Stable Subordinator Research Articles

Tempered fractional differential equations on hyperbolic space

Abstract In this paper we study linear fractional differential equations involving tempered Caputo-type derivatives in the hyperbolic space. We consider in detail the three-dimensional case for its simple and useful structure. We also discuss the probabilistic meaning of our results in relation to the distribution of an hyperbolic Brownian motion time-changed with the inverse of a tempered stable subordinator. The generalization to an arbitrary dimension n can be easily obtained. We also show that it is possible to construct a particular solution for the non-linear porous-medium type tempered equation by using elementary functions.

On integrals of birth–death processes at random time

In this paper, we consider a time-changed path integral of the homogeneous birth–death process. Here, the time changes according to an inverse stable subordinator. It is shown that its joint distribution with the time-changed birth–death process is governed by a fractional partial differential equation. In a linear case, the explicit expressions for the Laplace transform of their joint generating function, means, variances and covariance are obtained. The limiting behavior of this integral process has been studied. Later, we consider the fractional integrals of linear birth–death processes and their time-changed versions. The mean values of these fractional integrals are obtained and analyzed. In a particular case, it is observed that the time-changed path integral of the linear birth–death process and the fractional integral of time-changed linear birth–death process have equal mean growth.

Fast Exact Simulation of the First Passage of a Tempered Stable Subordinator Across a Non-Increasing Function

Fast Exact Simulation of the First Passage of a Tempered Stable Subordinator Across a Non-Increasing Function

Explosion Rates for Continuous-State Branching Processes in a Lévy Environment

Here, we study the long-term behaviour of the non-explosion probability for continuous-state branching processes in a Lévy environment when the branching mechanism is given by the negative of the Laplace exponent of a subordinator. In order to do so, we study the law of this family of processes in the infinite mean case and provide necessary and sufficient conditions for the process to be conservative, i.e. that the process does not explode in finite time a.s. In addition, we establish precise rates for the non-explosion probabilities in the subcritical and critical regimes, first found by Palau et al. (ALEA Lat Am J Probab Math Stat 13(2):1235–1258, 2016) in the case when the branching mechanism is given by the negative of the Laplace exponent of a stable subordinator.

Bivariate tempered space-fractional Poisson process and shock models

Abstract We introduce a bivariate tempered space-fractional Poisson process (BTSFPP) by time-changing the bivariate Poisson process with an independent tempered $\alpha$ -stable subordinator. We study its distributional properties and its connection to differential equations. The Lévy measure for the BTSFPP is also derived. A bivariate competing risks and shock model based on the BTSFPP for predicting the failure times of items that undergo two random shocks is also explored. The system is supposed to break when the sum of two types of shock reaches a certain random threshold. Various results related to reliability, such as reliability function, hazard rates, failure density, and the probability that failure occurs due to a certain type of shock, are studied. We show that for a general Lévy subordinator, the failure time of the system is exponentially distributed with mean depending on the Laplace exponent of the Lévy subordinator when the threshold has a geometric distribution. Some special cases and several typical examples are also demonstrated.

Generalized fractional derivatives generated by Dickman subordinator and related stochastic processes

In this article, convolution-type fractional derivatives generated by Dickman subordinator and inverse Dickman subordinator are discussed. The Dickman subordinator and its inverse are generalizations of stable and inverse stable subordinators, respectively. The series representations of densities of the Dickman subordinator and inverse Dickman subordinator are also obtained, which could be helpful for computational purposes. Moreover, the space and time-fractional Poisson-Dickman processes, space-fractional Skellam Dickman process and non-homogenous Poisson-Dickman process are introduced and their main properties are studied.

Stable distributions and pseudo-processes related to fractional Airy functions

In this article, we study pseudo-processes related to odd-order heat-type equations composed with Lévy stable subordinators. The aim of the article is twofold. We first show that the pseudo-density of the subordinated pseudo-process can be represented as an expectation of damped oscillations with generalized gamma distributed parameters. This stochastic representation also arises as the solution to a fractional diffusion equation, involving a higher-order Riesz-Feller operator, which generalizes the odd-order heat-type equation. We then prove that, if the stable subordinator has a suitable exponent, the time-changed pseudo-process is identical in distribution to a genuine Lévy stable process. This result permits us to obtain a power series representation for the probability density function of an arbitrary asymmetric stable process of exponent ν > 1 and skewness parameter β, with 0 < | β | < 1 . The methods we use in order to carry out our analysis are based on the study of a fractional Airy function which emerges in the investigation of the higher-order Riesz-Feller operator.

On a time-changed variant of the generalized counting process

AbstractIn this paper, we time-change the generalized counting process (GCP) by an independent inverse mixed stable subordinator to obtain a fractional version of the GCP. We call it the mixed fractional counting process (MFCP). The system of fractional differential equations that governs its state probabilities is obtained using the Z transform method. Its one-dimensional distribution, mean, variance, covariance, probability generating function, and factorial moments are obtained. It is shown that the MFCP exhibits the long-range dependence property whereas its increment process has the short-range dependence property. As an application we consider a risk process in which the claims are modelled using the MFCP. For this risk process, we obtain an asymptotic behaviour of its finite-time ruin probability when the claim sizes are subexponentially distributed and the initial capital is arbitrarily large. Later, we discuss some distributional properties of a compound version of the GCP.

Spectral heat content on a class of fractal sets for subordinate killed Brownian motions

AbstractWe study the spectral heat content for a class of open sets with fractal boundaries determined by similitudes in , , with respect to subordinate killed Brownian motions via ‐stable subordinators and establish the asymptotic behavior of the spectral heat content as for the full range of . Our main theorems show that these asymptotic behaviors depend on whether the sequence of logarithms of the coefficients of the similitudes is arithmetic when , where is the interior Minkowski dimension of the boundary of the open set. The main tools for proving the theorems are the previous results on the spectral heat content for Brownian motions and the renewal theorem.

Correlation Structure of Time-Changed Generalized Mixed Fractional Brownian Motion

The generalized mixed fractional Brownian motion (gmfBm) is a Gaussian process with stationary increments that exhibits long-range dependence controlled by its Hurst indices. It is defined by taking linear combinations of a finite number of independent fractional Brownian motions with different Hurst indices. In this paper, we investigate the long-time behavior of gmfBm when it is time-changed by a tempered stable subordinator or a gamma process. As a main result, we show that the time-changed process exhibits a long-range dependence property under some conditions on the Hurst indices. The time-changed gmfBm can be used to model natural phenomena that exhibit long-range dependence, even when the underlying process is not itself long-range dependent.

Pseudoprocesses related to higher-order equations of vibrations of rods

In this paper we study Fresnel pseudoprocesses whose signed measure density is a solution to a higher-order extension of the equation of vibrations of rods. We also investigate space-fractional extensions of the pseudoprocesses related to the Riesz operator. The measure density is represented in terms of generalized Airy functions which include the classical Airy function as a particular case. We prove that the Fresnel pseudoprocess time-changed with an independent stable subordinator produces genuine stochastic processes. In particular, if the exponent of the subordinator is chosen in a suitable way, the time-changed pseudoprocess is identical in distribution to a mixture of stable processes. The case of a mixture of Cauchy distributions is discussed and we show that the symmetric mixture can be either unimodal or bimodal, while the probability density function of an asymmetric mixture can possibly have an inflection point.

The Rates of Convergence for Functional Limit Theorems with Stable Subordinators and for CTRW Approximations to Fractional Evolutions

From the initial development of probability theory to the present days, the convergence of various discrete processes to simpler continuous distributions remains at the heart of stochastic analysis. Many efforts have been devoted to functional central limit theorems (also referred to as the invariance principle), dealing with the convergence of random walks to Brownian motion. Though quite a lot of work has been conducted on the rates of convergence of the weighted sums of independent and identically distributed random variables to stable laws, the present paper is the first to supply the rates of convergence in the functional limit theorem for stable subordinators. On the other hand, there is a lot of activity on the convergence of CTRWs (continuous time random walks) to processes with memory (subordinated Markov process) described by fractional PDEs. Our second main result is the first one yielding rates of convergence in such a setting. Since CTRW approximations may be used for numeric solutions of fractional equations, we obtain, as a direct consequence of our results, the estimates for error terms in such numeric schemes.

Risk process with mixture of tempered stable inverse subordinators: Analysis and synthesis

Abstract We propose two fractional risk models, where the classical risk process is time-changed by the mixture of tempered stable inverse subordinators. We characterize the risk processes by deriving the marginal distributions and establish the moments and covariance structure. We study the main characteristics of these models such as ruin probability and time to ruin and illustrate the results with Monte Carlo simulations. The data suggest that the ruin time can be approximated by the inverse gaussian distribution and its generalizations.

Competing risks and shock models governed by a generalized bivariate Poisson process

AbstractWe propose a stochastic model for the failure times of items subject to two external random shocks occurring as events in an underlying bivariate counting process. This is a special formulation of the competing risks model, which is of interest in reliability theory and survival analysis. Specifically, we assume that a system, or an item, fails when the sum of the two types of shock reaches a critical random threshold. In detail, the two kinds of shock occur according to a bivariate space-fractional Poisson process, which is a two-dimensional vector of independent homogeneous Poisson processes time-changed by an independent stable subordinator. Various results are given, such as analytic hazard rates, failure densities, the probability that the failure occurs due to a specific type of shock, and the survival function. Some special cases and ageing notions related to the NBU characterization are also considered. In this way we generalize certain results in the literature, which can be recovered when the underlying process reduces to the homogeneous Poisson process.

The tempered space-fractional Cattaneo equation

We consider the time-fractional Cattaneo equation involving the tempered Caputo space-fractional derivative. There is an increasing interest in the recent literature for the applications of the fractional-type Cattaneo equations to heat transfer models. Our main aim is to discuss the role played by a fractional tempered operator in this framework. We show that the fundamental solution coincides with the probability law of a time-changed Brownian motion, obtained by means of a tempered stable subordinator. We find the characteristic function of this process and we explain the main differences with previous stochastic treatments of the time-fractional Cattaneo equation. We also provide the solution of a Dirichlet problem for the tempered fractional Cattaneo equation by means of the H-Fox function.

Skellam and time-changed variants of the generalized fractional counting process

In this paper, we study a Skellam type variant of the generalized counting process (GCP), namely, the generalized Skellam process. Some of its distributional properties such as the probability mass function, probability generating function, mean, variance and covariance are obtained. Its fractional version, namely, the generalized fractional Skellam process (GFSP) is considered by time-changing it with an independent inverse stable subordinator. It is observed that the GFSP is a Skellam type version of the generalized fractional counting process (GFCP) which is a fractional variant of the GCP. It is shown that the one-dimensional distributions of the GFSP are not infinitely divisible. An integral representation for its state probabilities is obtained. We establish its long-range dependence property by using its variance and covariance structure. Also, we consider two time-changed versions of the GFCP. These are obtained by time-changing the GFCP by an independent Lévy subordinator and its inverse. Some particular cases of these time-changed processes are discussed by considering specific Lévy subordinators.

Scaling limits of branching random walks and branching-stable processes

AbstractBranching-stable processes have recently appeared as counterparts of stable subordinators, when addition of real variables is replaced by branching mechanisms for point processes. Here we are interested in their domains of attraction and describe explicit conditions for a branching random walk to converge after a proper magnification to a branching-stable process. This contrasts with deep results obtained during the past decade on the asymptotic behavior of branching random walks and which involve either shifting without rescaling, or demagnification.

Optimal sizing of the sediment replenishment capacity based on robust ergodic control of subordinator-driven dynamics

A countermeasure against sediment depletion in rivers is sand replenishment. The cost-efficient sizing of its capacity is critical for sustainable environmental management. However, only temporally discrete replenishment is possible, and sediment transport is intermittent and uncertain. In this paper, we approach this new design problem using a robust ergodic control. First, a stochastic differential equation driven by a (tempered) stable subordinator describes the sediment storage dynamics. Sand replenishment is considered an impulse control, wherein observation/replenishment chances arrive at Poisson jump times. The subordinator is ambiguous since the decision-maker does not precisely know its Lévy measure. Using a dynamic risk measure that balances replenishment cost, disutility triggered by sediment depletion, and penalization of ambiguity, we derived the Hamilton–Jacobi–Bellman–Isaacs (HJBI) equation to determine the optimal storage capacity. We obtained that the HJBI equation has an analytical solution when a stable subordinator is used. Furthermore, we prove it admits a unique continuous viscosity solution and that it is optimal and applied the proposed model to real data.

Multivariate tempered stable additive subordination for financial models

We study a class of multivariate tempered stable distributions and introduce the associated class of tempered stable Sato subordinators. These Sato subordinators are used to build additive inhomogeneous processes by subordination of a multiparameter Brownian motion. The resulting process is additive and time inhomogeneous and it is a generalization of multivariate Lévy processes with good fit properties on financial data. We specify the model to have unit time normal inverse Gaussian distribution and we discuss the ability of the model to fit time inhomogeneous correlations on real data.

Towards control of dam and reservoir systems with forward–backward stochastic differential equations driven by clustered jumps

AbstractWe deal with a new maximum principle‐based stochastic control model for river management through operating a dam and reservoir system. The model is based on coupled forward–backward stochastic differential equations (FBSDEs) derived from jump‐driven streamflow dynamics and reservoir water balance. A continuous‐time branching process with immigration driven by a tempered stable subordinator efficiently describes clustered inflow streamflow dynamics. This is a completely new attempt in hydrology and control engineering. Applying a stochastic maximum principle to the dynamics based on an objective functional for designing cost‐efficient control of dam and reservoir systems leads to the FBSDEs as a system of optimality equations. The FBSDEs under a linear‐quadratic ansatz lead to a tractable model, while they are solved numerically in the other cases using a least‐squares Monte‐Carlo method. Optimal controls are found in the former, while only sub‐optimal ones are computable in the latter due to a hard state constraint. Model parameters are successfully identified from a real data of a river in Japan having a dam and reservoir system. We also show that the linear‐quadratic case can capture the real operation data of the system with underestimation of the outflow discharge. More complex cases with a realistic time horizon are analyzed numerically to investigate impacts of considering the environmental flows and seasonal operational purposes. Key challenges towards more sophisticated modeling and analysis with jump‐driven FBSDEs are discussed as well.