On some extensions of generalized counting processes

Abstract: We study different fractional extensions of the Poisson process and generalized counting processes by introducing time-change represented by the inverse to the sums of stable and tempered stable subordinators. We state the governing equations for probability distributions and probability generating functions which involve fractional derivatives of different orders. Closed form expressions for probability distributions and probability generating functions are also provided for several considered models.

• The paper introduces a novel framework that extends generalized counting processes via inverse sums of stable and tempered stable subordinators to capture non-Markovian dynamics.
• It employs fractional derivatives and generalized Caputo-Djrbashian operators to derive explicit solutions using (multivariate) Mittag-Leffler functions.
• The results provide practical insights for modeling complex systems in fields like risk theory, anomalous diffusion, and non-Markovian queuing.

Extensions of Generalized Counting Processes via Inverse Sums of Stable and Tempered Stable Subordinators

Introduction and Context

This work investigates a comprehensive class of fractional and generalized counting processes, focusing on their fractional extensions effected by random time-changes. The primary mechanism is a Bochner-type subordination where the operational time is modeled as the inverse of sums of independent stable or tempered stable subordinators. This construction enables the introduction of multiple fractional dynamics, and, via a mixture of convolution-type (Caputo-Djrbashian and Riemann-Liouville) derivatives, accommodates governing equations reflecting non-Markovian and non-local effects. The analysis culminates not only in governing equations but also in explicit solutions employing (multivariate) Mittag-Leffler functions. The framework encompasses fractional and generalized Poisson processes, generalized counting processes (GCP), non-homogeneous variants, and links to risk models in actuarial science.

Generalized Fractional Calculus and Its Stochastic Interpretation

The foundation for the analysis is the use of generalized derivatives $\mathcal{D}_t^f$ and $\mathbb{D}_t^f$ associated with a Bernstein function $f$, which serves as the Laplace exponent of a subordinator. This allows the description of governing equations for subordinators and their inverses in terms of convolution-type derivatives. Notably, the Laplace transform of the transition probabilities for processes time-changed by these inverse subordinators is governed by fractional order equations, and the Laplace transform of the inverse subordinator density is the eigenfunction for the generalized Caputo-Djrbashian derivative, with explicit connection to Mittag-Leffler functions in the classical stable case.

Time-Change via Sums of Stable Subordinators

A principal contribution is the extension of the time-change mechanism to include inverse processes driven by sums of stable subordinators, specifically: $$\mathcal{H}^{\nu}(t) = H_1^{2\nu}(t) + (2\lambda)^{1/\nu} H_2^{\nu}(t)$$ and its inverse

$$\mathcal{L}^\nu(t) = \inf\{s > 0: \mathcal{H}^{\nu}(s) \geq t\}.$$

The density $\ell_\nu(x, t)$ of $\mathcal{L}^\nu(t)$ satisfies an evolution equation involving a generalized telegraph-type operator: $$\left(\mathbb{D}_t^{2\nu} + 2\lambda \mathbb{D}_t^\nu \right) \ell_\nu(x, t) = -\frac{\partial}{\partial x} \ell_\nu(x, t)$$ with explicit boundary terms. The Laplace transform in space variable admits a closed form in terms of two-parameter Mittag-Leffler functions and, more generally, multivariate variants.

Fractional and Generalized Fractional Counting Processes

Fractional Poisson Process Time-Changed by Inverse Sums

For the time-changed fractional Poisson process: $N^\nu(t) = N(\mathcal{L}^\nu(t)),$ the probability laws admit governing equations: $$\frac{d^{2\nu}}{dt^{2\nu}} p_k^\nu(t) + 2\lambda \frac{d^\nu}{dt^\nu} p_k^\nu(t) = -\Lambda(p_k^\nu(t) - p_{k-1}^\nu(t))$$ solved with initial data. Explicit solutions, depending on parameter regimes ($\mathbb{D}_t^f$0, general case), are given via generalized Mittag-Leffler functions: $\mathbb{D}_t^f$1 and similarly with multivariate Mittag-Leffler functions for the less degenerate case.

The probability generating function is

$\mathbb{D}_t^f$2

where $\mathbb{D}_t^f$3 is expressed via the aforementioned Laplace transforms.

Generalized Fractional Counting Processes (GCP)

For generalized counting processes permitting jumps of size up to $\mathbb{D}_t^f$4, and with rates $\mathbb{D}_t^f$5, the time-changed fractional generalization $\mathbb{D}_t^f$6 is solved explicitly. The probability laws inherit similar governing equations: $\mathbb{D}_t^f$7 and the explicit expressions for transition probabilities and generating functions are constructed by layering the sum-of-random-variables structure over the explicit laws for the fractional Poisson case.

Extension to Linear Combinations and Sums

The model is further generalized to linear combinations of $\mathbb{D}_t^f$8 stable subordinators of possibly different indices: $\mathbb{D}_t^f$9 with corresponding evolution equations involving sums of fractional derivatives of different orders: $f$0 and all subsequent counting processes and generating functions follow analogous formulas, involving multivariate Mittag-Leffler functions parameterized by the mixture.

Extensions to Tempered Stable Subordinators

Stable subordinators with tempered tails are included under the form: $f$1 with corresponding generalized Caputo and Riemann-Liouville derivatives. The sum of two such subordinators, or mixed stable-tempered stable subordinator sums, is shown to lead to governing equations with subordinate tempered operators: $f$2 and, again, probabilistic results for time-changed counting processes extend naturally, with explicit Laplace-domain solutions.

Application to Ruin Probabilities in Risk Theory

The methodology developed extends to practical models in risk theory. For a classical risk model where the claim arrivals follow a generalized counting process and claim sizes are gamma-distributed, an explicit formula is derived for the non-ruin probability $f$3 for arbitrary initial capital $f$4. The expression is given by a convergent series of convolutions involving the two-parameter Mittag-Leffler function: $f$5 where $f$6 denotes convolution. This extends previously known results both analytically and numerically, offering a computable descriptor for risk evaluation under complex arrival laws.

Theoretical and Practical Implications

This class of processes situates itself at the intersection of classical stochastic processes, fractional calculus, and renewal theory. The models derived provide a tractable approach to systems with intermediate memory—interpolating between Poissonian (Markovian) and highly non-Markovian regimes—by tuning the orders of fractional derivatives. The explicit forms for transition laws and generating functions, especially via generalized and multivariate Mittag-Leffler functions, facilitate both analytical inspection and numerical implementation. In risk and insurance mathematics, these extensions permit the modeling of premium and claim arrivals beyond the limitations of simplistic Poisson/Event assumption, including bursty, self-similar, or heavy-tailed patterns observed in empirical data.

For future developments, the framework admits further extensions—multivariate counting processes, regime-switching subordinators, and applications to anomalous diffusion in physics or non-Markovian queuing models in computing and communication networks. Algorithms based on numerical inversion of Laplace transforms and series expansions for the multivariate Mittag-Leffler functions will be essential for practical use.

Conclusion

The paper systematically extends the theory of fractional and generalized counting processes via operational time-changes determined by inverse sums of stable and tempered stable subordinators. This approach captures complex temporal dependencies and introduces a family of evolution equations involving multiple fractional derivatives of varying orders. The authors deliver closed-form solutions—explicitly in terms of (generalized, multivariate) Mittag-Leffler functions—for the laws and generating functions of these processes, and demonstrate applications to risk models, significantly broadening the analytical scope of non-classical counting processes in probability and applied mathematics. The framework is robust and adaptable, offering considerable potential for both theoretical exploration and practical modeling in multiple domains.