Non-Homogeneous Generalized Counting Process

• NGCP is a stochastic integer-valued process with independent increments and time-varying jump rates that allow multiple jump amplitudes.
• Its representation as a weighted sum of non-homogeneous Poisson processes facilitates tractable analysis and martingale characterizations.
• NGCP models exhibit over-dispersion and long-range dependence, offering valuable applications in finance, ruin theory, and high-frequency data analysis.

A Non-Homogeneous Generalized Counting Process (NGCP) is a stochastic integer-valued process characterized by independent increments, arbitrary time-varying jump rates, and the possibility of arrivals of multiple amplitudes. The class subsumes several important families including non-homogeneous Poisson processes, compound Poisson processes, Cox processes with random intensities, and their time-fractional analogs. The process exhibits flexible over-dispersion and can be generalized via subordinator-induced time-changes and fractionalizations yielding long-range dependence or heavy-tailed statistics (Kataria et al., 2022, Tathe et al., 27 Jul 2024, Tathe et al., 18 Nov 2025, Wang et al., 2023, Laskin, 2023).

1. Formal Definition and Construction

Fix $k\geq 1$. The NGCP, denoted $\{M(t)\}_{t\geq 0}$, is a nondecreasing, integer-valued process starting at $M(0)=0$, allowing up-jumps of size $j=1,\dots,k$. The instantaneous rates $\lambda_j(t)\colon [0,\infty)\to[0,\infty)$ determine the probability of a jump of amplitude $j$ at time $t$. Specifically, the transition probabilities for increments over a vanishingly small interval $h$ are

$$\Pr\{M(t+h)-M(t)=j\} = \lambda_j(t) h + o(h), \quad 1\leq j\leq k,$$

$$\Pr\{M(t+h)-M(t)=0\} = 1 - \sum_{j=1}^k \lambda_j(t) h + o(h),$$

with independent increments (Kataria et al., 2022, Tathe et al., 27 Jul 2024, Tathe et al., 18 Nov 2025). This process can always be represented in law as a weighted sum of $k$ independent non-homogeneous Poisson processes: $$M(t) = \sum_{j=1}^k j N_j(t), \qquad N_j(t) \sim \mathrm{Poisson}(\Lambda_j(t)),\;\;\Lambda_j(t) = \int_0^t \lambda_j(s) ds.$$ This representation is fundamental for analytical tractability and for establishing martingale properties (Tathe et al., 18 Nov 2025).

2. Marginal and Increment Distributions

The marginal distribution $q_n(t) = \Pr\{M(t) = n\}$ is given by a superposition over all integer partitions of $n$ into jumps of size $1$ to $k$: $$q_n(t) = \sum_{x_1+\dots+x_k=n;\,x_j\geq 0} \prod_{j=1}^k \frac{\left(\Lambda_j(t)\right)^{x_j}}{x_j!} e^{-\Lambda_j(t)}.$$ For increments over intervals $[v,v+t]$, the shifted cumulatives $A_j(v+t)-A_j(v)$ define the law of $I(t,v)=M(v+t)-M(v)$. The generating function is

$$G(u,t) = \mathbb{E}[u^{M(t)}] = \exp\left\{ \sum_{j=1}^k \Lambda_j(t) (u^j-1) \right\},$$

which encodes all moment and recurrence information (Kataria et al., 2022, Tathe et al., 27 Jul 2024, Tathe et al., 18 Nov 2025).

3. Governing Equations and Recurrence Relations

The Kolmogorov forward (difference–differential) equations for $q_n(t)$ are: $$\frac{d}{dt}q_n(t) = -\left(\sum_{j=1}^k \lambda_j(t)\right) q_n(t) + \sum_{j=1}^{\min\{k,n\}} \lambda_j(t) q_{n-j}(t),\qquad q_0(0)=1, q_n(0)=0\, (n\geq 1).$$ From the generating function, the upward recurrence is

$$q_0(t) = e^{-\Lambda(t)},\qquad q_n(t) = \frac{1}{n} \sum_{j=1}^{\min\{k,n\}} j\,\Lambda_j(t)\,q_{n-j}(t),\quad n\geq 1.$$

These equations yield closed forms for the first and higher order moments (Kataria et al., 2022, Tathe et al., 27 Jul 2024).

4. Moments, Covariance and Over-Dispersion

For $M(t)$ constructed as above,

$$\mathbb{E}[M(t)]=\sum_{j=1}^k j \Lambda_j(t),\qquad \mathrm{Var}[M(t)] = \sum_{j=1}^k j^2 \Lambda_j(t).$$

Covariances are

$$\mathrm{Cov}[M(s),M(t)] = \sum_{j=1}^k j^2 \Lambda_j(s),\quad 0\leq s\leq t.$$

The variance always exceeds the mean under nontrivial multi-jump $k>1$, leading to intrinsic over-dispersion. For classical (single-jump) non-homogeneous Poisson processes ($k=1$), these collapse to familiar laws (Kataria et al., 2022, Tathe et al., 27 Jul 2024, Laskin, 2023).

5. Martingale Characterizations and Related Constructions

The compensated process $M(t)-A(t)$, with $A(t) = \sum_{j=1}^k j \Lambda_j(t)$, is a martingale. The stochastic exponential martingale

$$X_u(t) = \exp\left\{ \sum_{j=1}^k u j N_j(t) - \sum_{j=1}^k (e^{u j} - 1) \Lambda_j(t) \right\}$$

is martingale equivalent to the compensated form (Tathe et al., 18 Nov 2025). These characterizations underpin uniqueness and are extended to time-changed variants (e.g., fractional NGCPs via inverse stable subordinators) and Skellam-type differences, yielding processes with signed integer states.

6. Extensions: Fractional and Cox-Driven NGCP

Fractional generalizations employ Caputo derivatives and/or time-changed arguments. For instance, Laskin’s fractional NGCP employs state probabilities $P_{\mu,\beta}(n,t)$ satisfying

$${_0}D_t^\mu P_{\mu,\beta}(n,t) = \lambda_{\mu+\beta} t^\beta \left[P_{\mu,\beta}(n-1,t) - P_{\mu,\beta}(n,t)\right],$$

with explicit solutions in terms of the Kilbas–Saigo three-parameter Mittag-Leffler function $E_{\alpha,m,l}$ (Laskin, 2023). These processes admit stretched-exponential interarrival distributions, long-range dependence through heavy-tailed waiting times, and recapitulate Poisson/fractional Poisson limits.

Another major extension is the NGCP constructed by marginalizing Poisson-driven Cox processes, with a stochastic intensity modeled as

$$\Lambda(t) = \beta_0 + w Y(t), \qquad Y(t)\sim \text{NPP with rate }\gamma(t),$$

yielding a marginal counting process with over-dispersion and likelihood

$$p_X(t_1,\dots,t_M) = \left[\sum_{j=0}^M c_j^{(M)} w^j \beta_0^{M-j}\right] \exp\left\{-\beta_0 T - \int_0^T \lambda(u) du\right\},$$

with recursively-defined $c_j^{(M)}$ coefficients. The likelihood facilitates direct optimization or Bayesian inference without latent variable augmentation (Wang et al., 2023).

7. Special and Limiting Cases

Specializations of the NGCP include:

• $k=1$ with arbitrary $\lambda_1(t)$: standard non-homogeneous Poisson process.
• All $\lambda_j(t)\equiv\lambda$ constant: homogeneous Generalized Counting Process.
• $w\to 0$ in Cox-driven NGCP: homogeneous Poisson process.
• $\beta_0=0$ in Cox-driven NGCP: pure shot-noise over-dispersed process.
• Constant $\gamma(t)$ in Cox-driven models: compound-Poisson mixtures.

Fractional and time-changed versions involve stable or tempered stable subordinators, leading to processes with nontrivial dependence structures, heavy tails, and anomalous scaling.

8. Applications, Properties and Connections

Various NGCP frameworks (including Skellam-type differences and fractional extensions) have been applied to problems in high-frequency finance, ruin theory, order statistics, and modeling of phenomena with bursty, correlated, or over-dispersed arrival patterns (Tathe et al., 27 Jul 2024, Kataria et al., 2022). The running average process, first passage statistics, and compound versions are actively studied for such applications.

The process interpolates between Poissonian, compound, and fractional laws, providing a unified approach to count processes with arbitrary jump amplitudes and non-stationary, path-dependent intensities. The unifying weighted-sum representation serves as a foundational structural property, connecting NGCPs to a wide array of martingale and time-changed processes.

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References:

• "A Non-homogeneous Count Process: Marginalizing a Poisson Driven Cox Process" (Wang et al., 2023).
• "Generalized Counting Process: its Non-Homogeneous and Time-Changed Versions" (Kataria et al., 2022).
• "Martingale Characterizations of Non-Homogeneous Counting Processes and Their Fractional Variants" (Tathe et al., 18 Nov 2025).
• "Non-Homogeneous Generalized Fractional Skellam Process" (Tathe et al., 27 Jul 2024).
• "Fractional non-homogeneous counting process" (Laskin, 2023).