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.

Stochastic processes with random time and more general compositions of processes are quite popular topics of recent studies both in the theory of stochastic processes and in various applied areas. Specifically, in financial mathematics, models with random clock (or time change) allow to capture more realistically the relationship between calendar time and financial markets activity. Models with random time appear in reliability and queuing theory, biological, ecological and medical research, note also that for solving some problems of statistical estimation sampling of a stochastic process at random times or on the trajectory of another process can be used. Some examples of applications are described, for example, in [

In the present paper we study various compositions of Poisson and Gamma processes. We only consider the cases when processes used for compositions are independent.

The Poisson process directed by a Gamma process and, in reverse, the Gamma process directed by a Poisson process one can encounter, for example, in the book by W. Feller [

Time-changed Poisson processes have been investigated extensively in the literature. We mention, for example, the recent comprehensive study undertaken in [

The most intensively studied models of Poisson processes with time-change are two fractional extensions of Poisson process, namely, the space-fractional and the time-fractional Poisson processes, obtained by choosing a stable subordinator or its inverse process in the role of time correspondingly; the literature devoted to these processes is rather voluminous, we refer, e.g., to the recent papers [

Interesting models of processes are based on the use of the difference of two Poisson processes, so-called Skellam processes, and their generalizations via time change. Investigation of these models can be found, for example, in [

In the present paper we study time-changed Poisson and Skellam processes, where the role of time is played by compound Poisson-Gamma subordinators and their inverse (or first passage time) processes. Some motivation for our study is presented in Remarks

We obtain explicitly the probability distributions of considered time-changed processes and their first and second order moments.

In particular, for the case, where time-change is taken by means of compound Poisson-exponential subordinator and its inverse process, corresponding probability distributions of time-changed Poisson and Skellam processes are presented in terms of generalized Mittag-Leffler functions.

We also find the relation, in the form of differential equation, between the distribution of Poisson process, time-changed by Poisson-exponential process, and the distribution of Poisson processes time-changed by inverse Poisson-exponential process.

The paper is organized as follows. In Section

In this section we recall definitions of processes, which will be considered in the paper (see, e.g. [

The Poisson process

The Gamma process

The Skellam process is defined as

The probability mass function of

Skellam processes are considered, for example, in [

We will consider Skellam processes with time change

To represent distributions and other characteristics of processes considered in the next sections, we will use some special functions, besides the modified Bessel function introduced above. Namely, we will use the Wright function

The first example of compositions of Poisson and Gamma processes which we consider in the paper is the compound Poisson-Gamma process. This is a well known process, however here we would like to focus on some of its important features.

Let

The transition probability measure of the process

In particular, when

The transition probability measure of

Measures (

for the range

the limiting case when

the case

for

Probability distributions of the above subordinators can be written in the closed form in the case (iii); in the case (i) for

In the paper [

With the present paper we intend to complement the study undertaken in [

We also study Skellam processes with time-change by means of compound Poisson-Gamma subordinators, in this part our results are close to the corresponding results of the paper [

In our paper we develop (following [

It would be also interesting to study the above mentioned processes within the framework of Bochner subordination via semigroup approach. We address this topic for future research, as well as the study of other characteristics of the processes considered in the present paper and their comparison with related results existing in the literature.

Note that in our exposition for convenience we will write Lévy measures (

It is well known that the composition of two independent stable subordinators is again a stable subordinator. If

In the paper [

Let

In the case when

The probability mass function of the process

For

For

The governing equation for the probabilities

We obtain statements of Theorem

Moments of the process

The very detailed study of time-changed processes

Note also that in order to compute the first two moments and covariance function of time-changed Lévy processes the following result, stated in [

If

and if

Let

Note that the marginal laws of the time-changed processes obtained in Theorems

We now consider Skellam processes

Let the process

For the case

Using conditioning arguments, we can write:

The mean, variance and covariance function for Skellam process

Consider now the time-changed Skellam process of type II, where the role of time is played by the subordinator

Using the independence of

The moments of

We first consider the process

Define the inverse process (or first passage time):

The covariance function of the process

The proof of Lemma

It is known that, generally, the inverse subordinator is a process with non-stationary, non-independent increments (see, e.g., [

Note that, for inverse processes, the important role is played by the function

In our case we obtain

Let

The probabilities

The Laplace transform is obtained as follows:

We now state the relationship, in the form of a system of differential equations, between the marginal distributions of the processes

Introduce the following notations:

Firstly we represent the derivative

Therefore, (

We note that in addition to (

Consider now Skellam processes with time change, where the role of time is played by the inverse process

Let the Skellam process

Using conditioning arguments, we write:

The expressions for mean, variance and covariance function for the Skellam process

Consider the time-changed Skellam process of type II:

Analogously to the proof of Theorem

The moments of

Consider now the compound Poisson–Erlang process

For this case the inverse process

Laplace transform of the process

Using the arguments similar to those in [

The details of derivation of (

Consider the time-changed process

Proof is similar to that for Theorem

The first two moments of the process

Let the Skellam process

Proof is analogous to that of Theorem

Consider the time-changed Skellam process of type II:

Proof is analogous to that of Theorem

Covariance structure of the Skellam processes considered in this section appears to be of complicated form and we postpone this issue for future research.

The authors are grateful to the referees for their valuable comments and suggestions which helped to improve the paper.

Let

First two moments of

To calculate the covariance we need to find the expression for

We will use Theorem 3.1 from [

Denote

Let

The inverse Laplace transform can be found by the following calculations:

Therefore, for the covariance of the process

Using the expression for

We present some details of the derivation of the expression for probability density of the inverse Poisson–Erlang process

The inverse Poisson–Erlang process was considered in [

For convenience of a reader and to make the paper self-contained, we present here some details of calculations following the general approach developed in [

Introduce the Laplace transforms related to the process

The above formula holds, in fact, for more general compound Poisson processes. In the case of compound Poisson process with jumps having the p.d.f.

For the case of Poisson–Erlang process,

One special case when inversion of (

Here we present a reverse check, namely, we check that the double Laplace transform of the p.d.f. (

We next obtain the expressions for the moments of the process