Package 'StochFracPoisson' reference manual

Title:	Stochastic Poisson Processes and Fractional Counting Models
Description:	Implementation of advanced stochastic counting processes. Main models include the Fractional Counting Process at Levy times (Garg et al. (2025) <doi:10.1007/s10955-025-03515-9>), the Generalized Iterated Poisson Process (Soni & Pathak (2024) <doi:10.1007/s10959-024-01362-0>), the Generalized Fractional Risk Process (Soni & Pathak (2024) <doi:10.1007/s11009-024-10111-z>), and the Tempered Space-Time Fractional Negative Binomial Process (Garg et al. (2025) <doi:10.1007/s11009-025-10179-1>).
Authors:	Shikhar Tyagi [aut, cre] (ORCID: <https://orcid.org/0000-0003-1606-0844>), Vrijesh Tripathi [aut]
Maintainer:	Shikhar Tyagi <[email protected]>
License:	MIT + file LICENSE
Version:	0.1.0
Built:	2026-09-29 10:28:58 UTC
Source:	https://github.com/cran/StochFracPoisson

Alternative Generalized Fractional Risk Process (AGFRP)

Description

Computes the alternative GFRP with premium function epsilon $\hat{R}^\beta(t) = \nu + \eta(1 + \rho)\epsilon(W_\beta(t)) - \sum_{j=1}^{N^\beta(t)} X_j$

Usage

alternative_gfrp(
  nu,
  eta,
  rho,
  beta,
  lambda,
  t,
  epsilon = function(x) x,
  claim_dist = "gamma",
  claim_params = list(),
  n_sim = 1000
)
alternative_gfrp(
  nu,
  eta,
  rho,
  beta,
  lambda,
  t,
  epsilon = function(x) x,
  claim_dist = "gamma",
  claim_params = list(),
  n_sim = 1000
)

Arguments

nu

Initial capital (> 0)

eta

Mean claim size (> 0)

rho

Safety loading parameter (>= 0)

beta

Fractional parameter (0 < beta < 1)

lambda

Vector of rates lambda_1, ..., lambda_k for the GFCP

t

Time (> 0)

epsilon

Premium function (default = identity)

claim_dist

Distribution of claim sizes ("gamma", "exponential", "lognormal")

claim_params

Parameters for claim distribution

n_sim

Number of simulations (default = 1000)

Value

Simulated values of the alternative risk process at time t

Examples

alternative_gfrp(100, 10, 0.1, 0.7, c(1.0, 0.5), 1.0,
                   function(x) x^0.5, "gamma", list(shape = 2, rate = 0.2))
alternative_gfrp(100, 10, 0.1, 0.7, c(1.0, 0.5), 1.0,
                   function(x) x^0.5, "gamma", list(shape = 2, rate = 0.2))

Classical Bell Polynomials

Description

Computes the classical Bell polynomials B_n(x)

Usage

bell_polynomial(n, x, n_terms = 50)
bell_polynomial(n, x, n_terms = 50)

Arguments

n

Polynomial order (non-negative integer)

x

Argument

n_terms

Number of terms in series expansion (default = 50)

Value

Value of the Bell polynomial

Examples

bell_polynomial(5, 1.0)
bell_polynomial(5, 1.0)

Check Long-Range Dependence of GIPP

Description

Checks if the GIPP exhibits long-range dependence

Usage

check_lrd_gipp(beta, lambda, lambda_beta, s, t_max = 100)
check_lrd_gipp(beta, lambda, lambda_beta, s, t_max = 100)

Arguments

beta

Fractional parameter (0 < beta < 1)

lambda

Parameter of outer Poisson process (> 0)

lambda_beta

Parameter of inner fractional Poisson process (> 0)

s

Fixed time for correlation analysis

t_max

Maximum time for asymptotic analysis

Value

Logical indicating LRD property

Examples

check_lrd_gipp(0.7, 1.0, 0.5, 0.5, 100)
check_lrd_gipp(0.7, 1.0, 0.5, 0.5, 100)

Check Martingale Property of GFRP

Description

Checks if the GFRP is a martingale, submartingale, or supermartingale

Usage

check_martingale_gfrp(rho)
check_martingale_gfrp(rho)

Arguments

rho

Safety loading parameter (>= 0)

Value

String indicating martingale property

Examples

check_martingale_gfrp(0.1)
check_martingale_gfrp(0.1)

Compound Generalized Fractional Counting Process (CGFCP)

Description

Computes the probability mass function of the CGFCP $C^\beta(t) = \sum_{j=1}^{N^\beta(t)} X_j$ From Soni and Pathak (2024)

Usage

compound_gfcp(
  beta,
  lambda,
  t,
  n,
  jump_dist = "poisson",
  jump_params = list(),
  n_terms = 50
)
compound_gfcp(
  beta,
  lambda,
  t,
  n,
  jump_dist = "poisson",
  jump_params = list(),
  n_terms = 50
)

Arguments

beta

Fractional parameter (0 < beta < 1)

lambda

Vector of rates lambda_1, ..., lambda_k for the GFCP

t

Time (> 0)

n

Vector of non-negative integers

jump_dist

Distribution of jump sizes ("poisson", "geometric")

jump_params

Parameters for jump distribution

n_terms

Number of terms in series expansion (default = 50)

Value

Vector of probabilities P(C(t) = n)

Examples

compound_gfcp(0.7, c(1.0, 0.5), 1.0, 0:10, "poisson", list(lambda = 2))
compound_gfcp(0.7, c(1.0, 0.5), 1.0, 0:10, "poisson", list(lambda = 2))

Covariance of GIPP

Description

Computes the covariance of the GIPP

Usage

cov_gipp(beta, lambda, lambda_beta, s, t)
cov_gipp(beta, lambda, lambda_beta, s, t)

Arguments

beta

Fractional parameter (0 < beta < 1)

lambda

Parameter of outer Poisson process (> 0)

lambda_beta

Parameter of inner fractional Poisson process (> 0)

s

First time (0 < s <= t)

t

Second time

Value

Covariance value

Examples

cov_gipp(0.7, 1.0, 0.5, 0.5, 1.0)
cov_gipp(0.7, 1.0, 0.5, 0.5, 1.0)

Covariance of Inverse Stable Subordinator

Description

Computes the covariance of the inverse beta-stable subordinator Based on Leonenko et al. (2014)

Usage

cov_inverse_stable(s, t, beta)
cov_inverse_stable(s, t, beta)

Arguments

s

First time

t

Second time (s <= t)

beta

Stability parameter (0 < beta < 1)

Value

Covariance value

Examples

cov_inverse_stable(0.5, 1, 0.8)
cov_inverse_stable(0.5, 1, 0.8)

Covariance of TSTFNBP

Description

Computes the covariance of the TSTFNBP

Usage

cov_tstfnbp(beta, alpha, mu, lambda1, beta1, lambda, s, t)
cov_tstfnbp(beta, alpha, mu, lambda1, beta1, lambda, s, t)

Arguments

beta

Fractional parameter (0 < beta < 1)

alpha

Stability parameter (0 < alpha < 1)

mu

Tempering parameter (> 0)

lambda1

Gamma subordinator rate (> mu*alpha)

beta1

Gamma subordinator shape (> 0)

lambda

Poisson parameter (> 0)

s

First time (0 < s <= t)

t

Second time

Value

Covariance value

Examples

cov_tstfnbp(0.7, 0.8, 0.5, 2, 1, 1.0, 0.5, 1.0)
cov_tstfnbp(0.7, 0.8, 0.5, 2, 1, 1.0, 0.5, 1.0)

Create FCP Object

Description

Creates an object representing the Fractional Counting Process

Usage

create_fcp(mu, zeta, theta, lambda, t, n, vartheta = NULL, n_terms = 50)
create_fcp(mu, zeta, theta, lambda, t, n, vartheta = NULL, n_terms = 50)

Arguments

mu

Parameter mu (0 < mu <= 1)

zeta

Parameter zeta (> 0)

theta

Parameter theta (0 < theta <= 1)

lambda

Parameter lambda (> 0)

t

Time (> 0)

n

Vector of non-negative integers

vartheta

Parameter vartheta (>= mu*zeta, default = zeta)

n_terms

Number of terms in series expansion (default = 50)

Value

An object of class fcp containing parameters, probabilities, mean, and variance.

Examples

fcp_obj <- create_fcp(0.8, 1.0, 0.5, 1.0, 1.0, 0:10)
fcp_obj <- create_fcp(0.8, 1.0, 0.5, 1.0, 1.0, 0:10)

Create GIPP Object

Description

Creates an object representing the GIPP

Usage

create_gipp(beta, lambda, lambda_beta, t, k, n_terms = 50)
create_gipp(beta, lambda, lambda_beta, t, k, n_terms = 50)

Arguments

beta

Fractional parameter (0 < beta < 1)

lambda

Parameter of outer Poisson process (> 0)

lambda_beta

Parameter of inner fractional Poisson process (> 0)

t

Time (> 0)

k

Vector of non-negative integers

n_terms

Number of terms in series expansion (default = 50)

Value

An object of class gipp containing parameters, probabilities, mean, and variance.

Examples

gipp_obj <- create_gipp(0.7, 1.0, 0.5, 1.0, 0:10)
gipp_obj <- create_gipp(0.7, 1.0, 0.5, 1.0, 0:10)

Create TCFCP Object

Description

Creates an object representing the Time-Changed Fractional Counting Process

Usage

create_tcfcp(
  mu,
  zeta,
  theta,
  lambda,
  t,
  n,
  subordinator_type = "gamma",
  subordinator_params = list(),
  vartheta = NULL,
  n_terms = 50
)
create_tcfcp(
  mu,
  zeta,
  theta,
  lambda,
  t,
  n,
  subordinator_type = "gamma",
  subordinator_params = list(),
  vartheta = NULL,
  n_terms = 50
)

Arguments

mu

Parameter mu (0 < mu <= 1)

zeta

Parameter zeta (> 0)

theta

Parameter theta (0 < theta <= 1)

lambda

Parameter lambda (> 0)

t

Time (> 0)

n

Vector of non-negative integers

subordinator_type

Type of subordinator

subordinator_params

List of parameters for the subordinator

vartheta

Parameter vartheta (>= mu*zeta, default = zeta)

n_terms

Number of terms in series expansion (default = 50)

Value

An object of class tcfcp containing parameters, probabilities, mean, and variance.

Examples

tcfcp_obj <- create_tcfcp(0.8, 1.0, 0.5, 1.0, 1.0, 0:10, "gamma", list(shape = 2, rate = 1))
tcfcp_obj <- create_tcfcp(0.8, 1.0, 0.5, 1.0, 1.0, 0:10, "gamma", list(shape = 2, rate = 1))

PDF of Gamma Subordinator

Description

Computes the probability density function of the gamma subordinator

Usage

dgamma_subordinator(x, t, shape, rate)
dgamma_subordinator(x, t, shape, rate)

Arguments

x

Value

t

Time

shape

Shape parameter (beta1)

rate

Rate parameter (lambda1)

Value

PDF value

Examples

dgamma_subordinator(1, 1, 2, 1)
dgamma_subordinator(1, 1, 2, 1)

PDF of Tempered Mittag-Leffler Lévy Process

Description

Computes the probability density function of the tempered Mittag-Leffler Lévy process Based on Kumar et al. (2019b)

Usage

dtempered_mittag_leffler_levy(x, t, alpha, mu, lambda1, beta1, n_terms = 50)
dtempered_mittag_leffler_levy(x, t, alpha, mu, lambda1, beta1, n_terms = 50)

Arguments

x

Value

t

Time

alpha

Stability parameter (0 < alpha < 1)

mu

Tempering parameter (> 0)

lambda1

Gamma subordinator rate (> 0)

beta1

Gamma subordinator shape (> 0)

n_terms

Number of terms in series expansion (default = 50)

Value

PDF value

Examples

dtempered_mittag_leffler_levy(1, 1, 0.8, 0.5, 2, 1)
dtempered_mittag_leffler_levy(1, 1, 0.8, 0.5, 2, 1)

Exponential Generating Function of Fractional Bell Polynomials

Description

Computes the exponential generating function $\sum_{m=0}^\infty (u^m / m!) B_m^\beta(x) = L_\beta(x(e^u - 1))$

Usage

egf_fractional_bell(beta, x, u, n_terms = 50)
egf_fractional_bell(beta, x, u, n_terms = 50)

Arguments

beta

Fractional parameter (0 < beta < 1)

x

Argument

u

Generating function variable

n_terms

Number of terms in series expansion (default = 50)

Value

Value of the exponential generating function

Examples

egf_fractional_bell(0.7, 1.0, 0.5)
egf_fractional_bell(0.7, 1.0, 0.5)

Fractional Differential Equation for CGFCP

Description

Computes the fractional differential equation governing the CGFCP

Usage

fde_cgfcp(
  beta,
  lambda,
  t,
  n,
  jump_dist = "poisson",
  jump_params = list(),
  h = 1e-06
)
fde_cgfcp(
  beta,
  lambda,
  t,
  n,
  jump_dist = "poisson",
  jump_params = list(),
  h = 1e-06
)

Arguments

beta

Fractional parameter (0 < beta < 1)

lambda

Vector of rates lambda_1, ..., lambda_k for the GFCP

t

Time (> 0)

n

Value at which to evaluate the derivative

jump_dist

Distribution of jump sizes

jump_params

Parameters for jump distribution

h

Step size for numerical differentiation (default = 1e-6)

Value

Value of the fractional derivative

Examples

fde_cgfcp(0.7, c(1.0, 0.5), 1.0, 5, "poisson", list(lambda = 2))
fde_cgfcp(0.7, c(1.0, 0.5), 1.0, 5, "poisson", list(lambda = 2))

First Passage Time Density of GIPP

Description

Computes the density of the first passage time for GIPP

Usage

first_passage_density_gipp(beta, lambda, lambda_beta, t, k, n_terms = 50)
first_passage_density_gipp(beta, lambda, lambda_beta, t, k, n_terms = 50)

Arguments

beta

Fractional parameter (0 < beta < 1)

lambda

Parameter of outer Poisson process (> 0)

lambda_beta

Parameter of inner fractional Poisson process (> 0)

t

Time (> 0)

k

Threshold level (positive integer)

n_terms

Number of terms in series expansion (default = 50)

Value

Density value at t

Examples

first_passage_density_gipp(0.7, 1.0, 0.5, 1.0, 3)
first_passage_density_gipp(0.7, 1.0, 0.5, 1.0, 3)

First Passage Time Distribution of FCP

Description

Computes the survival function of the first passage time P(T_w > t) where T_w = inf{t >= 0: N(t) >= w}

Usage

first_passage_time_fcp(
  mu,
  zeta,
  theta,
  lambda,
  t,
  w,
  vartheta = NULL,
  n_terms = 50
)
first_passage_time_fcp(
  mu,
  zeta,
  theta,
  lambda,
  t,
  w,
  vartheta = NULL,
  n_terms = 50
)

Arguments

mu

Parameter mu (0 < mu <= 1)

zeta

Parameter zeta (> 0)

theta

Parameter theta (0 < theta <= 1)

lambda

Parameter lambda (> 0)

t

Time (> 0)

w

Threshold level (positive integer)

vartheta

Parameter vartheta (>= mu*zeta, default = zeta)

n_terms

Number of terms in series expansion (default = 50)

Value

Survival probability P(T_w > t)

Examples

first_passage_time_fcp(0.8, 1.0, 0.5, 1.0, 1.0, 3)
first_passage_time_fcp(0.8, 1.0, 0.5, 1.0, 1.0, 3)

First Passage Time Distribution of GIPP

Description

Computes the survival function of the first passage time for GIPP

Usage

first_passage_time_gipp(beta, lambda, lambda_beta, t, k, n_terms = 50)
first_passage_time_gipp(beta, lambda, lambda_beta, t, k, n_terms = 50)

Arguments

beta

Fractional parameter (0 < beta < 1)

lambda

Parameter of outer Poisson process (> 0)

lambda_beta

Parameter of inner fractional Poisson process (> 0)

t

Time (> 0)

k

Threshold level (positive integer)

n_terms

Number of terms in series expansion (default = 50)

Value

Survival probability P(T_k > t)

Examples

first_passage_time_gipp(0.7, 1.0, 0.5, 1.0, 3)
first_passage_time_gipp(0.7, 1.0, 0.5, 1.0, 3)

First Passage Time Distribution of TCFCP

Description

Computes the survival function of the first passage time for TCFCP

Usage

first_passage_time_tcfcp(
  mu,
  zeta,
  theta,
  lambda,
  t,
  w,
  subordinator_type = "gamma",
  subordinator_params = list(),
  vartheta = NULL,
  n_terms = 50
)
first_passage_time_tcfcp(
  mu,
  zeta,
  theta,
  lambda,
  t,
  w,
  subordinator_type = "gamma",
  subordinator_params = list(),
  vartheta = NULL,
  n_terms = 50
)

Arguments

mu

Parameter mu (0 < mu <= 1)

zeta

Parameter zeta (> 0)

theta

Parameter theta (0 < theta <= 1)

lambda

Parameter lambda (> 0)

t

Time (> 0)

w

Threshold level (positive integer)

subordinator_type

Type of subordinator

subordinator_params

List of parameters for the subordinator

vartheta

Parameter vartheta (>= mu*zeta, default = zeta)

n_terms

Number of terms in series expansion (default = 50)

Value

Survival probability P(T_w > t)

Examples

first_passage_time_tcfcp(0.8, 1.0, 0.5, 1.0, 1.0, 3, "gamma", list(shape = 2, rate = 1))
first_passage_time_tcfcp(0.8, 1.0, 0.5, 1.0, 1.0, 3, "gamma", list(shape = 2, rate = 1))

First Waiting Time Distribution of FCP

Description

Computes the first waiting time distribution of the FCP

Usage

first_waiting_time_fcp(mu, zeta, theta, lambda, tau, vartheta = NULL)
first_waiting_time_fcp(mu, zeta, theta, lambda, tau, vartheta = NULL)

Arguments

mu

Parameter mu (0 < mu <= 1)

zeta

Parameter zeta (> 0)

theta

Parameter theta (0 < theta <= 1)

lambda

Parameter lambda (> 0)

tau

Time (> 0)

vartheta

Parameter vartheta (>= mu*zeta, default = zeta)

Value

First waiting time density at tau

Examples

first_waiting_time_fcp(0.8, 1.0, 0.5, 1.0, 0.5)
first_waiting_time_fcp(0.8, 1.0, 0.5, 1.0, 0.5)

First Waiting Time Distribution of TCFCP

Description

Computes the first waiting time distribution of the TCFCP

Usage

first_waiting_time_tcfcp(
  mu,
  zeta,
  theta,
  lambda,
  tau,
  subordinator_type = "gamma",
  subordinator_params = list(),
  vartheta = NULL,
  n_terms = 50
)
first_waiting_time_tcfcp(
  mu,
  zeta,
  theta,
  lambda,
  tau,
  subordinator_type = "gamma",
  subordinator_params = list(),
  vartheta = NULL,
  n_terms = 50
)

Arguments

mu

Parameter mu (0 < mu <= 1)

zeta

Parameter zeta (> 0)

theta

Parameter theta (0 < theta <= 1)

lambda

Parameter lambda (> 0)

tau

Time (> 0)

subordinator_type

Type of subordinator

subordinator_params

List of parameters for the subordinator

vartheta

Parameter vartheta (>= mu*zeta, default = zeta)

n_terms

Number of terms in series expansion (default = 50)

Value

First waiting time density at tau

Examples

first_waiting_time_tcfcp(0.8, 1.0, 0.5, 1.0, 0.5, "gamma", list(shape = 2, rate = 1))
first_waiting_time_tcfcp(0.8, 1.0, 0.5, 1.0, 0.5, "gamma", list(shape = 2, rate = 1))

Fractional Bell Polynomials

Description

Computes the fractional Bell polynomials $B_n^\beta(x)$ Uses log-space computations to prevent numerical overflow for large orders. From Soni and Pathak (2024)

Usage

fractional_bell_polynomial(n, beta, x, n_terms = 50)
fractional_bell_polynomial(n, beta, x, n_terms = 50)

Arguments

n

Polynomial order (non-negative integer)

beta

Fractional parameter (0 < beta < 1)

x

Argument

n_terms

Number of terms in series expansion (default = 50)

Value

Value of the fractional Bell polynomial

Examples

fractional_bell_polynomial(5, 0.7, 1.0)
fractional_bell_polynomial(5, 0.7, 1.0)

Fractional Counting Process (FCP)

Description

Computes the probability mass function of the Fractional Counting Process based on the three-parameter Mittag-Leffler function (Prabhakar function) From Garg et al. (2025)

Usage

fractional_counting_process(
  mu,
  zeta,
  theta,
  lambda,
  t,
  n,
  vartheta = NULL,
  n_terms = 50
)
fractional_counting_process(
  mu,
  zeta,
  theta,
  lambda,
  t,
  n,
  vartheta = NULL,
  n_terms = 50
)

Arguments

mu

Parameter mu (0 < mu <= 1)

zeta

Parameter zeta (> 0)

theta

Parameter theta (0 < theta <= 1)

lambda

Parameter lambda (> 0)

t

Time (> 0)

n

Vector of non-negative integers

vartheta

Parameter vartheta (>= mu*zeta, default = zeta)

n_terms

Number of terms in series expansion (default = 50)

Value

Vector of probabilities P(N(t) = n)

Examples

fractional_counting_process(0.8, 1.0, 0.5, 1.0, 1.0, 0:10)
fractional_counting_process(0.8, 1.0, 0.5, 1.0, 1.0, 0:10)

Fractional Stirling Numbers of the Second Kind

Description

Computes the fractional Stirling numbers of the second kind $S_\beta(m, k)$ Uses log-space computation to prevent numerical overflow. From Soni and Pathak (2024)

Usage

fractional_stirling2(m, k, beta)
fractional_stirling2(m, k, beta)

Arguments

m

Upper index (non-negative integer)

k

Lower index (non-negative integer)

beta

Fractional parameter (0 < beta < 1)

Value

Value of the fractional Stirling number

Examples

fractional_stirling2(5, 3, 0.7)
fractional_stirling2(5, 3, 0.7)

Gamma Subordinator

Description

Simulates a gamma subordinator G(t) with shape parameter beta1*t and rate lambda1

Usage

gamma_subordinator(t, shape, rate, n = 1)
gamma_subordinator(t, shape, rate, n = 1)

Arguments

t

Time

shape

Shape parameter (beta1)

rate

Rate parameter (lambda1)

n

Number of simulations

Value

Simulated values of the gamma subordinator

Examples

gamma_subordinator(1, 2, 1, 10)
gamma_subordinator(1, 2, 1, 10)

Generalized Fractional Risk Process (GFRP)

Description

Computes the Generalized Fractional Risk Process for insurance applications $R^\beta(t) = \nu + \eta(1 + \rho) \sum_{i=1}^k i \lambda_i W_\beta(t) - \sum_{j=1}^{N^\beta(t)} X_j$ From Soni and Pathak (2024)

Usage

generalized_fractional_risk_process(
  nu,
  eta,
  rho,
  beta,
  lambda,
  t,
  claim_dist = "gamma",
  claim_params = list(),
  n_sim = 1000
)
generalized_fractional_risk_process(
  nu,
  eta,
  rho,
  beta,
  lambda,
  t,
  claim_dist = "gamma",
  claim_params = list(),
  n_sim = 1000
)

Arguments

nu

Initial capital (> 0)

eta

Mean claim size (> 0)

rho

Safety loading parameter (>= 0)

beta

Fractional parameter (0 < beta < 1)

lambda

Vector of rates lambda_1, ..., lambda_k for the GFCP

t

Time (> 0)

claim_dist

Distribution of claim sizes ("gamma", "exponential", "lognormal")

claim_params

Parameters for claim distribution

n_sim

Number of simulations (default = 1000)

Value

Simulated values of the risk process at time t

Examples

generalized_fractional_risk_process(100, 10, 0.1, 0.7, c(1.0, 0.5), 1.0, 
                                     "gamma", list(shape = 2, rate = 0.2))
generalized_fractional_risk_process(100, 10, 0.1, 0.7, c(1.0, 0.5), 1.0, 
                                     "gamma", list(shape = 2, rate = 0.2))

Generalized Fractional Stirling Numbers of the Second Kind

Description

Computes the generalized fractional Stirling numbers $S_{\mu,\vartheta}(p, i)$ From Laskin (2024)

Usage

generalized_fractional_stirling2(p, i, mu, vartheta)
generalized_fractional_stirling2(p, i, mu, vartheta)

Arguments

p

Upper index (non-negative integer)

i

Lower index (non-negative integer)

mu

Parameter mu (0 < mu <= 1)

vartheta

Parameter vartheta (> 0)

Value

Value of the generalized fractional Stirling number

Examples

generalized_fractional_stirling2(5, 3, 0.8, 1.0)
generalized_fractional_stirling2(5, 3, 0.8, 1.0)

Generalized Iterated Poisson Process (GIPP)

Description

Computes the probability mass function of the Generalized Iterated Poisson Process Q(t) = N(N_beta(t), lambda) - composition of HPP with TFPP. From Soni and Pathak (2024)

Usage

generalized_iterated_poisson(beta, lambda, lambda_beta, t, k, n_terms = 50)
generalized_iterated_poisson(beta, lambda, lambda_beta, t, k, n_terms = 50)

Arguments

beta

Fractional parameter (0 < beta < 1)

lambda

Parameter of outer Poisson process (> 0)

lambda_beta

Parameter of inner fractional Poisson process (> 0)

t

Time (> 0)

k

Vector of non-negative integers

n_terms

Number of terms in series expansion (default = 50)

Value

Vector of probabilities P(Q(t) = k)

Examples

generalized_iterated_poisson(0.7, 1.0, 0.5, 1.0, 0:10)
generalized_iterated_poisson(0.7, 1.0, 0.5, 1.0, 0:10)

Generalized Fractional Counting Process (GFCP)

Description

Computes the probability mass function of the GFCP (with jumps of size 1, ..., k) using PGF inversion via FFT.

Usage

gfcp(beta, lambda, t, n)
gfcp(beta, lambda, t, n)

Arguments

beta

Fractional parameter (0 < beta < 1)

lambda

Vector of rates lambda_1, ..., lambda_k

t

Time (> 0)

n

Vector of non-negative integers

Value

Vector of probabilities P(N(t) = n)

Examples

gfcp(0.7, c(1.0, 0.5), 1.0, 0:10)
gfcp(0.7, c(1.0, 0.5), 1.0, 0:10)

Incomplete Beta Function

Description

Computes the incomplete beta function $B(a, b; x) = \int_0^x t^{a-1} (1-t)^{b-1} dt$

Usage

incomplete_beta(a, b, x)
incomplete_beta(a, b, x)

Arguments

a

Parameter a > 0

b

Parameter b > 0

x

Upper limit of integration (0 <= x <= 1)

Value

Value of the incomplete beta function

Examples

incomplete_beta(0.5, 0.5, 0.5)
incomplete_beta(0.5, 0.5, 0.5)

Inverse Stable Subordinator

Description

Simulates an inverse beta-stable subordinator E_beta(t) = inf{r >= 0: S_beta(r) > t} Using the exact relationship E_beta(t) = t^beta * S_beta(1)^(-beta).

Usage

inverse_stable_subordinator(t, beta, n = 1)
inverse_stable_subordinator(t, beta, n = 1)

Arguments

t

Time

beta

Stability parameter (0 < beta < 1)

n

Number of simulations

Value

Simulated values of the inverse stable subordinator

Examples

inverse_stable_subordinator(1, 0.8, 10)
inverse_stable_subordinator(1, 0.8, 10)

Numerical Inversion of Probability Generating Function

Description

Computes the probability mass function (PMF) of a discrete random variable from its PGF using Fast Fourier Transform (FFT).

Usage

invert_pgf(pgf_func, max_n, r = 0.85, M = NULL)
invert_pgf(pgf_func, max_n, r = 0.85, M = NULL)

Arguments

pgf_func

A function that evaluates the PGF at complex arguments

max_n

Maximum value of the random variable to compute the probability for

r

Radius of the contour circle (0 < r < 1, default = 0.85)

M

Number of points on the circle (must be a power of 2, default = NULL)

Value

Vector of probabilities P(X = 0), ..., P(X = max_n)

Laplace Transform of FCP

Description

Computes the Laplace transform of the FCP (expectation of exp(-s * N(t)))

Usage

lt_fcp(mu, zeta, theta, lambda, t, s, vartheta = NULL)
lt_fcp(mu, zeta, theta, lambda, t, s, vartheta = NULL)

Arguments

mu

Parameter mu (0 < mu <= 1)

zeta

Parameter zeta (> 0)

theta

Parameter theta (0 < theta <= 1)

lambda

Parameter lambda (> 0)

t

Time (> 0)

s

Laplace parameter

vartheta

Parameter vartheta (>= mu*zeta, default = zeta)

Value

Laplace transform value

Examples

lt_fcp(0.8, 1.0, 0.5, 1.0, 1.0, 0.5)
lt_fcp(0.8, 1.0, 0.5, 1.0, 1.0, 0.5)

Laplace Transform of TCFCP

Description

Computes the Laplace transform of the TCFCP (expectation of exp(-s * Z(t)))

Usage

lt_tcfcp(
  mu,
  zeta,
  theta,
  lambda,
  t,
  s,
  subordinator_type = "gamma",
  subordinator_params = list(),
  vartheta = NULL,
  n_terms = 50
)
lt_tcfcp(
  mu,
  zeta,
  theta,
  lambda,
  t,
  s,
  subordinator_type = "gamma",
  subordinator_params = list(),
  vartheta = NULL,
  n_terms = 50
)

Arguments

mu

Parameter mu (0 < mu <= 1)

zeta

Parameter zeta (> 0)

theta

Parameter theta (0 < theta <= 1)

lambda

Parameter lambda (> 0)

t

Time (> 0)

s

Laplace parameter

subordinator_type

Type of subordinator

subordinator_params

List of parameters for the subordinator

vartheta

Parameter vartheta (>= mu*zeta, default = zeta)

n_terms

Number of terms in series expansion (default = 50)

Value

Laplace transform value

Examples

lt_tcfcp(0.8, 1.0, 0.5, 1.0, 1.0, 0.5, "gamma", list(shape = 2, rate = 1))
lt_tcfcp(0.8, 1.0, 0.5, 1.0, 1.0, 0.5, "gamma", list(shape = 2, rate = 1))

Laplace Transform of Tempered Mittag-Leffler Lévy Process

Description

Computes the Laplace transform E(exp(-u * M(t)))

Usage

lt_tempered_mittag_leffler_levy(u, t, alpha, mu, lambda1, beta1)
lt_tempered_mittag_leffler_levy(u, t, alpha, mu, lambda1, beta1)

Arguments

u

Laplace transform parameter

t

Time

alpha

Stability parameter (0 < alpha < 1)

mu

Tempering parameter (> 0)

lambda1

Gamma subordinator rate (> 0)

beta1

Gamma subordinator shape (> 0)

Value

Laplace transform value

Examples

lt_tempered_mittag_leffler_levy(1, 1, 0.8, 0.5, 2, 1)
lt_tempered_mittag_leffler_levy(1, 1, 0.8, 0.5, 2, 1)

Laplace Transform of TSTFNBP

Description

Computes the Laplace transform of the TSTFNBP (expectation of exp(-u * Q(t)))

Usage

lt_tstfnbp(beta, alpha, mu, lambda1, beta1, lambda, t, u, n_terms = 50)
lt_tstfnbp(beta, alpha, mu, lambda1, beta1, lambda, t, u, n_terms = 50)

Arguments

beta

Fractional parameter (0 < beta < 1)

alpha

Stability parameter (0 < alpha < 1)

mu

Tempering parameter (> 0)

lambda1

Gamma subordinator rate (> mu*alpha)

beta1

Gamma subordinator shape (> 0)

lambda

Poisson parameter (> 0)

t

Time (> 0)

u

Laplace parameter

n_terms

Number of terms in series expansion (default = 50)

Value

Laplace transform value

Examples

lt_tstfnbp(0.7, 0.8, 0.5, 2, 1, 1.0, 1.0, 0.5)
lt_tstfnbp(0.7, 0.8, 0.5, 2, 1, 1.0, 1.0, 0.5)

Mean of CGFCP

Description

Computes the mean of the CGFCP

Usage

mean_cgfcp(beta, lambda, t, jump_dist = "poisson", jump_params = list())
mean_cgfcp(beta, lambda, t, jump_dist = "poisson", jump_params = list())

Arguments

beta

Fractional parameter (0 < beta < 1)

lambda

Vector of rates lambda_1, ..., lambda_k for the GFCP

t

Time (> 0)

jump_dist

Distribution of jump sizes ("poisson", "geometric")

jump_params

Parameters for jump distribution

Value

Mean value

Examples

mean_cgfcp(0.7, c(1.0, 0.5), 1.0, "poisson", list(lambda = 2))
mean_cgfcp(0.7, c(1.0, 0.5), 1.0, "poisson", list(lambda = 2))

Mean of Fractional Counting Process

Description

Computes the mean of the FCP

Usage

mean_fcp(mu, zeta, theta, lambda, t, vartheta = NULL)
mean_fcp(mu, zeta, theta, lambda, t, vartheta = NULL)

Arguments

mu

Parameter mu (0 < mu <= 1)

zeta

Parameter zeta (> 0)

theta

Parameter theta (0 < theta <= 1)

lambda

Parameter lambda (> 0)

t

Time (> 0)

vartheta

Parameter vartheta (>= mu*zeta, default = zeta)

Value

Mean value

Examples

mean_fcp(0.8, 1.0, 0.5, 1.0, 1.0)
mean_fcp(0.8, 1.0, 0.5, 1.0, 1.0)

Mean of GFRP

Description

Computes the mean of the GFRP

Usage

mean_gfrp(nu, eta, rho, beta, lambda, t)
mean_gfrp(nu, eta, rho, beta, lambda, t)

Arguments

nu

Initial capital (> 0)

eta

Mean claim size (> 0)

rho

Safety loading parameter (>= 0)

beta

Fractional parameter (0 < beta < 1)

lambda

Vector of rates lambda_1, ..., lambda_k for the GFCP

t

Time (> 0)

Value

Mean value

Examples

mean_gfrp(100, 10, 0.1, 0.7, c(1.0, 0.5), 1.0)
mean_gfrp(100, 10, 0.1, 0.7, c(1.0, 0.5), 1.0)

Mean of GIPP

Description

Computes the mean of the GIPP

Usage

mean_gipp(beta, lambda, lambda_beta, t)
mean_gipp(beta, lambda, lambda_beta, t)

Arguments

beta

Fractional parameter (0 < beta < 1)

lambda

Parameter of outer Poisson process (> 0)

lambda_beta

Parameter of inner fractional Poisson process (> 0)

t

Time (> 0)

Value

Mean value

Examples

mean_gipp(0.7, 1.0, 0.5, 1.0)
mean_gipp(0.7, 1.0, 0.5, 1.0)

Mean of Inverse Stable Subordinator

Description

Computes the mean of the inverse beta-stable subordinator

Usage

mean_inverse_stable(t, beta)
mean_inverse_stable(t, beta)

Arguments

t

Time

beta

Stability parameter (0 < beta < 1)

Value

Mean value

Examples

mean_inverse_stable(1, 0.8)
mean_inverse_stable(1, 0.8)

Mean of Time-Changed FCP

Description

Computes the mean of the TCFCP

Usage

mean_tcfcp(
  mu,
  zeta,
  theta,
  lambda,
  t,
  subordinator_type = "gamma",
  subordinator_params = list(),
  vartheta = NULL
)
mean_tcfcp(
  mu,
  zeta,
  theta,
  lambda,
  t,
  subordinator_type = "gamma",
  subordinator_params = list(),
  vartheta = NULL
)

Arguments

mu

Parameter mu (0 < mu <= 1)

zeta

Parameter zeta (> 0)

theta

Parameter theta (0 < theta <= 1)

lambda

Parameter lambda (> 0)

t

Time (> 0)

subordinator_type

Type of subordinator

subordinator_params

List of parameters for the subordinator

vartheta

Parameter vartheta (>= mu*zeta, default = zeta)

Value

Mean value

Examples

mean_tcfcp(0.8, 1.0, 0.5, 1.0, 1.0, "gamma", list(shape = 2, rate = 1))
mean_tcfcp(0.8, 1.0, 0.5, 1.0, 1.0, "gamma", list(shape = 2, rate = 1))

Mean of TSTFNBP

Description

Computes the mean of the TSTFNBP

Usage

mean_tstfnbp(beta, alpha, mu, lambda1, beta1, lambda, t)
mean_tstfnbp(beta, alpha, mu, lambda1, beta1, lambda, t)

Arguments

beta

Fractional parameter (0 < beta < 1)

alpha

Stability parameter (0 < alpha < 1)

mu

Tempering parameter (> 0)

lambda1

Gamma subordinator rate (> mu*alpha)

beta1

Gamma subordinator shape (> 0)

lambda

Poisson parameter (> 0)

t

Time (> 0)

Value

Mean value

Examples

mean_tstfnbp(0.7, 0.8, 0.5, 2, 1, 1.0, 1.0)
mean_tstfnbp(0.7, 0.8, 0.5, 2, 1, 1.0, 1.0)

Moment of Gamma Subordinator

Description

Computes the q-th moment of the gamma subordinator

Usage

mgamma_subordinator(q, t, shape, rate)
mgamma_subordinator(q, t, shape, rate)

Arguments

q

Order of moment

t

Time

shape

Shape parameter (beta1)

rate

Rate parameter (lambda1)

Value

q-th moment value

Examples

mgamma_subordinator(2, 1, 2, 1)
mgamma_subordinator(2, 1, 2, 1)

Moment Generating Function of FCP

Description

Computes the moment generating function of the Fractional Counting Process

Usage

mgf_fcp(mu, zeta, theta, lambda, t, s, vartheta = NULL)
mgf_fcp(mu, zeta, theta, lambda, t, s, vartheta = NULL)

Arguments

mu

Parameter mu (0 < mu <= 1)

zeta

Parameter zeta (> 0)

theta

Parameter theta (0 < theta <= 1)

lambda

Parameter lambda (> 0)

t

Time (> 0)

s

MGF argument

vartheta

Parameter vartheta (>= mu*zeta, default = zeta)

Value

MGF value

Examples

mgf_fcp(0.8, 1.0, 0.5, 1.0, 1.0, 0.5)
mgf_fcp(0.8, 1.0, 0.5, 1.0, 1.0, 0.5)

Moment Generating Function of GIPP

Description

Computes the moment generating function of the GIPP

Usage

mgf_gipp(beta, lambda, lambda_beta, t, u)
mgf_gipp(beta, lambda, lambda_beta, t, u)

Arguments

beta

Fractional parameter (0 < beta < 1)

lambda

Parameter of outer Poisson process (> 0)

lambda_beta

Parameter of inner fractional Poisson process (> 0)

t

Time (> 0)

u

MGF argument

Value

MGF value

Examples

mgf_gipp(0.7, 1.0, 0.5, 1.0, 0.5)
mgf_gipp(0.7, 1.0, 0.5, 1.0, 0.5)

Mittag-Leffler Function

Description

Computes the Mittag-Leffler function $E_{\alpha,\beta}(z) = \sum_{k=0}^\infty z^k / \Gamma(\alpha k + \beta)$ Supports complex arguments and uses log-space computation for numerical stability.

Usage

mittag_leffler(alpha, beta = 1, z, n_terms = 100, tolerance = 1e-10)
mittag_leffler(alpha, beta = 1, z, n_terms = 100, tolerance = 1e-10)

Arguments

alpha

Parameter alpha > 0

beta

Parameter beta > 0 (default = 1)

z

Complex argument (can be a vector)

n_terms

Number of terms in series expansion (default = 100)

tolerance

Convergence tolerance (default = 1e-10)

Value

Value of the Mittag-Leffler function

Examples

mittag_leffler(0.8, 1, -1)
mittag_leffler(1, 1, 1)  # Should equal exp(1)
mittag_leffler(0.8, 1, -1)
mittag_leffler(1, 1, 1)  # Should equal exp(1)

Prabhakar (Three-Parameter) Mittag-Leffler Function

Description

Computes the generalized three-parameter Mittag-Leffler function $E_{\alpha,\beta}^{\gamma}(z) = \sum_{k=0}^\infty (\gamma)_k z^k / (k! \Gamma(\alpha k + \beta))$ Supports complex arguments and uses log-space computation for numerical stability.

Usage

mittag_leffler_prabhakar(
  alpha,
  beta,
  gamma,
  z,
  n_terms = 100,
  tolerance = 1e-10
)
mittag_leffler_prabhakar(
  alpha,
  beta,
  gamma,
  z,
  n_terms = 100,
  tolerance = 1e-10
)

Arguments

alpha

Parameter alpha > 0

beta

Parameter beta > 0

gamma

Parameter gamma > 0

z

Complex argument (can be a vector)

n_terms

Number of terms in series expansion (default = 100)

tolerance

Convergence tolerance (default = 1e-10)

Value

Value of the Prabhakar Mittag-Leffler function

Examples

mittag_leffler_prabhakar(0.8, 1, 1, -1)
mittag_leffler_prabhakar(0.8, 1, 1, -1)

Net Profit Condition for GFRP

Description

Checks if the net profit condition is satisfied

Usage

net_profit_condition_gfrp(eta, rho, beta, lambda)
net_profit_condition_gfrp(eta, rho, beta, lambda)

Arguments

eta

Mean claim size (> 0)

rho

Safety loading parameter (>= 0)

beta

Fractional parameter (0 < beta < 1)

lambda

Vector of rates lambda_1, ..., lambda_k for the GFCP

Value

Logical indicating if net profit condition is satisfied

Examples

net_profit_condition_gfrp(10, 0.1, 0.7, c(1.0, 0.5))
net_profit_condition_gfrp(10, 0.1, 0.7, c(1.0, 0.5))

Partial Ordinary Bell Polynomials

Description

Computes the partial ordinary Bell polynomials $\hat{B}_{j,i}(a)$ Used in the asymptotic expansion of tempered Mittag-Leffler moments.

Usage

partial_ordinary_bell(j, i, a)
partial_ordinary_bell(j, i, a)

Arguments

j

Index (non-negative integer)

i

Index (non-negative integer)

a

Coefficients vector

Value

Value of the partial ordinary Bell polynomial

Examples

partial_ordinary_bell(3, 2, c(0.5, 0.3, 0.2))
partial_ordinary_bell(3, 2, c(0.5, 0.3, 0.2))

CDF of Gamma Subordinator

Description

Computes the cumulative distribution function of the gamma subordinator

Usage

pgamma_subordinator(x, t, shape, rate)
pgamma_subordinator(x, t, shape, rate)

Arguments

x

Value

t

Time

shape

Shape parameter (beta1)

rate

Rate parameter (lambda1)

Value

CDF value

Examples

pgamma_subordinator(1, 1, 2, 1)
pgamma_subordinator(1, 1, 2, 1)

Probability Generating Function of CGFCP

Description

Computes the probability generating function of the CGFCP

Usage

pgf_cgfcp(
  beta,
  lambda,
  t,
  u,
  jump_dist = "poisson",
  jump_params = list(),
  n_terms = 50
)
pgf_cgfcp(
  beta,
  lambda,
  t,
  u,
  jump_dist = "poisson",
  jump_params = list(),
  n_terms = 50
)

Arguments

beta

Fractional parameter (0 < beta < 1)

lambda

Vector of rates lambda_1, ..., lambda_k for the GFCP

t

Time (> 0)

u

PGF argument (|u| <= 1)

jump_dist

Distribution of jump sizes

jump_params

Parameters for jump distribution

n_terms

Number of terms in series expansion (default = 50)

Value

PGF value

Examples

pgf_cgfcp(0.7, c(1.0, 0.5), 1.0, 0.5, "poisson", list(lambda = 2))
pgf_cgfcp(0.7, c(1.0, 0.5), 1.0, 0.5, "poisson", list(lambda = 2))

Probability Generating Function of FCP

Description

Computes the probability generating function of the Fractional Counting Process

Usage

pgf_fcp(mu, zeta, theta, lambda, t, s, vartheta = NULL)
pgf_fcp(mu, zeta, theta, lambda, t, s, vartheta = NULL)

Arguments

mu

Parameter mu (0 < mu <= 1)

zeta

Parameter zeta (> 0)

theta

Parameter theta (0 < theta <= 1)

lambda

Parameter lambda (> 0)

t

Time (> 0)

s

PGF argument (|s| <= 1)

vartheta

Parameter vartheta (>= mu*zeta, default = zeta)

Value

PGF value

Examples

pgf_fcp(0.8, 1.0, 0.5, 1.0, 1.0, 0.5)
pgf_fcp(0.8, 1.0, 0.5, 1.0, 1.0, 0.5)

Probability Generating Function of GIPP

Description

Computes the probability generating function of the GIPP

Usage

pgf_gipp(beta, lambda, lambda_beta, t, u)
pgf_gipp(beta, lambda, lambda_beta, t, u)

Arguments

beta

Fractional parameter (0 < beta < 1)

lambda

Parameter of outer Poisson process (> 0)

lambda_beta

Parameter of inner fractional Poisson process (> 0)

t

Time (> 0)

u

PGF argument (|u| <= 1)

Value

PGF value

Examples

pgf_gipp(0.7, 1.0, 0.5, 1.0, 0.5)
pgf_gipp(0.7, 1.0, 0.5, 1.0, 0.5)

Probability Generating Function of TSTFNBP

Description

Computes the probability generating function of the TSTFNBP

Usage

pgf_tstfnbp(beta, alpha, mu, lambda1, beta1, lambda, t, u, n_terms = 50)
pgf_tstfnbp(beta, alpha, mu, lambda1, beta1, lambda, t, u, n_terms = 50)

Arguments

beta

Fractional parameter (0 < beta < 1)

alpha

Stability parameter (0 < alpha < 1)

mu

Tempering parameter (> 0)

lambda1

Gamma subordinator rate (> mu*alpha)

beta1

Gamma subordinator shape (> 0)

lambda

Poisson parameter (> 0)

t

Time (> 0)

u

PGF argument (|u| <= 1)

n_terms

Number of terms in series expansion (default = 50)

Value

PGF value

Examples

pgf_tstfnbp(0.7, 0.8, 0.5, 2, 1, 1.0, 1.0, 0.5)
pgf_tstfnbp(0.7, 0.8, 0.5, 2, 1, 1.0, 1.0, 0.5)

Pochhammer Symbol

Description

Computes the Pochhammer symbol $(a)_n = \Gamma(a+n) / \Gamma(a)$ Uses log-gamma for numerical stability and preventing overflow.

Usage

pochhammer(a, n)
pochhammer(a, n)

Arguments

a

Real number

n

Non-negative integer

Value

Value of the Pochhammer symbol

Examples

pochhammer(1, 5)  # Should equal 120 = 5!
pochhammer(1, 5)  # Should equal 120 = 5!

S3 Print Method for FCP

Description

S3 Print Method for FCP

Usage

## S3 method for class 'fcp'
print(x, ...)
## S3 method for class 'fcp'
print(x, ...)

Arguments

x

FCP object

...

Additional arguments

Value

No return value, called for side effects.

S3 Print Method for GIPP

Description

S3 Print Method for GIPP

Usage

## S3 method for class 'gipp'
print(x, ...)
## S3 method for class 'gipp'
print(x, ...)

Arguments

x

GIPP object

...

Additional arguments

Value

No return value, called for side effects.

S3 Print Method for TCFCP

Description

S3 Print Method for TCFCP

Usage

## S3 method for class 'tcfcp'
print(x, ...)
## S3 method for class 'tcfcp'
print(x, ...)

Arguments

x

TCFCP object

...

Additional arguments

Value

No return value, called for side effects.

Ruin Probability for GFRP

Description

Estimates the ruin probability using coherent path simulation

Usage

ruin_probability_gfrp(
  nu,
  eta,
  rho,
  beta,
  lambda,
  t_max,
  claim_dist = "gamma",
  claim_params = list(),
  n_grid = 100,
  n_sim = 1000
)
ruin_probability_gfrp(
  nu,
  eta,
  rho,
  beta,
  lambda,
  t_max,
  claim_dist = "gamma",
  claim_params = list(),
  n_grid = 100,
  n_sim = 1000
)

Arguments

nu

Initial capital (> 0)

eta

Mean claim size (> 0)

rho

Safety loading parameter (>= 0)

beta

Fractional parameter (0 < beta < 1)

lambda

Vector of rates lambda_1, ..., lambda_k for the GFCP

t_max

Maximum time horizon

claim_dist

Distribution of claim sizes

claim_params

Parameters for claim distribution

n_grid

Number of grid points for path discretization (default = 100)

n_sim

Number of simulations (default = 1000)

Value

Estimated ruin probability

Examples

ruin_probability_gfrp(100, 10, 0.1, 0.7, c(1.0, 0.5), 10, 
                        "gamma", list(shape = 2, rate = 0.2), n_sim = 100)
ruin_probability_gfrp(100, 10, 0.1, 0.7, c(1.0, 0.5), 10, 
                        "gamma", list(shape = 2, rate = 0.2), n_sim = 100)

Simulate Sample Paths of CGFCP

Description

Simulates sample paths of the Compound Generalized Fractional Counting Process.

Usage

simulate_cgfcp(
  beta,
  lambda,
  t_max,
  jump_dist = "poisson",
  jump_params = list(),
  n_paths = 1,
  n_grid = 100
)
simulate_cgfcp(
  beta,
  lambda,
  t_max,
  jump_dist = "poisson",
  jump_params = list(),
  n_paths = 1,
  n_grid = 100
)

Arguments

beta

Fractional parameter (0 < beta < 1)

lambda

Vector of rates lambda_1, ..., lambda_k

t_max

Maximum simulation time

jump_dist

Distribution of jump sizes ("poisson", "geometric")

jump_params

Parameters for jump distribution

n_paths

Number of paths to simulate (default = 1)

n_grid

Number of grid points for path discretization (default = 100)

Value

A list of paths, each path is a list containing times, states, and the subordinator path

Examples

simulate_cgfcp(0.7, c(1.0, 0.5), 10, "poisson", list(lambda = 2), n_paths = 2)
simulate_cgfcp(0.7, c(1.0, 0.5), 10, "poisson", list(lambda = 2), n_paths = 2)

Simulate Sample Paths of FCP

Description

Simulates sample paths of the Fractional Counting Process. Uses exact Chambers-Mallows-Stuck simulation for FPP (zeta = vartheta = 1) and numerical CDF inversion for general FCP.

Usage

simulate_fcp(mu, zeta, theta, lambda, t_max, n_paths = 1, vartheta = NULL)
simulate_fcp(mu, zeta, theta, lambda, t_max, n_paths = 1, vartheta = NULL)

Arguments

mu

Parameter mu (0 < mu <= 1)

zeta

Parameter zeta (> 0)

theta

Parameter theta (0 < theta <= 1)

lambda

Parameter lambda (> 0)

t_max

Maximum simulation time

n_paths

Number of paths to simulate (default = 1)

vartheta

Parameter vartheta (>= mu*zeta, default = zeta)

Value

A list of paths, each path is a list containing times and states

Examples

simulate_fcp(0.8, 1.0, 0.5, 1.0, 10, n_paths = 2)
simulate_fcp(0.8, 1.0, 0.5, 1.0, 10, n_paths = 2)

Simulate Sample Paths of GFRP

Description

Simulates sample paths of the Generalized Fractional Risk Process.

Usage

simulate_gfrp(
  nu,
  eta,
  rho,
  beta,
  lambda,
  t_max,
  claim_dist = "gamma",
  claim_params = list(),
  n_paths = 1,
  n_grid = 100
)
simulate_gfrp(
  nu,
  eta,
  rho,
  beta,
  lambda,
  t_max,
  claim_dist = "gamma",
  claim_params = list(),
  n_paths = 1,
  n_grid = 100
)

Arguments

nu

Initial capital (> 0)

eta

Mean claim size (> 0)

rho

Safety loading parameter (>= 0)

beta

Fractional parameter (0 < beta < 1)

lambda

Vector of rates lambda_1, ..., lambda_k for the GFCP

t_max

Maximum simulation time

claim_dist

Distribution of claim sizes ("gamma", "exponential", "lognormal")

claim_params

Parameters for claim distribution

n_paths

Number of paths to simulate (default = 1)

n_grid

Number of grid points for path discretization (default = 100)

Value

A list of paths, each path is a list containing times, states (capital), subordinator (W_beta), and claims (accumulated claims)

Examples

simulate_gfrp(100, 10, 0.1, 0.7, c(1.0, 0.5), 10, "gamma", list(shape = 2, rate = 0.2), n_paths = 2)
simulate_gfrp(100, 10, 0.1, 0.7, c(1.0, 0.5), 10, "gamma", list(shape = 2, rate = 0.2), n_paths = 2)

Simulate Sample Paths of GIPP

Description

Simulates sample paths of the Generalized Iterated Poisson Process. Q(t) = N(N_beta(t), lambda).

Usage

simulate_gipp(beta, lambda, lambda_beta, t_max, n_paths = 1, n_grid = 100)
simulate_gipp(beta, lambda, lambda_beta, t_max, n_paths = 1, n_grid = 100)

Arguments

beta

Fractional parameter (0 < beta < 1)

lambda

Parameter of outer Poisson process (> 0)

lambda_beta

Parameter of inner fractional Poisson process (> 0)

t_max

Maximum simulation time

n_paths

Number of paths to simulate (default = 1)

n_grid

Number of grid points for path discretization (default = 100)

Value

A list of paths, each path is a list containing times, states, and inner_process (TFPP path)

Examples

simulate_gipp(0.7, 1.0, 0.5, 10, n_paths = 2)
simulate_gipp(0.7, 1.0, 0.5, 10, n_paths = 2)

Simulate Sample Paths of TCFCP

Description

Simulates sample paths of the Time-Changed Fractional Counting Process. First simulates the Lévy subordinator path on a time grid, then time-changes an independent FPP/FCP path.

Usage

simulate_tcfcp(
  mu,
  zeta,
  theta,
  lambda,
  t_max,
  subordinator_type = "gamma",
  subordinator_params = list(),
  n_paths = 1,
  vartheta = NULL,
  n_grid = 100
)
simulate_tcfcp(
  mu,
  zeta,
  theta,
  lambda,
  t_max,
  subordinator_type = "gamma",
  subordinator_params = list(),
  n_paths = 1,
  vartheta = NULL,
  n_grid = 100
)

Arguments

mu

Parameter mu (0 < mu <= 1)

zeta

Parameter zeta (> 0)

theta

Parameter theta (0 < theta <= 1)

lambda

Parameter lambda (> 0)

t_max

Maximum simulation time

subordinator_type

Type of subordinator ("gamma", "tempered_mittag_leffler")

subordinator_params

List of parameters for the subordinator

n_paths

Number of paths to simulate (default = 1)

vartheta

Parameter vartheta (>= mu*zeta, default = zeta)

n_grid

Number of grid points for path discretization (default = 100)

Value

A list of paths, each path is a list containing times, states, and the subordinator path

Examples

simulate_tcfcp(0.8, 1.0, 0.5, 1.0, 10, "gamma", list(shape = 2, rate = 1), n_paths = 2)
simulate_tcfcp(0.8, 1.0, 0.5, 1.0, 10, "gamma", list(shape = 2, rate = 1), n_paths = 2)

Simulate Sample Paths of TSTFNBP

Description

Simulates sample paths of the Tempered Space-Time Fractional Negative Binomial Process. First simulates the TMLLP subordinator path, then time-changes an independent FPP path.

Usage

simulate_tstfnbp(
  beta,
  alpha,
  mu,
  lambda1,
  beta1,
  lambda,
  t_max,
  n_paths = 1,
  n_grid = 100
)
simulate_tstfnbp(
  beta,
  alpha,
  mu,
  lambda1,
  beta1,
  lambda,
  t_max,
  n_paths = 1,
  n_grid = 100
)

Arguments

beta

Fractional parameter (0 < beta < 1)

alpha

Stability parameter (0 < alpha < 1)

mu

Tempering parameter (> 0)

lambda1

Gamma subordinator rate (> mu*alpha)

beta1

Gamma subordinator shape (> 0)

lambda

Poisson parameter (> 0)

t_max

Maximum simulation time

n_paths

Number of paths to simulate (default = 1)

n_grid

Number of grid points for path discretization (default = 100)

Value

A list of paths, each path is a list containing times, states, and the subordinator path

Examples

simulate_tstfnbp(0.7, 0.8, 0.5, 2, 1, 1.0, 10, n_paths = 2)
simulate_tstfnbp(0.7, 0.8, 0.5, 2, 1, 1.0, 10, n_paths = 2)

Classical Stirling Numbers of the Second Kind

Description

Computes the classical Stirling numbers of the second kind S(m, k)

Usage

stirling2(m, k)
stirling2(m, k)

Arguments

m

Upper index (non-negative integer)

k

Lower index (non-negative integer)

Value

Value of the Stirling number

Examples

stirling2(5, 3)
stirling2(5, 3)

Tempered Mittag-Leffler Lévy Process

Description

Simulates the tempered Mittag-Leffler Lévy process $M_{\alpha,\mu,\lambda_1,\beta_1}(t) = S_{\alpha,\mu}(G_{\lambda_1,\beta_1}(t))$

Usage

tempered_mittag_leffler_levy(t, alpha, mu, lambda1, beta1, n = 1)
tempered_mittag_leffler_levy(t, alpha, mu, lambda1, beta1, n = 1)

Arguments

t

Time

alpha

Stability parameter (0 < alpha < 1)

mu

Tempering parameter (> 0)

lambda1

Gamma subordinator rate (> 0)

beta1

Gamma subordinator shape (> 0)

n

Number of simulations

Value

Simulated values of the tempered Mittag-Leffler Lévy process

Examples

tempered_mittag_leffler_levy(1, 0.8, 0.5, 2, 1, 10)
tempered_mittag_leffler_levy(1, 0.8, 0.5, 2, 1, 10)

Fractional Moment of Tempered Mittag-Leffler Lévy Process

Description

Computes the q-th order moment of the tempered Mittag-Leffler Lévy process based on asymptotic results from Kumar et al. (2019b)

Usage

tempered_mittag_leffler_moment(q, alpha, mu, lambda1, beta1, t)
tempered_mittag_leffler_moment(q, alpha, mu, lambda1, beta1, t)

Arguments

q

Order of moment (> 0)

alpha

Stability parameter (0 < alpha < 1)

mu

Tempering parameter (> 0)

lambda1

Gamma subordinator rate (> 0)

beta1

Gamma subordinator shape (> 0)

t

Time (> 0)

Value

Asymptotic value of the q-th moment

Examples

tempered_mittag_leffler_moment(2, 0.8, 0.5, 2, 1, 1)
tempered_mittag_leffler_moment(2, 0.8, 0.5, 2, 1, 1)

Tempered Space-Time Fractional Negative Binomial Process (TSTFNBP)

Description

Computes the probability mass function of the TSTFNBP $Q_{\alpha,\mu,\lambda_1,\beta_1}^\beta(t, \lambda) = N_\beta(M_{\alpha,\mu,\lambda_1,\beta_1}(t), \lambda)$ From Garg et al. (2025)

Usage

tempered_st_fnbp(beta, alpha, mu, lambda1, beta1, lambda, t, n, n_terms = 50)
tempered_st_fnbp(beta, alpha, mu, lambda1, beta1, lambda, t, n, n_terms = 50)

Arguments

beta

Fractional parameter (0 < beta < 1)

alpha

Stability parameter (0 < alpha < 1)

mu

Tempering parameter (> 0)

lambda1

Gamma subordinator rate (> mu*alpha)

beta1

Gamma subordinator shape (> 0)

lambda

Poisson parameter (> 0)

t

Time (> 0)

n

Vector of non-negative integers

n_terms

Number of terms in series expansion (default = 50)

Value

Vector of probabilities P(Q(t) = n)

Examples

tempered_st_fnbp(0.7, 0.8, 0.5, 2, 1, 1.0, 1.0, 0:10)
tempered_st_fnbp(0.7, 0.8, 0.5, 2, 1, 1.0, 1.0, 0:10)

Tempered Stable Subordinator

Description

Simulates a tempered alpha-stable subordinator. Uses exact rejection-sampling via tilting of stable random variables.

Usage

tempered_stable_subordinator(t, alpha, mu, n = 1)
tempered_stable_subordinator(t, alpha, mu, n = 1)

Arguments

t

Time

alpha

Stability parameter (0 < alpha < 1)

mu

Tempering parameter (> 0)

n

Number of simulations

Value

Simulated values of the tempered stable subordinator

Examples

tempered_stable_subordinator(1, 0.8, 0.5, 10)
tempered_stable_subordinator(1, 0.8, 0.5, 10)

Time-Changed Fractional Counting Process (TCFCP)

Description

Computes the probability mass function of the Time-Changed Fractional Counting Process obtained by subordinating the FCP with a Lévy subordinator. From Garg et al. (2025)

Usage

time_changed_fcp(
  mu,
  zeta,
  theta,
  lambda,
  t,
  n,
  subordinator_type = "gamma",
  subordinator_params = list(),
  vartheta = NULL,
  n_terms = 50
)
time_changed_fcp(
  mu,
  zeta,
  theta,
  lambda,
  t,
  n,
  subordinator_type = "gamma",
  subordinator_params = list(),
  vartheta = NULL,
  n_terms = 50
)

Arguments

mu

Parameter mu (0 < mu <= 1)

zeta

Parameter zeta (> 0)

theta

Parameter theta (0 < theta <= 1)

lambda

Parameter lambda (> 0)

t

Time (> 0)

n

Vector of non-negative integers

subordinator_type

Type of subordinator ("gamma", "tempered_mittag_leffler")

subordinator_params

List of parameters for the subordinator

vartheta

Parameter vartheta (>= mu*zeta, default = zeta)

n_terms

Number of terms in series expansion (default = 50)

Value

Vector of probabilities P(Z(t) = n)

Examples

time_changed_fcp(0.8, 1.0, 0.5, 1.0, 1.0, 0:10, 
                 "gamma", list(shape = 2, rate = 1))
time_changed_fcp(0.8, 1.0, 0.5, 1.0, 1.0, 0:10, 
                 "gamma", list(shape = 2, rate = 1))

Variance of CGFCP

Description

Computes the variance of the CGFCP

Usage

var_cgfcp(beta, lambda, t, jump_dist = "poisson", jump_params = list())
var_cgfcp(beta, lambda, t, jump_dist = "poisson", jump_params = list())

Arguments

beta

Fractional parameter (0 < beta < 1)

lambda

Vector of rates lambda_1, ..., lambda_k for the GFCP

t

Time (> 0)

jump_dist

Distribution of jump sizes ("poisson", "geometric")

jump_params

Parameters for jump distribution

Value

Variance value

Examples

var_cgfcp(0.7, c(1.0, 0.5), 1.0, "poisson", list(lambda = 2))
var_cgfcp(0.7, c(1.0, 0.5), 1.0, "poisson", list(lambda = 2))

Variance of Fractional Counting Process

Description

Computes the variance of the FCP

Usage

var_fcp(mu, zeta, theta, lambda, t, vartheta = NULL)
var_fcp(mu, zeta, theta, lambda, t, vartheta = NULL)

Arguments

mu

Parameter mu (0 < mu <= 1)

zeta

Parameter zeta (> 0)

theta

Parameter theta (0 < theta <= 1)

lambda

Parameter lambda (> 0)

t

Time (> 0)

vartheta

Parameter vartheta (>= mu*zeta, default = zeta)

Value

Variance value

Examples

var_fcp(0.8, 1.0, 0.5, 1.0, 1.0)
var_fcp(0.8, 1.0, 0.5, 1.0, 1.0)

Variance of GFRP

Description

Computes the variance of the GFRP

Usage

var_gfrp(
  nu,
  eta,
  rho,
  beta,
  lambda,
  t,
  claim_dist = "gamma",
  claim_params = list()
)
var_gfrp(
  nu,
  eta,
  rho,
  beta,
  lambda,
  t,
  claim_dist = "gamma",
  claim_params = list()
)

Arguments

nu

Initial capital (> 0)

eta

Mean claim size (> 0)

rho

Safety loading parameter (>= 0)

beta

Fractional parameter (0 < beta < 1)

lambda

Vector of rates lambda_1, ..., lambda_k for the GFCP

t

Time (> 0)

claim_dist

Distribution of claim sizes

claim_params

Parameters for claim distribution

Value

Variance value

Examples

var_gfrp(100, 10, 0.1, 0.7, c(1.0, 0.5), 1.0, "gamma", list(shape = 2, rate = 0.2))
var_gfrp(100, 10, 0.1, 0.7, c(1.0, 0.5), 1.0, "gamma", list(shape = 2, rate = 0.2))

Variance of GIPP

Description

Computes the variance of the GIPP

Usage

var_gipp(beta, lambda, lambda_beta, t)
var_gipp(beta, lambda, lambda_beta, t)

Arguments

beta

Fractional parameter (0 < beta < 1)

lambda

Parameter of outer Poisson process (> 0)

lambda_beta

Parameter of inner fractional Poisson process (> 0)

t

Time (> 0)

Value

Variance value

Examples

var_gipp(0.7, 1.0, 0.5, 1.0)
var_gipp(0.7, 1.0, 0.5, 1.0)

Variance of Time-Changed FCP

Description

Computes the variance of the TCFCP using the law of total variance.

Usage

var_tcfcp(
  mu,
  zeta,
  theta,
  lambda,
  t,
  subordinator_type = "gamma",
  subordinator_params = list(),
  vartheta = NULL
)
var_tcfcp(
  mu,
  zeta,
  theta,
  lambda,
  t,
  subordinator_type = "gamma",
  subordinator_params = list(),
  vartheta = NULL
)

Arguments

mu

Parameter mu (0 < mu <= 1)

zeta

Parameter zeta (> 0)

theta

Parameter theta (0 < theta <= 1)

lambda

Parameter lambda (> 0)

t

Time (> 0)

subordinator_type

Type of subordinator

subordinator_params

List of parameters for the subordinator

vartheta

Parameter vartheta (>= mu*zeta, default = zeta)

Value

Variance value

Examples

var_tcfcp(0.8, 1.0, 0.5, 1.0, 1.0, "gamma", list(shape = 2, rate = 1))
var_tcfcp(0.8, 1.0, 0.5, 1.0, 1.0, "gamma", list(shape = 2, rate = 1))

Variance of TSTFNBP

Description

Computes the variance of the TSTFNBP

Usage

var_tstfnbp(beta, alpha, mu, lambda1, beta1, lambda, t)
var_tstfnbp(beta, alpha, mu, lambda1, beta1, lambda, t)

Arguments

beta

Fractional parameter (0 < beta < 1)

alpha

Stability parameter (0 < alpha < 1)

mu

Tempering parameter (> 0)

lambda1

Gamma subordinator rate (> mu*alpha)

beta1

Gamma subordinator shape (> 0)

lambda

Poisson parameter (> 0)

t

Time (> 0)

Value

Variance value

Examples

var_tstfnbp(0.7, 0.8, 0.5, 2, 1, 1.0, 1.0)
var_tstfnbp(0.7, 0.8, 0.5, 2, 1, 1.0, 1.0)

Generalized Wright Function

Description

Computes the generalized Wright function $\psi_{p,q}[(a_i, \alpha_i)_{1,p}; (b_j, \beta_j)_{1,q}](z)$ Uses log-space computation for numerical stability.

Usage

wright_function(a, alpha, b, beta, z, n_terms = 100, tolerance = 1e-10)
wright_function(a, alpha, b, beta, z, n_terms = 100, tolerance = 1e-10)

Arguments

a

Vector of a_i parameters

alpha

Vector of alpha_i parameters

b

Vector of b_j parameters

beta

Vector of beta_j parameters

z

Complex argument (can be a vector)

n_terms

Number of terms in series expansion (default = 100)

tolerance

Convergence tolerance (default = 1e-10)

Value

Value of the generalized Wright function

Examples

wright_function(c(1), c(1), c(1), c(1), 1)
wright_function(c(1), c(1), c(1), c(1), 1)

Package 'StochFracPoisson'

Help Index

Alternative Generalized Fractional Risk Process (AGFRP)

Description

Usage

Arguments

Value

Examples

Classical Bell Polynomials

Description

Usage

Arguments

Value

Examples

Check Long-Range Dependence of GIPP

Description

Usage

Arguments

Value

Examples

Check Martingale Property of GFRP

Description

Usage

Arguments

Value

Examples

Compound Generalized Fractional Counting Process (CGFCP)

Description

Usage

Arguments

Value

Examples

Covariance of GIPP

Description

Usage

Arguments

Value

Examples

Covariance of Inverse Stable Subordinator

Description

Usage

Arguments

Value

Examples

Covariance of TSTFNBP

Description

Usage

Arguments

Value

Examples

Create FCP Object

Description

Usage

Arguments

Value

Examples

Create GIPP Object

Description

Usage

Arguments

Value

Examples

Create TCFCP Object

Description

Usage

Arguments

Value

Examples

PDF of Gamma Subordinator

Description

Usage

Arguments

Value

Examples

PDF of Tempered Mittag-Leffler Lévy Process

Description

Usage

Arguments

Value

Examples