Convolution-type derivatives, hitting-times of subordinators and time-changed C0C_0-semigroups

In this paper we will take under consideration subordinators and their inverse processes (hitting-times). We will present in general the governing equations of such processes by means of convolution-type integro-differential operators similar to the fractional derivatives. Furthermore we will discuss the concept of time-changed $C_0$-semigroup in case the time-change is performed by means of the hitting-time of a subordinator. We will show that such time-change give rise to bounded linear operators not preserving the semigroup property and we will present their governing equations by using again integro-differential operators. Such operators are non-local and therefore we will investigate the presence of long-range dependence.Comment: Final version, Potential analysis, 201

Long-memory Gaussian processes governed by generalized Fokker-Planck equations

It 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 $\alpha$-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 a long-range dependence. Moreover, we consider further extensions, by means of the so-called convolution-type derivative.Comment: 24, accepted for publicatio

Correlation properties of continuous-time autoregressive processes delayed by the inverse of the stable subordinator

We define the delayed Lévy-driven continuous-time autoregressive process via the inverse of the stable subordinator. We derive correlation structure for the observed non-stationary delayed Lévy-driven continuous-time autoregressive processes of order p, emphasizing low orders, and we show they exhibit long-range dependence property. Distributional properties are discussed as wel

Fractional Risk Process in Insurance

Important models in insurance, for example the Carm{\'e}r--Lundberg theory and the Sparre Andersen model, essentially rely on the Poisson process. The process is used to model arrival times of insurance claims. This paper extends the classical framework for ruin probabilities by proposing and involving the fractional Poisson process as a counting process and addresses fields of applications in insurance. The interdependence of the fractional Poisson process is an important feature of the process, which leads to initial stress of the surplus process. On the other hand we demonstrate that the average capital required to recover a company after ruin does not change when switching to the fractional Poisson regime. We finally address particular risk measures, which allow simple evaluations in an environment governed by the fractional Poisson process

Heavy-tailed fractional Pearson diffusions

We define heavy-tailed fractional reciprocal gamma and Fisher-Snedecor diffusions by a nonMarkovian time change in the corresponding Pearson diffusions. Pearson diffusions are governed by the backward Kolmogorov equations with space-varying polynomial coefficients and are widely used in applications. The corresponding fractional reciprocal gamma and Fisher-Snedecor diffusions are governed by the fractional backward Kolmogorov equations and have heavy-tailed marginal distributions in the steady state. We derive the explicit expressions for the transition densities of the fractional reciprocal gamma and Fisher-Snedecor diffusions and strong solutions of the associated Cauchy problems for the fractional backward Kolmogorov equation

Approximation of heavy-tailed fractional Pearson diffusions in Skorokhod topology

Continuous time random walks (CTRWs) have random waiting times between particle jumps. We establish fractional diffusion approximation via correlated CTRWs. Instead of a random walk modeling particle jumps in the classical CTRW model, we use discrete-time Markov chain with correlated steps. The waiting times are selected from the domain of attraction of a stable law