Generalized fractional calculus and some models of generalized counting processes

In the paper we consider models of generalized counting processes time-changed by a general inverse subordinator, we characterize their distributions and present governing equations for them. The equations are given in terms of the generalized fractional derivatives, namely, convolutions-type derivatives with respect to Bern\v{s}tein functions. Some particular examples are presented.


Introduction
Stochastic processes with random time-change has become a well-established and highly ramified branch of the modern theory of stochastic processes which attracts more and more attention due to various applications in financial, biological, ecological, physical, technical and other fields of research.
Rich class of models is provided by time-changed Poisson processes, of which the most intensively studied are two fractional extensions of the Poisson process, namely, the space-fractional and time-fractional Poisson processes obtained by choosing a stable subordinator or its inverse in the role of time correspondingly.We refer, for example, to papers [8], [18], [20], [1], [2], [16], [19], among many others (see also references therein).In particular, in paper [20] a general class of time-changed Poisson processes N f (t) = N(H f (t)), t > 0, was introduced and studied, where N(t) is a Poisson process and H f (t) is an arbitrary subordinator with Laplace exponent f, independent of N(t), their distributional properties, hitting times and governing equations were presented (see, also [8]).In paper [5] Poisson processes time changed by general inverse subordinators were studied, the governing equations for their marginal distributions were presented and some other properties were described.Poisson process itself, being in a sense a core object concerning applicability to count data and simple tractability, however, as a reverse side of its simplicity, is a rather restrictive model.Therefore, it has been quite natural to search for its extensions and generalizations to provide some new useful features and properties needed for applications.For example, time changed processes N(H f (t)) allow for jumps of arbitrary size and other interesting properties ( [20], [8]).
In the present paper we consider several particular models of generalized counting processes time-changed by a general inverse subordinator, characterize their distributions and present governing equations for them, which are obtained following the technique presented in [5].The equations are given in terms of the generalized Caputo-Djrbashian derivatives, which called also convolutions-type derivatives with respect to Bernštein functions.These generalized derivatives were introduced in [14] and [21] and have been widely used to describe various advanced stochastic models.
The convolution-type derivatives allow to study the properties of subordinators and their inverses in the unifying manner ( [21,17]), in particular, the governing equations for their densities can be given in terms of convolution-type derivatives.These properties will be the key tools for our study: by considering several models of time-changed processes, we elucidate the technique leading from the equations for the densities of inverse subordinators and their Laplace transforms to the equations for probabilities of processes time-changed by inverse subordinators and equations for some related functions.
The paper is organized as follows.Section 2 collects the basic definitions and facts on the convolution-type derivatives needed in our study.We also consider, as a warmup example, the time changed Poisson process N(Y f (t)),t > 0, where Y f is the inverse to the subordinator with Laplace exponent f, independent of N(t).We partly extend the results from [5] for this process.This simplest case allows to demonstrate very transparently the technique underpinning the derivations of the corresponding results for more general models considered in the next sections.Namely, in Sections 3 and 4 we study the processes N ψ,f (t) = N H ψ (Y f (t)) , t > 0, and M ψ,f (t) = M H ψ (Y f (t)) , t > 0, where M is the generalized counting process.Time change is done by the independent subordinator H ψ and inverse subordinator Y f related to Bernštein functions ψ and f correspondingly.We present the differential-difference equations for probabilities of the processes N ψ,f and M ψ,f , which are given in terms of the convolutions-type derivative with respect to function f and the difference operator related to function ψ.Expressions for the probabilities of these processes are also presented in terms of the Laplace transform of the process Y f .Some other properties of these processes are described and a comparison with related results in the literature are given.As examples, we consider the processes N ψ,f and M ψ,f , where H ψ is a compound Poisson-Gamma process.

Preliminaries
Generalized fractional derivatives.We review the main definitions and some facts on the generalized fractional derivatives, which will be important for our study.(For more details see, e.g., [17,21].) Let f (x) be a Bernštein function: ν(ds) is a non-negative measure on (0, ∞) (the Lévy measure for f (x)) such that The generalized Caputo-Djrbashian (C-D) derivative, or convolution-type derivative, with respect to the Bernštein function f is defined on the space of absolutely continuous functions as follows ([21], Definition 2.4): where ν(s) = a + ν(s, ∞) is the tail of the Lévy measure ν(s) of the function f .In the case where f (x) = x α , x > 0, α ∈ (0, 1), the derivative (2) reduces to the fractional C-D derivative: For the Laplace transform of the derivative (2) the following relation holds ( [21], Lemma 2.5): for u such that |u(t)| ≤ Me s 0 t , M and s 0 are some constants.Similarly to the C-D fractional derivative, the convolution type derivative can be alternatively defined via its Laplace transform.
The generalization of the classical Riemann-Liouville (R-L) fractional derivative is introduced in [21] by means of another convolution-type derivative with respect to f given by the following formula: The derivatives D f t and D f t are related as follows (see, [21], Proposition 2.7): Bernštein functions are associated in a natural way with subordinators.
Let H(t), t ≥ 0, be a subordinator, that is, nondecreasing Lévy process.Its Laplace transform is of the form: where the function f , called the Laplace exponent, is a Bernštein function.Consider a subordinator H f with the Laplace exponent f given by ( 1), and let Y f be its inverse process defined as It was shown in [21] that the distribution of the inverse process Y f has a density ℓ f (t, x) = P{Y f (t) ∈ dx}/dx provided that the following condition holds: The Laplace transform of the density with respect to t has the form ( [21]): r) .
The density ℓ f (t, u) of the inverse process Y f satisfies the following equation ( [21], Theorem 4.1): subject to The space Laplace transform of the density ℓ f (t, x) is an eigenfunction of the operator D f t , that is, satisfies the equation (see, [5,14,17]).In the case where f , where E α (•) is the Mittag-Leffler function.Equations ( 6)-( 9) are important for the study of the processes time changed by inverse subordinators as can be seen in what follows.We first consider the Poisson process with an inverse subordinator as a simplest example which demonstrates the technique to be applied further to more general models.
Poisson process time changed by an inverse subordinator.Let N(t) be the Poisson process with intensity λ, and Y f (t) be the inverse subordinator. and where The probability generating function of the process N f has the form and satisfies the equation with G f (u, 0) = 1.
Proof.Proof of equation (10).For the probabilities p f x (t) we have: We take the generalized R-L convolution-type derivative D f t given by (4) and use the equation for the density ℓ f (t, u) of the inverse subordinator: with We obtain: Using the relation between convolution derivatives of C-D and R-L types, we have: note also that From ( 17), taking into account ( 19)-( 20), we finally obtain: (11): We used above that the distribution of the Poisson process N with the rate λ can be writen as follows: For the probability generating function we have therefore, we obtain formula (12).Now we use the fact, that lf (t, λ) is an eigenfunction of the operator D f t (see (9)), from which we conclude that G f (u, t), being given by lf (t, λ(1 − u)), satisfies equation (13).

Models of time-changed Poisson processes
Consider the time-changed Poisson processes N ψ (t) = N H ψ (t) , t > 0, where N(t) is the Poisson process with intensity λ and H ψ (t) is the subordinator with Bernštein function ψ(u), independent of N(t).This class of processes was introduced and studied in [20] and called by the authors Poisson processes with Bernštein intertimes.It was shown in [20] that the distributions of N ψ (t), t > 0, can be presented as follows: and satisfy the difference-differential equations: which can be also written as with the usual initial conditions: The last equation can be represented in the form: where B is the backshift operator: , and it is supposed that p −1 (t) = 0.
The probability generating function of the process N ψ is of the form ( [20]): We refer for more detail on these processes to [20].Consider the process N ψ time-changed by an independent inverse subordinator Y f .We can state the following result.
and the probabilities p ψ,f k satisfy the following equation with the usual initial conditions.The probability generating function of the process N ψ,f has the form and satisfies the equation with G ψ,f (u, 0) = 1.
Proof.To prove equation ( 25) we perform the calculations as follows: Now it is left to note that Let us state equation (26).We have We repeat the same lines as in the proof of Theorem 1.We take the generalized R-L derivative D f t and use equations ( 22)-( 23): Using the relation between generalized derivatives of C-D and R-L types, we can write 0); and we have (see (7)): Therefore, we obtain Using the expression for G ψ (u, t) given by ( 24), we calculate G ψ,f : and since lf (t, ψ(λ(1 − u))) is an eigenfunction of D f t with the eigenvalue ψ(λ(1 − u)) (see ( 9)), it follows that G ψ,f (u, t) satisfies equation (28).Remark 2. We notice at once that equations (26) for the probabilities of the process N ψ,f (t) = N(H ψ (Y f (t))) mimic the corresponding equations for the process N ψ (t) = N(H ψ (t)), that is, the process before the time change by an inverse subordinator, only the ordinary derivative in time is changed for the generalized fractional derivative.This can be anticipated quite straightforwardly, in view of the technique used for the proof, and holds also for other models, like those below and in the next section.On the other side, the equations (26) tell us, that for the probabilities of the processes with double time change like N H ψ (Y f (t)) , the action (in time) of the operator D f t (which is related to Y f ) is equal to the action (in space) of the operator −ψ (λ (I − B)) (which is related to H ψ , and depends also of the outer process N).Some more insight on these operators can be found within the approach applied in the paper [1].Let Y f = Y β be the inverse stable subordinator with a parameter β, that is, f (λ) = λ β , then equation (26) becomes where D β t is the Caputo-Djrbashian fractional derivative (3).If we suppose furthermore that H ψ = H α is the stable subordinator with a parameter α, that is, ψ(λ) = λ α , then we come to the equation Equations ( 30) and (31) were stated in paper [1], where the particular representation of the operator ψ (λ (I − B)) was used, and the equations for the probabilities of time changed processes were stated using the interplay between the operator ψ (λ (I − B)) and the fractional operator D β t in the Fourier domain.The approach in [1] can be extended for the case where we deal with the generalized fractional operator D f t , as soon as we know that its eigenfunction is given by lf (t, λ) (see (9)).We refer for more detail on this approach to [1].
Consider a generalization of counting processes with Bernštein intertimes, which was introduced in the paper [8]: where H ψ i , i = 1, . . .n are n independent subordinators with Bernštein functions ψ i , independent of the Poisson process N. In [8] the following result was stated.
Proof of Theorem 3 is obtained by the same reasoning as that of Theorem 2, using Proposition 1.
Example 1.Consider the time-changed process N GN (t) = N 1 (G N (t)), t > 0, where N 1 (t) is the Poisson process with intensity λ 1 , and G N (t) = G(N(t)), t > 0, is the compound Poisson-Gamma subordinator with parameters λ, α, β, that is, with the Laplace exponent ψ GN (u) = λβ α (β −α − (β + u) −α ) .In the case when α = 1 we have the compound Poisson-exponential subordinator, which we will denote as E N (t), and the corresponding time-changed process as N E (t).The detailed study of the processes N GN (t) and N E (t) was presented in [3], [4].We now consider the processes The next theorem is obtained as a corollary of Theorem 2.
Theorem 4. The process N GN,f (t) has probability distribution function and p GN,f k (t) satisfy the following equation where ψ E (u) = λu β+u , and p E,f k (t) satisfy the following equation
Following the lines of the previos section, we consider the time-changed process M ψ,f (t) = M(H ψ (Y f (t)), that is, with double time-change by an independent subordinator H ψ and an inverse subordinator Y f , which are independent of M. To the best of our knowledge, such general case has not been presented in the literature.In the next theorem we characterize its probabilities pψ,f n (t) = P M ψ,f (t) = n and probability generating function.Theorem 5.The process M ψ,f has probability distribution function and pψ,f n (t) satisfy the following equation with initial conditions The probability generating function of the process M ψ,f is of the form and satisfies the equation with Gψ,f (u, 0) = 1.
Proof.For calculating pψ,f n (t) we use the formula (34): This implies formula (35), since To derive equation (36) we follow the same lines as is the proof of Theorem 2, but we use now instead of the equation ( 22) (or ( 23)) for probabilites of the process N ψ = N(H ψ ), the following equation for the probabilities pψ n (t) of the process M ψ = M(H ψ )(see [13]): Thus, we obtain and then apply the same reasonings as those after formula (29) to come to (36).Using the expression for the probability generating function Gψ (u, t) of the time-changed process M ψ (t) = M(H ψ (t)) (see [11]): we calculate Gψ,f as follows: that is, (37) holds.Equation (38) follows in view of (37) and ( 9).