Preface to the revised edition Remarks on notation 1. Basic examples 2. Characterization and existence 3. Stable processes and their extensions 4. The Levy-Ito decomposition of sample functions 5. Distributional properties of Levy processes 6. Subordination and density transformation 7. Recurrence and transience 8. Potential theory for Levy processes 9. Wiener-Hopf factorizations 10. More distributional properties Supplement Solutions to exercises References and author index Subject index.

Lévy processes and infinitely divisible distributions

Special Functions of Mathematical Physics and Chemistry. By Ian N. Sneddon. Pp. 164. 10s. 6d. 1956. (Oliver and Boyd)

The fractional Poisson process is a renewal process with Mittag-Leffler waiting times. Its distributions solve
a time-fractional analogue of the Kolmogorov forward equation for a Poisson process. This paper shows that a
traditional Poisson process, with the time variable replaced by an independent inverse stable subordinator, is also a
 fractional Poisson process. This result unifies the two main approaches in the stochastic theory of time-fractional
 diffusion equations. The equivalence extends to a broad class of renewal processes that include models for tempered
 fractional diffusion, and distributed-order (e.g., ultraslow) fractional diffusion. The paper also {discusses the relation between} the fractional Poisson process and Brownian time.

The Fractional Poisson Process and the Inverse Stable Subordinator

In this paper, we study the fractional Poisson process (FPP) time-changed by an independent Levy subordinator and the inverse of the Levy subordinator, which we call TCFPP-I and TCFPP-II, respectively. Various distributional properties of these processes are established. We show that, under certain conditions, the TCFPP-I has the long-range dependence property, and also its law of iterated logarithm is proved. It is shown that the TCFPP-II is a renewal process and its waiting time distribution is identified. The bivariate distributions of the TCFPP-II are derived. Some specific examples for both the processes are discussed. Finally, we present simulations of the sample paths of these processes.

Fractional Poisson Process Time-Changed by Lévy Subordinator and Its Inverse

In this article, we derive the state probabilities of different type of space- and time-fractional Poisson processes using z-transform. We work on tempered versions of time-fractional Poisson proce...

/pdf/tempered-fractional-poisson-processes-and-fractional-2rz18gie5z.pdf

Tempered fractional Poisson processes and fractional equations with Z-transform

In the paper we present the governing equations for marginal distributions of Poisson and Skellam processes time-changed by inverse subordinators. The equations are given in terms of convolution-type derivatives.

On the governing equations for Poisson and Skellam processes time-changed by inverse subordinators

In the paper we study the models of time-changed Poisson and Skellam-type processes, where the role of time is played by compound Poisson-Gamma subordinators and their inverse (or first passage time) processes. We obtain explicitly the probability distributions of considered time-changed processes and discuss their properties.

/pdf/compositions-of-poisson-and-gamma-processes-1rufv1ryzg.pdf

Compositions of Poisson and Gamma processes

In the paper we consider time-changed Poisson processes where the time is expressed by compound Poisson-Gamma subordinators $G(N(t))$ and derive the expressions for their hitting times. We also study the time-changed Poisson processes where the role of time is played by the processes of the form $G(N(t)+at)$ and by the iteration of such processes.

/pdf/properties-of-poisson-processes-directed-by-compound-poisson-3hl4dy799r.pdf

Properties of Poisson processes directed by compound Poisson-Gamma subordinators

/pdf/properties-of-poisson-processes-directed-by-compound-poisson-33odvj3oxu.pdf

Khrystyna Buchak

Papers

On the governing equations for Poisson and Skellam processes time-changed by inverse subordinators

Compositions of Poisson and Gamma processes

Properties of Poisson processes directed by compound Poisson-Gamma subordinators

Properties of Poisson processes directed by compound Poisson-Gamma subordinators.

On the governing equations for Poisson and Skellam processes time-changed by inverse subordinators