Stochastic Poisson Processes and Fractional Counting Models

Implementation of advanced stochastic counting processes. Main models include the Fractional Counting Process at Levy times (Garg et al. (2025) ), the Generalized Iterated Poisson Process (Soni & Pathak (2024) ), the Generalized Fractional Risk Process (Soni & Pathak (2024) ), and the Tempered Space-Time Fractional Negative Binomial Process (Garg et al. (2025) ).

StochFracPoisson

An R package implementing advanced stochastic counting processes based on recent research in fractional calculus and stochastic subordination.

Installation

## Installation

You can install the package from the local source directory:

```r
# Install from local source
install.packages("path/to/StochFracPoisson", repos = NULL, type = "source")

Overview

This package implements the following stochastic processes:

Fractional Counting Processes

• Fractional Counting Process (FCP): A generalized counting process using three-parameter Mittag-Leffler functions
• Time-Changed FCP (TCFCP): FCP subordinated by Lévy subordinators
• Compound Generalized FCP (CGFCP): Compound version with random jump sizes

Risk Processes

• Generalized Fractional Risk Process (GFRP): Insurance risk models with fractional counting processes

Negative Binomial Processes

• Tempered Space-Time Fractional Negative Binomial Process (TSTFNBP): Tempered Mittag-Leffler subordinated process

Iterated Processes

• Generalized Iterated Poisson Process (GIPP): Composition of Poisson processes with fractional Bell polynomials

Special Functions

The package includes implementations of:

• Mittag-Leffler functions (standard and Prabhakar three-parameter)
• Gamma subordinators
• Inverse stable subordinators
• Tempered stable subordinators
• Tempered Mittag-Leffler Lévy processes
• Fractional Bell polynomials
• Fractional Stirling numbers of the second kind

Example Usage

library(StochFracPoisson)

# Fractional Counting Process
fcp <- fractional_counting_process(mu = 0.8, zeta = 1.0, theta = 0.5, 
                                   lambda = 1.0, t = 1.0, n = 0:10)

# Time-Changed FCP with gamma subordinator
tcfcp <- time_changed_fcp(mu = 0.8, zeta = 1.0, theta = 0.5, lambda = 1.0,
                          subordinator_type = "gamma", 
                          subordinator_params = list(shape = 2, rate = 1),
                          t = 1.0, n = 0:10)

# Generalized Iterated Poisson Process
gipp <- generalized_iterated_poisson(beta = 0.7, lambda = 1.0, 
                                     lambda_beta = 0.5, t = 1.0, k = 0:10)

# Simulate sample paths
sim_data <- simulate_fcp(mu = 0.8, zeta = 1.0, theta = 0.5, lambda = 1.0,
                         t_max = 10, n_paths = 5)

References

Please cite the following papers if you use this package:

1. Garg, S., Pathak, A.K. & Maheshwari, A. Fractional counting process at Lévy times and its applications. J Stat Phys 192, 130 (2025). https://doi.org/10.1007/s10955-025-03515-9

2. Soni, R., Pathak, A.K. Generalized Iterated Poisson Process and Applications. J Theor Probab 37, 3216–3245 (2024). https://doi.org/10.1007/s10959-024-01362-0

3. Soni, R., Pathak, A.K. Generalized Fractional Risk Process. Methodol Comput Appl Probab 26, 42 (2024). https://doi.org/10.1007/s11009-024-10111-z

4. Garg, S., Pathak, A.K. & Maheshwari, A. Tempered Space-Time Fractional Negative Binomial Process. Methodol Comput Appl Probab 27, 51 (2025). https://doi.org/10.1007/s11009-025-10179-1

License

MIT License

Contributing

Contributions are welcome! Please feel free to submit a Pull Request.