Properties of trajectories of a multifractional Rosenblatt process

Abstract

Abstract. A Rosenblatt process and its multifractional counterpart are considered. For a multifractional Rosenblatt process, we investigate the local properties of its trajectories, namely the continuity and localizability. We prove the existence of square integrable local times for both processes. IntroductionStochastic processes with long memory (in other words, with long range dependence) remain an extensively developing topic over more than a half of a century because of their numerous applications in modelling various natural phenomena, transferring information in computer nets, dynamics of prices of financial assets, etc.The long range dependence phenomena is modelled most often with the help of a fractional Brownian motion. In the monograph by Mishura [5], a detailed survey of the literature devoted to the fractional Brownian motion as well as main results of this topic are given. A disadvantage of a fractional Brownian motion that restricts an area of its possible applications is that this process has light tails of the normal distribution. There is a number of studies devoted to processes with long range dependence that have heavier tails. These are, in particular, stable processes (see the book by Samorodnitsky and Taqqu [9]).An interesting class of processes with long range dependence that have "moderate" tails ("moderate" means that the tails are heavier than those of the normal distribution but lighter than those of a stable distribution), known as Hermite processes, appears in the so-called noncentral limit theorem for strongly dependent random variables proved in the papers by Dobrushin and Major [3] and Taqqu [10,11]. The most studied among these processes is the Hermite process of rank 2 defined in the paper by Rosenblatt [8] (the latter is also known as the Rosenblatt process). Among publications devoted to the Rosenblatt process, we mention papers by Pipiras [7], where a wavelet expansion is constructed for this process; Tudor [13], where stochastic analysis with respect to the Rosenblatt process is developed; Albin [1], where the distribution of the maximum of this process is found; and Tudor and Torres [12], where an application of the Rosenblatt process in finance mathematics is considered (namely, the Rosenblatt process is considered in [12] as a model of price evolutions).2010 Mathematics Subject Classification. Primary 60G22; Secondary 60J55, 60B10.

Universality for Persistence Exponents of Local Times of Self-Similar Processes with Stationary Increments

“…We thus give a slightly more explicit version of the argument. Existence of square integrable (in space) local times for R has been shown in [32]. Let us show continuity of the cumulative local time process ( t ) t≥0 .…”

Fractal dimensions of the Rosenblatt process

“…We need the following properties of the local time of the Rosenblatt process. Its existence was shown in [42] and one has the representation:…”

“…been established for stationary Gaussian processes, like the fractional Brownian motion, by Berman in [8]. His analytic approach, which is based on properties of the Fourier transform of the underlying process, has been adapted to the Rosenblatt setting in [42] where existence of the local time of Z was first established. Hölder regularity was then recovered in the recent paper [26].…”

“…The authors believe that many of the results above can be extended to some generalizations of the Rosenblatt process, for instance, when the time and space sets are N and d dimensional, or when the Hurst index is a function of time, as in [42]. To ease the presentation only the case N = d = 1 and H ∈ (1/2, 1)fixed is considered.…”

Universality for persistence exponents of local times of self-similar processes with stationary increments

“…We thus give a slightly more explicit version of the argument. Existence of square integrable (in space) local times for R has been shown in [She11].…”