Subordinated Processes with Infinite Variance

Abstract

In this paper we consider three models of subordinated processes. A subordinated process, called also a time-changed process, is defined as a superposition of two independent stochastic processes. To construct such stochastic system we replace the time of a given process (called also an external process) by another process which becomes the “operational time”. In the literature one can find different models that are constructed as a superposition of two stochastic processes. The most classical example is the Laplace motion, also known as variance gamma process, is stated as a Brownian motion time-changed by the gamma subordinator. In this paper the considered systems are constructed by replacing the time of the symmetric $\alpha$-stable Lévy motion with another stochastic process, namely the $\alpha _S$-stable, tempered $\alpha _T$-stable and gamma subordinator. We discuss the main characteristics of each introduced processes. We examine the characteristic function, the codifference, the probability density function, asymptotic tail behaviour and the fractional order moments. To make the application of these processes possible we propose a simulation procedure. Finally, we demonstrate how to estimate the tail index of the external process, i.e. alpha-stable Levy motion and by using Monte Carlo method we show the efficiency of the proposed estimation method.