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Showing papers on "Subordinator published in 2017"


Journal ArticleDOI
TL;DR: In this article, the existence and uniqueness of solutions for general fractional-time parabolic equations of mixture type, and their probabilistic representations in terms of the corresponding inverse subordinators with or without drifts, were studied.
Abstract: In this paper, we study the existence and uniqueness of solutions for general fractional-time parabolic equations of mixture type, and their probabilistic representations in terms of the corresponding inverse subordinators with or without drifts. An explicit relation between occupation measure for Markov processes time-changed by inverse subordinator in open sets and that of the original Markov process in the open set is also given.

100 citations


Journal ArticleDOI
TL;DR: This paper shows that the probability density function of an inverse tempered stable subordinator solves a tempered time-fractional diffusion equation, and its “folded” density solves a temperamental telegraph equation.
Abstract: The inverse tempered stable subordinator is a stochastic process that models power law waiting times between particle movements, with an exponential tempering that allows all moments to exist. This paper shows that the probability density function of an inverse tempered stable subordinator solves a tempered time-fractional diffusion equation, and its “folded” density solves a tempered time-fractional telegraph equation. Two explicit formulae for the density function are developed, and applied to compute explicit solutions to tempered fractional Cauchy problems, where a tempered fractional derivative replaces the first derivative in time. Several examples are given, including tempered fractional diffusion equations on bounded or unbounded domains, and the probability distribution of a tempered fractional Poisson process. It is shown that solutions to the tempered fractional diffusion equation have a cusp at the origin.

57 citations


Journal ArticleDOI
TL;DR: In this article, a partial integro-differential equation that describes the dynamics of the arbitrage-free price of the variance swap is formulated, under appropriate assumptions for the first four cumulants of the driving subordinator, a Veceř-type theorem is proved.
Abstract: In this paper a couple of variance dependent instruments in the financial market are studied. Firstly, a number of aspects of the variance swap in connection to the Barndorff-Nielsen and Shephard model are studied. A partial integro-differential equation that describes the dynamics of the arbitrage-free price of the variance swap is formulated. Under appropriate assumptions for the first four cumulants of the driving subordinator, a Veceř-type theorem is proved. The bounds of the arbitrage-free variance swap price are also found. Finally, a price-weighted index modulated by market variance is introduced. The large-basket limit dynamics of the price index and the “error term” are derived. Empirical data driven numerical examples are provided in support of the proposed price index.

46 citations


Journal ArticleDOI
TL;DR: In this article, the authors studied the potential theory of non-negative harmonic functions under a weak scaling condition on the derivative of the Laplace exponent of the Dirichlet Laplacian and showed that these functions satisfy the scale invariant Harnack inequality.
Abstract: Let $W^D$ be a killed Brownian motion in a domain $D\subset {\mathbb R}^d$ and $S$ an independent subordinator with Laplace exponent $\phi$. The process $Y^D$ defined by $Y^D_t=W^D_{S_t}$ is called a subordinate killed Brownian motion. It is a Hunt process with infinitesimal generator $-\phi(-\Delta|_D)$, where $\Delta|_D$ is the Dirichlet Laplacian. In this paper we study the potential theory of $Y^D$ under a weak scaling condition on the derivative of $\phi$. We first show that non-negative harmonic functions of $Y^D$ satisfy the scale invariant Harnack inequality. Subsequently we prove two types of scale invariant boundary Harnack principles with explicit decay rates for non-negative harmonic functions of $Y^D$. The first boundary Harnack principle deals with a $C^{1,1}$ domain $D$ and non-negative functions which are harmonic near the boundary of $D$, while the second one is for a more general domain $D$ and non-negative functions which are harmonic near the boundary of an interior open subset of $D$. The obtained decay rates are not the same, reflecting different boundary and interior behaviors of $Y^D$.

36 citations


Journal ArticleDOI
TL;DR: In this paper, an estimator of the Gerber-shiu function via the empirical Fourier transform of the GSH was proposed, and the L 2 -consistency of the estimator under the assumption of high-frequency observation of the surplus process in a long term was evaluated.
Abstract: Consider an insurance surplus process driven by a Levy subordinator, which is observed at discrete time points. An estimator of the Gerber–Shiu function is proposed via the empirical Fourier transform of the Gerber–Shiu function. By evaluating its mean squared error, we show the L 2 -consistency of the estimator under the assumption of high-frequency observation of the surplus process in a long term. Simulation studies are also presented to show the finite sample performance of the proposed estimator.

31 citations


Journal ArticleDOI
TL;DR: In this article, a random walk S τ which is obtained from a simple random walk by a discrete time version of Bochner's subordination was considered and it was shown that S τ converges in the Skorohod space to the symmetric α-stable process B α.

26 citations


Journal ArticleDOI
TL;DR: In this paper, the authors study the Metropolis dynamics of the simplest mean-field spin glass model, the random energy model, and show that this dynamics exhibits aging by showing that the properly rescaled time change process between the metropolis dynamics and a suitably chosen "fast" Markov chain converges in distribution to a stable subordinator.
Abstract: We study the Metropolis dynamics of the simplest mean-field spin glass model, the random energy model. We show that this dynamics exhibits aging by showing that the properly rescaled time change process between the Metropolis dynamics and a suitably chosen ‘fast’ Markov chain converges in distribution to a stable subordinator. The rescaling might depend on the realization of the environment, but we show that its exponential growth rate is deterministic.

20 citations


Journal ArticleDOI
TL;DR: In this article, the authors unify and extend a number of approaches related to constructing multivariate Madan-Seneta Variance-Gamma models for option pricing, and derive an overarching model by subordinating multivariate Brownian motion to a subordinator from Thorin's (1977) [58, 59] class of generalised Gamma convolutions.

20 citations


Journal ArticleDOI
TL;DR: In this article, it was shown that a scaled subordination of a random walk converges to a multiple of a rotationally stable process if and only if the Laplace exponent of the corresponding subordinator varies regularly at zero.
Abstract: In this article subordination of random walks in $R^d$ is considered. We prove that subordination of random walks in the sense of [BSC12] yields the same process as subordination of Levy processes (in the sense of Bochner). Furthermore, we prove that appropriately scaled subordinate random walk converges to a multiple of a rotationally $2\alpha$-stable process if and only if the Laplace exponent of the corresponding subordinator varies regularly at zero with index $\alpha\in (0,1]$.

15 citations


Journal ArticleDOI
TL;DR: In this article, the authors presented a novel construction of Marshall-Olkin (MO) multivariate exponential distributions of failure times as distributions of the first passage times of the coordinates of mul-dimensional Levy subordinator processes above independent unit-mean exponential random variables.
Abstract: The paper presents a novel construction of Marshall-Olkin (MO) multivariate exponential distributions of failure times as distributions of the first passage times of the coordinates of mul- tidimensional Levy subordinator processes above independent unit-mean exponential random variables. A time-inhomogeneous version is also given that replaces Levy subordinators with additive subordinators. An attractive feature of MO distributions for applications, such as to portfolio credit risk, is its singular component that yields positive probabilities of simultaneous defaults of multiple obligors, capturing the default clustering phenomenon. The drawback of the original MO fatal shock construction of MO distributions is that it requires one to simulate 2 n 1 independent exponential random variables. In practice, the dimensionality is typically on the order of hundreds or thousands of obligors in a large credit portfolio, rendering the MO fatal shock construction infeasible to simulate. The subordinator construction reduces the problem of simulating a rich subclass of MO distributions to simulating an n-dimensional subordinator. When one works with the class of subordinators constructed from independent one-dimensional subordinators with known transition distributions, such as gamma and inverse Gaussian, or their Sato versions in the additive case, the simulation e↵ort is linear in n. To illustrate, the paper presents a simulation of 100,000 samples of a credit portfolio with 1,000 obligors that takes less than 18 seconds on a PC.

13 citations


Journal ArticleDOI
TL;DR: In this paper, a multivariate rapidly decreasing Levy process is proposed for quanto option pricing, which captures three characteristics observed in real-world market for stock prices and currencies: jumps, heavy tails and skewness.
Abstract: We develop a multivariate Levy model for the pricing of quanto options that captures three characteristics observed in real-world market for stock prices and currencies: jumps, heavy tails and skewness. The model is developed by using a bottom-up approach from a subordinator. We do so by replacing the time of a Brownian motion with a non-decreasing Levy process, rapidly decreasing subordinator. We refer to this model as a multivariate rapidly decreasing Levy process. We consider two benchmarks: Black-Scholes and normal tempered stable process, the later constructed using a classical tempered stable subordinator. We then compare using a time series of daily log-returns and market prices of European-style quanto options the relative performance of the rapidly decreasing Levy process to that of Black-Scholes and the normal tempered stable process. We find that the proposed modeling process is superior to the other two processes for pricing quanto options.

Journal ArticleDOI
TL;DR: In this article, the authors show that the fractionally integrated inverse stable subordinator (FIISS) is a scaling limit in the Skorokhod space of a renewal shot noise process with heavy-tailed, infinite mean "inter-shot" distribution and regularly varying response function.

Journal ArticleDOI
TL;DR: In this paper, the authors studied the convergence rate of a Markov process to its invariant measure with a subordinator and the corresponding Bernstein function and showed that subordination can dramatically change the speed of convergence to equilibrium.
Abstract: We are interested in the rate of convergence of a subordinate Markov process to its invariant measure. Given a subordinator and the corresponding Bernstein function (Laplace exponent), we characterize the convergence rate of the subordinate Markov process; the key ingredients are the rate of convergence of the original process and the (inverse of the) Bernstein function. At a technical level, the crucial point is to bound three types of moment (subexponential, algebraic, and logarithmic) for subordinators as time t tends to ∞. We also discuss some concrete models and we show that subordination can dramatically change the speed of convergence to equilibrium.

Journal ArticleDOI
TL;DR: In this paper, the authors studied the properties of solutions to generalized Fokker-planck equations through the lens of the probability density functions of anomalous diffusion processes and presented a combination of four criteria which serve as a theoretical basis for model selection, statistical inference and predictions for physical experiments on anomalously diffusing systems.
Abstract: We study properties of solutions to generalized Fokker–Planck equations through the lens of the probability density functions of anomalous diffusion processes. In particular, we examine solutions in terms of their cusping, travelling wave behaviours, and variance, within the framework of stochastic representations of generalized Fokker–Planck equations. We give our analysis in the cases of anomalous diffusion driven by the inverses of the stable, tempered stable and gamma subordinators, demonstrating the impact of changing the distribution of waiting times in the underlying anomalous diffusion model. We also analyse the cases where the underlying anomalous diffusion contains a Levy jump component in the parent process, and when a diffusion process is time changed by an uninverted Levy subordinator. On the whole, we present a combination of four criteria which serve as a theoretical basis for model selection, statistical inference and predictions for physical experiments on anomalously diffusing systems. We discuss possible applications in physical experiments, including, with reference to specific examples, the potential for model misclassification and how combinations of our four criteria may be used to overcome this issue.

Journal ArticleDOI
TL;DR: In this paper, the running infimum of a Levy process relative to its point of issue is known to have the same range that of the negative of a certain subordinator, and the problem of conditioning a subordinator to remain in a strip is considered.

Journal ArticleDOI
TL;DR: In this paper, a pseudo-Poisson process of the following simple type is considered: a Poissonian subordinator for a sequence of i.i.d. random variables with finite variance.
Abstract: We consider a pseudo-Poisson process of the following simple type. This process is a Poissonian subordinator for a sequence of i.i.d. random variables with finite variance. Further we consider sums of i.i.d. copies of a pseudo-Poisson process. For a family of distributions of these random sums, we prove the tightness (relative compactness) in the Skorokhod space. Under the conditions of the Central Limit Theorem for vectors, we establish the weak convergence in the functional Skorokhod space of the examined sums to the Ornstein–Uhlenbeck process.

Journal ArticleDOI
TL;DR: In this article, an extension of the space-fractional diffusion equation, containing both the fractional shift operator and the Feller integral, is shown to satisfy the transition density of the process obtained by time changing the stable process with an independent linear birth process with drift.

Journal ArticleDOI
TL;DR: In this paper, the stable Levy motion subordinated by the so-called inverse Gaussian process is studied and a step-by-step procedure of parameters estimation is proposed to obtain the governing fractional partial differential equations for the probability density function.
Abstract: In this paper we study the stable Levy motion subordinated by the so-called inverse Gaussian process. This process extends the well known normal inverse Gaussian (NIG) process introduced by Barndorff-Nielsen, which arises by subordinating ordinary Brownian motion (with drift) with inverse Gaussian process. The NIG process found many interesting applications, especially in financial data description. We discuss here the main features of the introduced subordinated process, such as distributional properties, existence of fractional order moments and asymptotic tail behavior. We show the connection of the process with continuous time random walk. Further, the governing fractional partial differential equations for the probability density function is also obtained. Moreover, we discuss the asymptotic distribution of sample mean square displacement, the main tool in detection of anomalous diffusion phenomena (Metzler et al., 2014). In order to apply the stable Levy motion time-changed by inverse Gaussian subordinator we propose a step-by-step procedure of parameters estimation. At the end, we show how the examined process can be useful to model financial time series.

Journal ArticleDOI
12 Oct 2017
TL;DR: In this article, a new class of models for pricing generalized variance swaps is proposed, in which the process for the asset price is a function of a general time-homogeneous diffusion process belonging to a symmetric pricing semigroup, time changed by a composition of a Levy subordinator and an absolutely continuous process.
Abstract: We propose a new class of models for pricing generalized variance swaps. We assume that, in the most general form, the process for the asset price is a function of a general time-homogeneous diffusion process belonging to a symmetric pricing semigroup, time changed by a composition of a Levy subordinator and an absolutely continuous process. We derive the analytical pricing formulas for various types of generalized variance swaps based on eigenfunction expansion method. We also numerically implement the model and test its sensitivity to some of the key parameters of the model.

Posted Content
TL;DR: In this paper, the authors derived characteristic function identities for conditional distributions of an r-trimmed Levy process given its r largest jumps up to a designated time t. Assuming the underlying Levy process is in the domain of attraction of a stable process as t goes to 0, these identities are applied to show joint convergence of the trimmed process divided by its large jumps to corresponding quantities constructed from a stable limiting process.
Abstract: We derive characteristic function identities for conditional distributions of an r-trimmed Levy process given its r largest jumps up to a designated time t. Assuming the underlying Levy process is in the domain of attraction of a stable process as t goes to 0, these identities are applied to show joint convergence of the trimmed process divided by its large jumps to corresponding quantities constructed from a stable limiting process. This generalises related results in the 1-dimensional subordinator case developed in Kevei & Mason (2014) and produces new discrete distributions on the infinite simplex in the limit.

Journal ArticleDOI
Ömer Önalan1
TL;DR: In this article, the authors presented a novel model to analyze the behavior of random asset price process under the assumption that the stock price pro-cess is governed by time-changed generalized mixed fractional Brownian motion with an inverse gamma subordinator.
Abstract: In this paper we present a novel model to analyze the behavior of random asset price process under the assumption that the stock price pro-cess is governed by time-changed generalized mixed fractional Brownian motion with an inverse gamma subordinator. This model is con-structed by introducing random time changes into generalized mixed fractional Brownian motion process. In practice it has been observed that many different time series have long-range dependence property and constant time periods. Fractional Brownian motion provides a very general model for long-term dependent and anomalous diffusion regimes. Motivated by this facts in this paper we investigated the long-range dependence structure and trapping events (periods of prices stay motionless) of CSCO stock price return series. The constant time periods phenomena are modeled using an inverse gamma process as a subordinator. Proposed model include the jump behavior of price process because the gamma process is a pure jump Levy process and hence the subordinated process also has jumps so our model can be capture the random variations in volatility. To show the effectiveness of proposed model, we applied the model to calculate the price of an average arithmetic Asian call option that is written on Cisco stock. In this empirical study first the statistical properties of real financial time series is investigated and then the estimated model parameters from an observed data. The results of empirical study which is performed based on the real data indicated that the results of our model are more accuracy than the results based on traditional models.

Dissertation
01 Jul 2017
TL;DR: In this paper, the biased random walk on the subcritical Galton-Watson trees with leaves is studied and conditions under which the biased randomly trapped random walk is ballistic, satisfies an annealed invariance principle and a quenched central limit theorem with environment dependent centring.
Abstract: In this thesis we study biased randomly trapped random walks. As our main motivation, we apply these results to biased walks on subcritical Galton-Watson trees conditioned to survive. This application was initially considered model in its own right. We prove conditions under which the biased randomly trapped random walk is ballistic, satisfies an annealed invariance principle and a quenched central limit theorem with environment dependent centring. We also study the regime in which the walk is sub-ballistic; in this case we prove convergence to a stable subordinator. Furthermore, we study the fluctuations of the walk in the ballistic but sub-diffusive regime. In this setting we show that the walk can be properly centred and rescaled so that it converges to a stable process. The biased random walk on the subcritical GW-tree conditioned to survive fits suitably into the randomly trapped random walk model; however, due to a lattice effect, we cannot obtain such strong limiting results. We prove conditions under which the walk is ballistic, satisfies an annealed invariance principle and a quenched central limit theorem with environment dependent centring. In these cases the trapping is weak enough that the lattice effect does not have an influence; however, in the sub-ballistic regime it is only possible to obtain converge along specific subsequences. We also study biased random walks on infinite supercritical GW-trees with leaves. In this setting we determine critical upper and lower bounds on the bias such that the walk satisfies a quenched invariance principle.

Journal ArticleDOI
TL;DR: In this article, the authors present an approach for modeling dependencies in exponential Levy market models with arbitrary margins originated from time changed Brownian motions, which is superior to traditional approaches based on pathwise subordination.
Abstract: We present an approach for modeling dependencies in exponential Levy market models with arbitrary margins originated from time changed Brownian motions. Using weak subordination of Buchmann et al. (2016), we face a new layer of dependencies, superior to traditional approaches based on pathwise subordination, since weakly subordinated processes are not required to have independent components considering multivariate stochastic time changes. We apply a subordinator being able to incorporate any joint or idiosyncratic information arrivals. We emphasize multivariate variance gamma and normal inverse Gaussian processes and state explicit formulae for the Levy characteristics. Using maximum likelihood, we estimate a multivariate variance gamma model on various market data and show that the model is highly preferable to traditional approaches. Consistent values of basket-options under given marginal pricing models are achieved using Esscher transform, generating a non-flat implied correlation surface.

Posted Content
TL;DR: In this paper, the jump-diffusion CIR process is studied and sufficient conditions on the Levy measure of the subordinator are provided under which the jump diffusion process is ergodic.
Abstract: We study the jump-diffusion CIR process, which is an extension of the Cox-Ingersoll-Ross model and whose jumps are introduced by a subordinator. We provide sufficient conditions on the Levy measure of the subordinator under which the jump-diffusion CIR process is ergodic and exponentially ergodic, respectively. Furthermore, we characterize the existence of the $\kappa$-moment ($\kappa>0$) of the jump-diffusion CIR process by an integrability condition on the Levy measure of the subordinator.

Journal ArticleDOI
TL;DR: In this paper, the authors derived the stochastic solution of a Cauchy problem for the distribution of a fractional diffusion process, named fractional Bessel-Riesz motion.
Abstract: This paper derives the stochastic solution of a Cauchy problem for the distribution of a fractional diffusion process. The governing equation involves the Bessel-Riesz derivative (in space) to model heavy tails of the distribution, and the Caputo-Djrbashian derivative (in time) to depicts the memory of the diffusion process. The solution is obtained as Brownian motion with time change in terms of the Bessel-Riesz subordinator on the inverse stable subordinator. This stochastic solution, named fractional Bessel-Riesz motion, provides a method to simulate a large class of stochastic motions with memory and heavy tails.

Posted Content
TL;DR: In this paper, the boundary Harnack inequality holds for non-negative functions harmonic in a smooth open set with respect to the Green function of a subordinator with Laplace exponent.
Abstract: Let $Z$ be a subordinate Brownian motion in ${\mathbb R}^d$, $d\ge 2$, via a subordinator with Laplace exponent $\phi$. We kill the process $Z$ upon exiting a bounded open set $D\subset {\mathbb R}^d$ to obtain the killed process $Z^D$, and then we subordinate the process $Z^D$ by a subordinator with Laplace exponent $\psi$. The resulting process is denoted by $Y^D$. Both $\phi$ and $\psi$ are assumed to satisfy certain weak scaling conditions at infinity. We study the potential theory of $Y^D$, in particular the boundary theory. First, in case that $D$ is a $\kappa$-fat bounded open set, we show that the Harnack inequality holds. If, in addition, $D$ satisfies the local exterior volume condition, then we prove the Carleson estimate. In case $D$ is a smooth open set and the lower weak scaling index of $\psi$ is strictly larger than $1/2$, we establish the boundary Harnack principle with explicit decay rate near the boundary of $D$. On the other hand, when $\psi(\lambda)=\lambda^{\gamma}$ with $\gamma\in (0,1/2]$, we show that the boundary Harnack principle near the boundary of $D$ fails for any bounded $C^{1,1}$ open set $D$. Our results give the first example where the Carleson estimate holds true, but the boundary Harnack principle does not. One of the main ingredients in the proofs is the sharp two-sided estimates of the Green function of $Y^D$. Under an additional condition on $\psi$, we establish sharp two-sided estimates of the jumping kernel of $Y^D$ which exhibit some unexpected boundary behavior. We also prove a boundary Harnack principle for non-negative functions harmonic in a smooth open set $E$ strictly contained in $D$, showing that the behavior of $Y^D$ in the interior of $D$ is determined by the composition $\psi\circ \phi$.

01 Jan 2017
TL;DR: In this paper, a negative binomial point process with parameter $r>0$ and L\'evy density was introduced, which is a new class of distributions on the infinite simplex.
Abstract: The Poisson-Kingman distributions, $\mathrm{PK}(\rho)$, on the infinite simplex, can be constructed from a Poisson point process having intensity density $\rho$ or by taking the ranked jumps up till a specified time of a subordinator with L\'evy density $\rho$, as proportions of the subordinator. As a natural extension, we replace the Poisson point process with a negative binomial point process having parameter $r>0$ and L\'evy density $\rho$, thereby defining a new class $\mathrm{PK}^{(r)}(\rho)$ of distributions on the infinite simplex. The new class contains the two-parameter generalisation $\mathrm{PD}(\alpha, \theta)$ of Pitman and Yor (1997) when $\theta>0$. It also contains a class of distributions derived from the trimmed stable subordinator. We derive properties of the new distributions, with particular reference to the two most well-known $\mathrm{PK}$ distributions: the Poisson-Dirichlet distribution $\mathrm{PK}(\rho_\theta)$ generated by a Gamma process with L\'evy density $\rho_\theta(x) = \theta e^{-x}/x$, $x>0$, $\theta > 0$, and the random discrete distribution, $\mathrm{PD}(\alpha,0)$, derived from an $\alpha$-stable subordinator.

Journal ArticleDOI
TL;DR: In this article, the authors consider a jump-type Cox-Ingersoll-Ross (CIR) process driven by a standard Wiener process and a subordinator, and study asymptotic properties of the maximum likelihood estimator (MLE) for its growth rate.

Book ChapterDOI
05 Feb 2017
TL;DR: In this paper, three models of subordinated processes, namely, the gamma subordinator, the alpha-stable and the gamma sub-process, are considered and a simulation procedure is proposed to estimate the tail index of the external process.
Abstract: In this paper we consider three models of subordinated processes. A subordinated process, called also a time-changed process, is defined as a superposition of two independent stochastic processes. To construct such stochastic system we replace the time of a given process (called also an external process) by another process which becomes the “operational time”. In the literature one can find different models that are constructed as a superposition of two stochastic processes. The most classical example is the Laplace motion, also known as variance gamma process, is stated as a Brownian motion time-changed by the gamma subordinator. In this paper the considered systems are constructed by replacing the time of the symmetric \(\alpha \)-stable Levy motion with another stochastic process, namely the \(\alpha _S\)-stable, tempered \(\alpha _T\)-stable and gamma subordinator. We discuss the main characteristics of each introduced processes. We examine the characteristic function, the codifference, the probability density function, asymptotic tail behaviour and the fractional order moments. To make the application of these processes possible we propose a simulation procedure. Finally, we demonstrate how to estimate the tail index of the external process, i.e. alpha-stable Levy motion and by using Monte Carlo method we show the efficiency of the proposed estimation method.

Posted Content
TL;DR: In this article, the authors studied the fractional Poisson process (FPP) time-changed by an independent L'evy subordinator and the inverse of the L\'evy sub-subordinator.
Abstract: In this paper, we study the fractional Poisson process (FPP) time-changed by an independent L\'evy subordinator and the inverse of the L\'evy subordinator, which we call TCFPP-I and TCFPP-II, respectively. Various distributional properties of these processes are established. We show that, under certain conditions, the TCFPP-I has the long-range dependence property and also its law of iterated logarithm is proved. It is shown that the TCFPP-II is a renewal process and its waiting time distribution is identified. Its bivariate distributions and also the governing difference-differential equation are derived. Some specific examples for both the processes are discussed. Finally, we present the simulations of the sample paths of these processes.