1,721,106 research outputs found
Long-memory Gaussian processes governed by generalized Fokker-Planck equations
No full textIt is well-known that the transition function of the Ornstein-Uhlenbeck process solves the Fokker-Planck equation. This standard setting has been recently generalized in different directions, for example, by considering the so-called a-stable driven Ornstein-Uhlenbeck, or by time-changing the original process with an inverse stable subordinator. In both cases, the corresponding partial differential equations involve fractional derivatives (of Riesz and Riemann-Liouville types, respectively) and the solution is not Gaussian. We consider here a new model, which cannot be expressed by a random time-change of the original process: we start by a Fokker-Planck equation (in Fourier space) with the time-derivative replaced by a new fractional differential operator. The resulting process is Gaussian and, in the stationary case, exhibits long-range dependence. Moreover, we consider further extensions, by means of the so-called convolution-type derivative
Lévy processes linked to the lower-incomplete gamma function
Full text linkWe start by defining a subordinator by means of the lower-incomplete gamma function. This can be considered as an approximation of the stable subordinator, easier to be handled in view of its finite activity. A tempered version is also considered in order to overcome the drawback of infinite moments. Then, we study Lévy processes that are time-changed by these subordinators with particular attention to the Brownian case. An approximation of the fractional derivative (as well as of the fractional power of operators) arises from the analysis of governing equations. Finally, we show that time-changing the fractional Brownian motion produces a model of anomalous diffusion, which exhibits a sub-diffusive behavior
Non-central moderate deviations for compound fractional Poisson processes
No full textThe term moderate deviations is often used in the literature to mean a class of large deviation principles that, in some sense, fills the gap between a convergence in probability to zero (governed by a large deviation principle) and a weak convergence to a centered Normal distribution. We talk about non-central moderate deviations when the weak convergence is towards a non-Gaussian distribution. In this paper we study non-central moderate deviations for compound fractional Poisson processes with light-tailed jumps
Stochastic applications of Caputo-type convolution operators with nonsingular kernels
No full textWe consider here convolution operators, in the Caputo sense, with nonsingular kernels. We prove that the solutions to some integro-differential equations with such operators (acting on the space variable) coincide with the transition densities of a particular class of Lévy subordinators (i.e. compound Poisson processes with non-negative jumps). We then extend these results to the case where the kernels of the operators have random parameters, with given distribution. This assumption allows greater flexibility in the choice of the kernel’s parameters and, consequently, of the jumps’ density function
Multivariate fractional Poisson processes and compound sums
Full text linkIn this paper we present multivariate space-time fractional Poisson processes by considering
common random time-changes of a (finite-dimensional) vector of independent
classical (nonfractional) Poisson processes. In some cases we also consider compound
processes. We obtain some equations in terms of some suitable fractional derivatives and
fractional difference operators, which provides the extension of known equations for the
univariate processes
A class of processes defined in the white noise space through generalized fractional operators
No full textThe generalization of fractional Brownian motion in the white and grey noise spaces has been recently carried over, following the Mandelbrot–Van Ness representation, through Riemann–Liouville type fractional operators. Our aim is to extend this construction by means of more general fractional derivatives and integrals, which we define as Fourier-multiplier operators and then specialize by means of Bernstein functions. More precisely, we introduce a general
class of kernel-driven processes which encompasses, as special cases, a number of models in the literature, including fractional Brownian motion, tempered fractional Brownian motion, Ornstein–Uhlenbeck process. The greater generality of our model, with respect to the previous ones, allows a higher flexibility and a wider applicability. We derive here some properties of this
class of processes (such as continuity, occupation density, variance asymptotics and persistence)
according to the conditions satisfied by the Fourier symbol of the operator or the Bernstein
function chosen. On the other hand, these processes are proved to display short- or long-range dependence, if obtained by means of a derivative or an integral type operator, respectively, regardless of the kernel used in their definition. Finally, this kind of construction allows us to define the corresponding noise and to solve a Langevin-type integral equation
Non-central moderate deviations for compound fractional Poisson processes
No full textThe term moderate deviations is often used in the literature to mean a class of large
deviation principles that, in some sense, fills the gap between a convergence in probability
to zero (governed by a large deviation principle) and a weak convergence to a
centered Normal distribution. We talk about non-central moderate deviations when the
weak convergence is towards a non-Gaussian distribution. In this paper we study noncentral
moderate deviations for compound fractional Poisson processes with light-tailed
jumps
Random time-change with inverses of multivariate subordinators: governing equations and fractional dynamics
No full textIt is well-known that compositions of Markov processes with inverse subordinators are governed
by integro-differential equations of generalized fractional type. This kind of processes are of wide
interest in statistical physics as they are connected to anomalous diffusions. In this paper we consider a
generalization; more precisely we mean componentwise compositions of -valued Markov processes
with the components of an independent multivariate inverse subordinator. As a possible application, we
present a model of anomalous diffusion in anisotropic medium, which is obtained as a weak limit of
suitable continuous-time random walks
Integro-differential equations linked to compound birth processes with infinitely divisible addends
Full text linkStochastic modelling of fatigue (and other material's deterioration), as well as of cumulative damage in risk theory, are often based on compound sums of independent random variables, where the number of addends is represented by an independent counting process. We consider here a cumulative model where, instead of a renewal process (as in the Poisson case), a linear birth (or Yule) process is used. This corresponds to the assumption that the frequency of “damage” increments accelerates according to the increasing number of “damages”. We start from the partial differential equation satisfied by its transition density, in the case of exponentially distributed addends, and then we generalize it by introducing a space derivative of convolution type (i.e., defined in terms of the Laplace exponent of a subordinator). Then we are concerned with the solution of integro-differential equations, under proper initial conditions, which, in a special case, reduce to a fractional one. Correspondingly, we analyze the related cumulative jump processes under a general infinitely divisible distribution of the (positive) jumps. Some special cases (such as the stable, tempered stable, gamma, and Poisson) are presented
Random time-changes and asymptotic results for a class of continuous-time Markov chains on integers with alternating rates
Full text linkWe consider continuous-time Markov chains on integers which allow transitions to adjacent states only, with alternating rates. This kind of processes are useful in the study of chain molecular diffusions. We give explicit formulas for probability generating functions, and also for means, variances and state probabilities of the random variables of the process. Moreover we study independent random time-changes with the inverse of the stable subordinator, the stable subordinator and the tempered stable subordinator. We also present some asymptotic results in the fashion of large deviations. These results give some generalizations of those presented in [Journal of Statistical Physics 154 (2014), 1352–1364]
- …
