Fractional Gamma and Gamma-Subordinated Processes

Abstract

We introduce and study fractional generalizations of the well-known Gamma process, in the following sense: the corresponding densities are proved to satisfy the same differential equation as the usual Gamma process, but with the shift operator replaced by its fractional version of order $\nu >0$. In the case $\nu >1$, the solution corresponds to the density of a Gamma process time-changed by an independent stable subordinator of index $1/\nu $. For $\nu $ less than one an analogous result holds, with the subordinator replaced by the inverse. In this case the fractional Gamma process is proved to be a non-stationary version of the standard one, with power law behavior of the expected value. Hence it can be considered a useful tool in modelling stochastic deterioration in the non-linear cases, a situation which often occurs in real data (see i.e., \cite{VAN} and the references therein).\ud As a consequence of the previous results, the fractional generalizations of some Gamma subordinated processes (i.e. the Variance Gamma, the Geometric Stable and the Negative Binomial) are introduced and the corresponding fractional differential equations are obtained. These processes are particularly relevant for a wide range of financial and technological applications

On the inverse gamma subordinator

“…Such a property turns out to be quite demanding from the analytical point of view. We conclude the discussion about H by recalling that, from the representation (5), we are able to evaluate the moments…”

“…Gamma subordinator is a well-known subordinator which has been considered in many fields of Applied Sciences. In Mathematical Finance for instance, a well-known process is the Variance Gamma Process (or Laplace motion) which can be obtained by considering a Brownian motion with a random time given by a Gamma subordinator ( [3], [5], [20], [25]). We recall that a subordinator is a Lévy process with non-negative and non-decreasing paths.…”

Fractional diffusion-type equations with exponential and logarithmic differential operators

“…The fractional shift operator has been introduced in [5], in the special case α ∈ (0, 1) and θ = −α.…”

Pseudo-differential operators and related additive geometric stable processes

“…which is valid on Dom(D α,θ ). In the spirit of [3], [4] and [5], we call (26) fractional logarithmic operator. Analogously, by considering a generic semigroup T t generated by A, subordination to Gamma process produces the new generator…”

“…In section 2 we recall some facts on timeinhomogeneous Markov processes and related propagators, following the line of [19], [20], [7], [8] and [9], and we underline new aspects on the special case of additive processes. In section 3 we write the generators of the GS processes as fractional operators of logarithmic type, which have been treated, in [3], [4] and [5], in the homogeneous case. In Section 4 and 5 we construct the two additive geometric stable processes and present all the related results, together with some relevant particular cases.…”