Poisson processes with randomized time have been intensively studied in the recent literature. The most popular models of such processes are represented by the space-fractional and time-fractional Poisson processes where a random time-change is introduced by a stable subordinator or its inverse correspondingly (we refer, for example, to [7, 12, 14, 3, 10, 1, 13], to mention only few, see also references therein).

In the paper [14] a general class of time-changed Poisson processes Nf(t)=N(Hf(t)) , t>0 0$]]>, has been introduced and studied, where N(t) is a Poisson process and Hf(t) is an arbitrary subordinator with Laplace exponent f, independent of N(t) . Distributional properties, hitting times and governing equations for such processes were presented in [14, 7]; the case of iterated time change and some further generalizations of the class of process Nf(t) were also considered in [7]. Hitting times for the iterated Poisson process were studied in [11, 6], and for population processes with random time in [4].

In the papers [5, 9] time-changed Poisson processes were studied for the case where the role of time is played by compound Poisson-Gamma subordinators and their inverse processes. In the present paper, we continue to investigate the properties of the processes N(GN(t)) , t>0 0$]]>, where GN(t)=G(N(t)) is the compound Poisson-Gamma process. Some motivation for considering this class of processes is given in [5] (see Section 3 therein). We derive the expressions for the hitting times and first passage times of the processes N(GN(t)) in Sections 3 and 4. We next study in Section 5 the time-change introduced by the process GN+a(t)=G(N(t)+at) , where the Gamma process G(t) and the Poisson process with a drift N(t)+at are independent. In Section 6 we consider some kinds of iterated Bessel transforms and their use for time-change in the Poisson process.

In this section we recall some results on time-changed Poisson processes, which will be used in the next sections. In the paper [14] the time-changed Poisson processes Nf(t)=N(Hf(t)) , t>0 0$]]>, have been studied, where N(t) is the Poisson process with intensity λ and Hf(t) is the subordinator with a Bernštein function f(u) , independent of N(t) . Their distributions are characterized as follows: (1) Pr{Nf[t,t+dt)=k}=dtλkk!∫0∞e−λsskν(ds)+o(dt),k≥1,1−dt∫0∞(1−e−λs)ν(ds)+o(dt),k=0, (2) pkf(t)=P{Nf(t)=k}=(−1)kk!dkduke−tf(λu)|u=1, the probability generating function of Nf(t) is given by (3) Gf(u,t)=e−tf(λ(1−u)),|u|≤1.

The time-changed Poisson processes Nf(t) , t>0 0$]]>, have independent stationary increments (see, e.g., the general result on subordinated Lévy processes given in Theorem 1.3.25 [2]).

It is also shown in [14] that the probabilities of the processes Nf(t) satisfy the difference-differential equations: (4) ddtpkf(t)=−f(λ)pkf(t)+ ∑m=1kλmm!pk−mf(t)∫0∞e−sλsmν(ds),k≥0,t>0, 0,\]]]> with the usual initial conditions: p0f(0)=1 , and pkf(0)=0 for k≥1 . The equation (4) can be also written in the following form (see [14], Remark 2.3): (5) ddtpkf(t)=−f(λ(I−B))pkf(t),k≥0,t>0, 0,\]]]> where B is the shift operator: Bpkf(t)=pk−1f(t) , and it is supposed that p−1(t)=0 .

Let N1(t) be the Poisson process with intensity λ1 , and let GN(t)=G(N(t)) , t>0 0$]]>, be the compound Poisson-Gamma subordinator with parameters λ,α,β , that is, with the Laplace exponent f(u)=λβα(β−α−(β+u)−α) and the Lévy measure ν(du)=λβα(Γ(α))−1uα−1e−βudu , λ,α,β>0 0$]]>. In the case when α=1 we have the compound Poisson-exponential subordinator, which we will denote as EN(t) .

Consider the time-changed process N1(GN(t))=N1(G(N(t))) , t>0 0$]]>, where the compound Poisson-Gamma process GN(t) is independent of N1(t) .

We recall the results on probability distributions of the process N1(GN(t)) , which were presented in our previous paper [5] and will be used in the next sections.

To represent the distribution function in the next theorem we will use the three-parameter generalized Mittag-Leffler function, which is defined as follows: (11) 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]).

Figures 1 and 2 show the behavior of the probabilities (6) and (12), for various choices of t (t=1,2,3) .

In this section we study the hitting times for the process N1(GN(s)) defined as Tk=inf{s:N1(GN(s))=k},k≥1.

In the next theorem we obtain the analytic expression for P{Tk<∞} .

In the paper [7] the authors show that in general case for the process Nf(t)=N(Hf(t)) the probabilities P{Tkf∈ds} can be represented as follows (see the formula (2.8) in [7]): (20) P{Tkf∈ds}= ∑j=0k−1(−1)jj!djduje−sf(λu)|u=1dsλk−j(k−j)!∫0∞e−λttk−jν(dt) To be able to use the above formula, it is necessary to calculate the derivatives (21) djduj1f(λ1u)|u=1. It is noted in [7] that evaluation of these derivatives seems possible only for a small subset of Bernštein functions. One example of such functions is f(u)=uα , the Bernštein function which corresponds to the stable subordinator. We show below another example of such functions.

In the case when α=1 the process GN(t) becomes EN(t) , the compound Poisson process with exponentially distributed jumps that has Laplace exponent f(u)=λuβ+u,λ>0,β>0 0,\beta >0$]]>. For this function it is possible to calculate the derivatives (21), since we easily find: (22) djduj1f(λ1u)=(−1)jj!βλλ1u−(j+1),j≥1,1f(λ1u)=β+λ1uλλ1u,j=0. Therefore, we can use the formula (20) for the process N1(EN(T)) .

By integrating the left part of the formula (27) by s we get: P{TkE<∞}=λβλ1+β ∑n=0∞∫0∞e−λs(λs)nn!dsP{Nβ(Gλ1(k))=n}=βλ1+β ∑n=0∞P{Nβ(Gλ1(k))=n}=βλ1+β, which coincides with the formula (23).

In this section we study first passage time for the process N1(GN(s)) defined as T˜k=inf{s≥0:N1(GN(s))≥k}. We have: P{T˜k<s}=P{N1(GN(s))≥k}= ∑j=k∞∫0∞e−λ1z(λ1z)jj!P{GN(s)∈dz}=e−λs ∑j=k∞(λ1λ1+β)j1j! ∑n=1∞(λsβα(λ1+β)α)nΓ(αn+j)n!Γ(αn), and, therefore, P{T˜k∈ds}/ds=ddse−λs ∑j=k∞(λ1λ1+β)j1j! ∑n=1∞(λsβα(λ1+β)α)nΓ(αn+j)n!Γ(αn)=e−λs ∑j=k∞(λ1λ1+β)j1j! ∑n=1∞sn−1(n−λs)(λsβα(λ1+β)α)nΓ(αn+j)n!Γ(αn). When α=1 , the process GN(t) becomes EN(t) , that is, the compound Poisson process with exponentially distributed jumps. For this process we denote the hitting times by T˜kE=inf{s≥0:N1(EN(s))≥k}. We obtain: P{T˜kE<s}=P{N1(GN(s))≥k}= ∑j=k∞∫0∞e−λ1z(λ1z)jj!P{EN(s)∈dz}=e−λs ∑j=k∞λ1jλβs(λ1+β)j+1E1,2j+1(λβsλ1+β). Therefore, P{T˜kE∈ds}/ds=ddse−λs ∑j=k∞λ1jλβs(λ1+β)j+1E1,2j+1(λβsλ1+β). Taking into account that dds(asE1,2γ(as))=aE1,1γ(as) , we obtain: P{T˜kE∈ds}/ds= ∑j=k∞λ1j(λ1+β)jddse−λsλβsλ1+βE1,2j+1(λβsλ1+β)= ∑j=k∞λ1j(λ1+β)je−λs[(−λλβsλ1+βE1,2j+1(λβsλ1+β)+λβλ1+βE1,1j+1(λβsλ1+β))]=e−λsλβλ1+β ∑j=k∞λ1j(λ1+β)j×[E1,1j+1(λβsλ1+β)−λsE1,2j+1(λβsλ1+β)].