A Ferguson–Klass–LePage series representation of multistable multifractional motions and related processes

Abstract

The study of non-stationary processes whose local form has controlled properties is a fruitful and important area of research, both in theory and applications. In [9], a particular way of constructing such processes was investigated, leading in particular to multifractional multistable processes, which were built using sums over Poisson processes. We present here a different construction of these processes, based on the Ferguson -Klass -LePage series representation of stable processes. We consider various particular cases of interest, including multistable Lévy motion, multistable reverse Ornstein-Uhlenbeck process, log-fractional multistable motion and linear multistable multifractional motion. We also show that the processes defined here have the same finite dimensional distributions as the corresponding processes constructed in [9]. Finally, we display numerical experiments showing graphs of synthesized paths of such processes.

Multistable Processes and Localizability

“…We then define multistable processes in terms of multistable integrals and give sufficient conditions for such processes to be localisable or strongly localisable, that is to have a local scaling limit at time t that is an α(t)-stable process. It turns out that these multistable processes differ significantly from those of [8,11] in the nature of their finite-dimensional distributions. We give a range of examples of these processes.…”

“…In this light, we may compare the definition of the multistable processes of Section 3 with the other definitions of multistable processes given in [8,11] which depend on defining a random field in terms of a suitable function f (t, v, x) and taking a diagonal section by setting v = t. It is shown in [11,Proposition 6.13] that the finite-dimensional distributions of the processes obtained by both the Poisson point representation [8] and the random series representation [11] are given by…”

“…We call a stochastic process {Y (t), t ∈ R} multistable if for almost all u, Y is localisable at u with Y u an α-stable process for some α = α(u), where 0 < α(u) ≤ 2. Various constructions of multistable processes are given in [8,9,11].…”

“…Some of these are considered in [8,9,11] using alternative definitions of multistable processes. It is convenient to make the convention that…”

“…Similarly, a number of constructions for multistable processes have been given recently, generalizing the various constructions of stable processes. One approach is based on Poisson point process [8] and another is based on sums of random series [11]. Here we first use characteristic functions to construct multistable integrals and measures.…”