let a ￡ (1,2), /3 ￡ (0,1 — ^), ￡ ￡ S(R), define fractional integral operator:=r7m fV^K#.^R.■^ \P) Jxthen I is a linear continuous map from : w,￡> = (w,/fO V^ e 5(R),//a - a.a. u ￡ S*(R).Then, the /3-fractional a-stable measure \iPa = fxa o T^1 is just the image measure of pia induced by the map Tp. And, for any <^ ￡ ^} under fia. Especially, on the probabilty space (S*(R),fia),=(cj,I01M)t t>0, is the general ,0-fractional a-stable processes, with integral representation:When a = 2, the general /3-fractional a-stable processes X@ reduce to fractional Brownian motions. And in the limit case of /3 —> 0, we obtain the a-stable Levy processes.Conclusion 3 Under the white noise analysis framework, by the commuta-tivity of translation operators with convolutions, the invariance of Lebesgue measure by translation and the homogeneity of kernel K_x = yts)x- > etc., we prove that the general /^-fractional a-stable processes Xf have stationary increments; self-similarity with Hurst parameter H = /3 + ^; finite -^-variation and 7-Holder continuous versions of its trajectories for any 7 < j3.Conclusion 4 Let a ￡ (1,2), (3 = (ft,--- ,/3n),0 < pk < 1 - ± (k = 1, ? ? ? , n). Let I13 be Riesz poly-potential operator, i.e.: x ^where ln{(3) := 2?nLi W C0S(Af)> k " I/I1"0 := ElLi kfe " ^l1"^- Hence> we could define naturally the /3-fractional a-stable anisotropic random fields as cc-stable white noise functional,where [0,t] := Yfk=il^^k\- Furthermore, the /^-fractional a-stable random fields have stationary increments, self-similarity with Hurst parameter H = (A+^, ? ? ? ,/3