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 [5].

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 [6], where distributions of these processes are presented as particular examples within the general framework of directed processes.

Time-changed Poisson processes have been investigated extensively in the literature. We mention, for example, the recent comprehensive study undertaken in [16] concerned with the processes N(Hf(t)) , where N(t) is a Poisson process and Hf(t) is an arbitrary subordinator with Laplace exponent f, independent of N(t) . The particular emphasis has been made on the cases where f(u)=uα , α∈(0,1) (space-fractional Poisson process), f(u)=(u+θ)α−θα , α∈(0,1) ,  θ>0 0\hspace{2.5pt}$]]>(tempered Poisson process) and f(λ)=log(1+λ) (negative binomial process), in the last case we have the Poisson process with Gamma subordinator.

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 [7, 15, 12] (and references therein), and to the paper [14], where the correlation structure of the time-changed Lévy processes has been investigated and the correlation of time-fractional Poisson process was discussed among the variety of other examples. The most recent results are concerned with non-homogeneous fractional Poisson processes (see, e.g. [13] and references therein).

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 [2], and we also refer to the paper [10], where fractional Skellam processes were introduced and studied.

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 1 and 2 in Section 3.

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 2 we recall definitions of processes which will be considered in the paper, in Section 3 we discuss the main features of the compound Poisson-Gamma process GN(t) . In Section 4 we study Poisson and Skellam-type processes time-changed by the process GN(t) ; in Section 5 we investigate time-changed Poisson and Skellam-type processes, where the role of time is played by the inverse process for GN(t) , we also discuss some properties of the inverse processes. Appendix contains details of the derivation of some results stated in Section 5.

In this section we recall definitions of processes, which will be considered in the paper (see, e.g. [1, 3]).

The Poisson process N(t) with intensity parameter λ>0 0$]]> is a Lévy process with values in N∪{0} such that each N(t) has Poisson distribution with parameter λt , that is, P{N(t)=n}=(λt)nn!e−λt; Lévy measure of N(t) can be written in the form ν(du)=λδ{1}(u)du , where δ{1}(u) is the Dirac delta centered at u=1 , the corresponding Bernštein function (or Laplace exponent of N(t) ) is f(u)=λ(1−e−u),u>0 0$]]>.

The Gamma process G(t) with parameters α,β>0 0$]]> is a Lévy process such that each G(t) follows Gamma distribution Γ(αt,β) , that is, has the density hG(t)(x)=βαtΓ(αt)xαt−1e−βx,x≥0; The Lévy measure of G(t) is ν(du)=αu−1e−βudu and the corresponding Bernštein function is f(u)=αlog(1+uβ).

The Skellam process is defined as S(t)=N1(t)−N2(t),t≥0, where N1(t),t≥0 , and N2(t),t≥0 , are two independent Poisson processes with intensities λ1>0 0$]]> and λ2>0 0$]]>, respectively.

The probability mass function of S(t) is given by (1) sk(t)=P(S(t)=k)=e−t(λ1+λ2)(λ1λ2)k/2I|k|(2tλ1λ2),k∈Z={0,±1,±2,…}, where Ik is the modified Bessel function of the first kind [20]: (2) Ik(z)= ∑n=0∞(z/2)2n+kn!(n+k)!. The Skellam process is a Lévy process, its Lévy measure is the linear combination of two Dirac measures: ν(du)=λ1δ{1}(u)du+λ2δ{−1}(du) , the corresponding Bernštein function is fS(θ)=∫−∞∞(1−e−θy)ν(dy)=λ1(1−e−θ)+λ2(1−eθ), and the moment generating function is given by EeθS(t)=e−t(λ1+λ2−λ1eθ−λ2e−θ),θ∈R.

Skellam processes are considered, for example, in [2], the Skellam distribution had been introduced and studied in [19] and [9].

We will consider Skellam processes with time change SI(t)=S(X(t))=N1(X(t))−N2(X(t)), where X(t) is a subordinator independent of N1(t) and N2(t) , and will call such processes time-changed Skellam processes of type I. We will also consider the processes of the form SII(t)=N1(X1(t))−N2(X2(t)), where N1(t) , N2(t) are two independent Poisson processes with intensities λ1>0 0$]]> and λ2>0 0$]]>, and X1(t) , X2(t) are two independent copies of a subordinator X(t) , which are also independent of N1(t) and N2(t) , and we will call the process SII(t) a time-changed Skellam process of type II.

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 (3) Φ(ρ,δ,z)= ∑k=0∞zkk!Γ(ρk+δ),z∈C,ρ∈(−1,0)∪(0,∞),δ∈C, for δ=0 , (3) is simplified as follows: (4) Φ(ρ,0,z)= ∑k=1∞zkk!Γ(ρk); the two-parameter generalized Mittag-Leffler function (5) Eρ,δ(z)= ∑k=0∞zkΓ(ρk+δ),z∈C,ρ∈(0,∞),δ∈(0,∞); and the three-parameter generalized Mittag-Leffler function (6) Eρ,δγ(z)= ∑k=0∞Γ(γ+k)Γ(γ)zkk!Γ(ρk+δ),z∈C,ρ,δ,γ∈C, with  Re(ρ)>0,Re(δ)>0,Re(γ)>0 0,\text{Re}(\delta )>0,\text{Re}(\gamma )>0$]]> (see, e.g., [8] for definitions and properties of these functions).

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 N(t) be a Poisson process and {Gn,n≥1} be a sequence of i.i.d. Gamma random variables independent of N(t) . Then compound Poisson process with Gamma distributed jumps is defined as Y(t)= ∑n=1N(t)Gn,t>0,Y(0)=0 0,\hspace{1em}Y(0)=0\]]]> (in the above definition it is meant that ∑n=10=def0 ). This process can be also represented as Y(t)=G(N(t)) , that is, as a time-changed Gamma process, where the role of time is played by Poisson process. Let us denote this process by GN(t) : GN(t)=G(N(t)). Let N(t) and G(t) have parameters λ and (α,β) correspondingly. The process GN(t)  is a Lévy process with Laplace exponent (or Bernštein function) of the form f(u)=λβα(β−α−(β+u)−α),λ>0,α>0,β>0, 0,\hspace{2.5pt}\alpha >0,\hspace{2.5pt}\beta >0,\]]]> and the corresponding Lévy measure is (7) ν(du)=λβα(Γ(α))−1uα−1e−βudu,λ>0,α>0,β>0. 0,\hspace{2.5pt}\alpha >0,\hspace{2.5pt}\beta >0.\]]]>

The transition probability measure of the process GN(t) can be written in the closed form: (8) P{GN(t)∈ds}=e−λtδ{0}(ds)+ ∑n=1∞e−λt(λtβα)nn!Γ(αn)sαn−1e−βsds=e−λtδ{0}(ds)+e−λt−βs1sΦ(α,0,λt(βs)α)ds, therefore, probability law of GN(t) has atom e−λt at zero, that is, has a discrete part P{GN(t)=0}=e−λt , and the density of the absolutely continuous part can be expressed in terms of the Wright function.

In particular, when α=n , n∈N , we have a Poisson–Erlang process, which we will denote by GN(n)(t) and for α=1 , we have a Poisson process with exponentially distributed jumps. We will denote this last process by EN(t) . Its Lévy measure ν(du)=λβe−βudu and Laplace exponent is f(u)=λuβ+u.

The transition probability measure of EN(t) is given by P{EN(t)∈ds}=e−λtδ{0}(ds)+e−λt−βs1sΦ(1,0,λtβs)ds=e−λtδ{0}(ds)+e−λt−βsλtβssI1(2λtβs)ds. Remark 1.

Measures (7) belong to the class of Lévy measures (9) ν(du)=cu−a−1e−budu,c>0,a<1,b>0. 0,\hspace{2.5pt}a<1,\hspace{2.5pt}b>0.\]]]> Note that: (i)

for the range a∈(0,1) and b>0 0$]]> we obtain the tempered Poisson processes,

Probability distributions of the above subordinators can be written in the closed form in the case (iii); in the case (i) for α=1/2 , when we have the inverse Gaussian subordinators; and in the case (iv).

In the paper [16] the deep and detailed investigation was performed for the time-changed Poisson processes where the role of time is played by the subordinators from the above cases (i)–(iii) (and as we have already pointed out in the introduction, the most studied in the literature is the case (ii)). The mentioned paper deals actually with the general time-changed processes of the form N(Hf(t)) , where N(t) is a Poisson process and Hf(t) is an arbitrary subordinator with Laplace exponent f, independent of N(t) . This general construction falls into the framework of Bochner subordination (see, e.g. book by Sato [18]) and can be studied by means of different approaches.

With the present paper we intend to complement the study undertaken in [16] with one more example. We consider time-change by means of subordinators corresponding to the above case (iv) and, therefore, subordination related to the measures (9) will be covered for the all range of parameters. We must also admit that the attractive feature of these processes is the closed form of their distributions which allows to perform the exact calculations for characteristics of corresponding time-changed processes.

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 [10].

In our paper we develop (following [16, 10] among others) an approach for studying time-changed Poisson process via investigation of their distributional properties, with the use of the form and properties of distributions of the processes involved.

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 (9) for the case of compound Poisson-Gamma subordinators in the form (7) with α>0 0$]]> and transparent meaning of parameters.

Let N1(t) be the Poisson process with intensity λ1 . Consider the time-changed process N1(GN(t))=N1(G(N(t))) , where GN(t) is compound Poisson-Gamma process (with parameters λ,α,β ) independent of N1(t) .

In the case when a=1 , that is, when the process GN(t) becomes EN(t) , the compound Poisson process with exponentially distributed jumps, probabilities pk(t) , k≥0 , can be represented via the generalized Mittag-Leffler function as stated in the next theorem.

We now consider Skellam processes S(t) with time change, where the role of time is played by compound Poisson-Gamma subordinators GN(t) with Laplace exponent f(u)=λβα(β−α−(β+u)−α) , λ>0 0$]]>, α>0 0$]]>, β>0 0$]]>.

Let the process S(t) have parameters λ1 and λ2 and let us consider first the time-changed Skellam process of type I, that is the process SI(t)=S(GN(t))=N1(GN(t))−N2(GN(t)), where N1(t) , N2(t) and GN(t) are mutually independent.

Consider now the time-changed Skellam process of type II, where the role of time is played by the subordinator X(t)=EN(t) with Laplace exponent f(u)=λuβ+u , that is, the process (23) SII(t)=N1(X1(t))−N2(X2(t)), where X1(t) and X2(t) are independent copies X(t) and independent of N1(t) , N2(t) . Theorem 4.

Let SII(t) be the time-changed Skellam process of type II given by (23). Its probability mass function is given by (24) P(SII(t)=k)=e−2λt(λtβ)2λ1k(λ1+β)k+1(λ2+β) ∑n=0∞(λ1λ2)n(λ1+β)n(λ2+β)n×E1,2n+k+1(λβtλ1+β)E1,2n+1(λβtλ2+β) for k∈Z , k≥0 , and when k<0 (25) P(SII(t)=k)=e−2λt(λtβ)2λ2|k|(λ1+β)(λ2+β)|k|+1 ∑n=0∞(λ1λ2)n(λ1+β)n(λ2+β)n×E1,2n+1(λβtλ1+β)E1,2n+|k|+1(λβtλ2+β). The moment generating function is EeθSII(t)=e−tλλ1(1−eθ)/(β+λ1(1−eθ))e−tλλ2(1−e−θ)/(β+λ2(1−e−θ)) for θ such that β+λ1(1−eθ)≠0 .