Showing papers on "Subordinator published in 2018"

Heat kernel estimates for time fractional equations

Abstract: In this paper, we establish existence and uniqueness of weak solutions to general time fractional equations and give their probabilistic representations. We then derive sharp two-sided estimates for fundamental solutions of a family of time fractional equations in metric measure spaces.

Marginal consistent dependence modelling using weak subordination for Brownian motions

Abstract: We present an approach for modelling dependencies in exponential Levy market models with arbitrary margins originated from time changed Brownian motions. Using weak subordination of Buchmann et al. [Bernoulli, 2017], 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 multivariate variance gamma models on various market data and show that these models are highly preferable to traditional approaches. Consistent values of basket-options under given marginal pricing models are achieved using the Esscher transform, generating a non-flat implied correlation ...

L\'{e}vy-walk-like Langevin dynamics

Abstract: Continuous time random walks and Langevin equations are two classes of stochastic models for describing the dynamics of particles in the natural world. While some of the processes can be conveniently characterized by both of them, more often one model has significant advantages (or has to be used) compared with the other one. In this paper, we consider the weakly damped Langevin system coupled with a new subordinator|$\alpha$-dependent subordinator with $1<\alpha<2$. We pay attention to the diffusion behaviour of the stochastic process described by this coupled Langevin system, and find the super-ballistic diffusion phenomena for the system with an unconfined potential on velocity but sub-ballistic superdiffusion phenomenon with a confined potential, which is like Levy walk for long times. One can further note that the two-point distribution of inverse subordinator affects mean square displacement of this coupled weakly damped Langevin system in essential.

Tempered fractional Langevin-Brownian motion with inverse $\beta$-stable subordinator

Abstract: Time-changed stochastic processes have attracted great attention and wide interests due to their extensive applications, especially in financial time series, biology and physics. This paper pays attention to a special stochastic process, tempered fractional Langevin motion, which is non-Markovian and undergoes ballistic diffusion for long times. The corresponding time-changed Langevin system with inverse $\beta$-stable subordinator is discussed in detail, including its diffusion type, moments, Klein-Kramers equation, and the correlation structure. Interestingly, this subordination could result in both subdiffusion and superdiffusion, depending on the value of $\beta$. The difference between the subordinated tempered fractional Langevin equation and the subordinated Langevin equation with external biasing force is studied for a deeper understanding of subordinator. The time-changed tempered fractional Brownian motion by inverse $\beta$-stable subordinator is also considered, as well as the correlation structure of its increments. Some properties of the statistical quantities of the time-changed process are discussed, displaying striking differences compared with the original process.

Time Fractional Poisson Equations: Representations and Estimates

Abstract: In this paper, we study existence and uniqueness of strong as well as weak solutions for general time fractional Poisson equations. We show that there is an integral representation of the solutions of time fractional Poisson equations with zero initial values in terms of semigroup for the infinitesimal spatial generator ${\cal L}$ and the corresponding subordinator associated with the time fractional derivative. This integral representation has an integral kernel $q(t, x, y)$, which we call the fundamental solution for the time fractional Poisson equation, if the semigrou for ${\cal L}$ has an integral kernel. We further show that $q(t, x, y)$ can be expressed as a time fractional derivative of the fundamental solution for the homogenous time fractional equation under the assumption that the associated subordinator admits a conjugate subordinator. Moreover, when the Laplace exponent of the associated subordinator satisfies the weak scaling property and its distribution is self-decomposable, we establish two-sided estimates for the fundamental solution $q(t,x, y)$ through explicit estimates of transition density functions of subordinators.

Estimates of Dirichlet heat kernels for subordinate Brownian motions

Abstract: In this paper, we discuss estimates of transition densities of subordinate Brownian motions in open subsets of Euclidean space. When $D$ is a $C^{1,1}$ domain, we establish sharp two-sided estimates for the transition densities of a large class of subordinate Brownian motions in $D$ whose scaling order is not necessarily strictly below $2$. Our estimates are explicit and written in terms of the dimension, the Euclidean distance between two points, the distance to the boundary and the Laplace exponent of the corresponding subordinator only.

Space‐time fractional Dirichlet problems

Abstract: This paper establishes explicit solutions for fractional diffusion problems on bounded domains. It also gives stochastic solutions, in terms of Markov processes time‐changed by an inverse stable subordinator whose index equals the order of the fractional time derivative. Some applications are given, to demonstrate how to specify a well‐posed Dirichlet problem for space‐time fractional diffusions in one or several variables. This solves an open problem in numerical analysis.

Space-fractional versions of the negative binomial and Polya-type processes

Abstract: In this paper, we introduce a space fractional negative binomial process (SFNB) by time-changing the space fractional Poisson process by a gamma subordinator. Its one-dimensional distributions are derived in terms of generalized Wright functions and their governing equations are obtained. It is a Levy process and the corresponding Levy measure is given. Extensions to the case of distributed order SFNB, where the fractional index follows a two-point distribution, are investigated in detail. The relationship with space fractional Polya-type processes is also discussed. Moreover, we define and study multivariate versions, which we obtain by time-changing a d-dimensional space-fractional Poisson process by a common independent gamma subordinator. Some applications to population’s growth and epidemiology models are explored. Finally, we discuss algorithms for the simulation of the SFNB process.

Gibbs Partitions, Riemann-Liouville Fractional Operators, Mittag-Leffler Functions, and Fragmentations Derived From Stable Subordinators

Abstract: Pitman (2003) (and subsequently Gnedin and Pitman (2006)) showed that a large class of random partitions of the integers derived from a stable subordinator of index $\alpha\in(0,1)$ have infinite Gibbs (product) structure as a characterizing feature. In particular, the Brownian case $\alpha=1/2,$ may be expressed in terms of Hermite functions. We, HJL (2007), for $\alpha\in(0,1),$ showed that the relevant quantities are Fox $H$ and Meijer $G$ functions, thus in principle allowing for the calculation of a myriad of partition distributions. The most notable member of this class are the $(\alpha,\theta)$ partitions, derived from mass partitions having a $\mathrm{PD}(\alpha,\theta)$ distribution, which are induced by mixing over variables with generalized Mittag-Leffler distributions, denoted by $\mathrm{ML}(\alpha,\theta).$ We provide further interpretations of the broader class. We start with representations in terms of Riemann-Liouville fractional integrals indexed by a stable density. This leads to connections to fractional calculus, wherein the interplay between special functions and probability theory, in particular as it relates to size biased sampling, is illustrated. A centerpiece of our work are results related to Mittag-Leffler functions which plays a key role in fractional calculus. Leading to connections to a mixed Poisson waiting time framework. We provide novel characterizations of general laws related to two nested families of $\mathrm{PD}(\alpha,\theta)$ mass partitions appearing in the literature, by Dong, Goldschmidt and Martin (2005) and Pitman (1999), that exhibit dual coagulation/fragmentation relations, constitute Markov chains, and are otherwise connected to the construction of various random trees and graphs. Simplifications in the Brownian case are also highlighted, indicating relations in the current literature.

Fractional Erlang Queues.

Abstract: We introduce a fractional generalization of the Erlang Queues $M/E_k/1$. Such process is obtained through a time-change via inverse stable subordinator of the classical queue process. We first exploit the (fractional) Kolmogorov forward equation for such process, then we use such equation to obtain an interpretation of this process in the queuing theory context. Then we also exploit the transient state probabilities and some features of this fractional queue model, such as the mean queue length, the distribution of the busy periods and some conditional distributions of the waiting times. Finally, we provide some algorithms to simulate their sample paths.

Negative Binomial Construction of Random Discrete 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.

A cross-linguistic rarity in synchrony and diachrony: Adverbial subordinator prefixes exist

Abstract: Abstract This article shows that a hitherto unattested construction type – namely, adverbial subordinator prefixes – is in fact attested in several languages. While Dryer’s 659-language convenience sample does not turn up any clear example of such a construction, we argue that this is in part due to arbitrary coding choices that a priori exclude potential constructions of this type. In order to document the existence of adverbial subordinator prefixes, we present a number of languages with different genealogical and areal affiliations, each of which shows solid synchronic evidence for what appears to be a universally dispreferred feature. Furthermore, we identify some diachronic pathways through which adverbial subordinator prefixes grammaticalize.

On the infinite divisibility of distributions of some inverse subordinators

Abstract: We consider the infinite divisibility of distributions of some well-known inverse subordinators. Using a tail probability bound, we establish that distributions of many of the inverse subordinators used in the literature are not infinitely divisible. We further show that the distribution of a renewal process time-changed by an inverse stable subordinator is not infinitely divisible, which in particular implies that the distribution of the fractional Poisson process is not infinitely divisible.

Calibration for Weak Variance-Alpha-Gamma Processes

Abstract: The weak variance-alpha-gamma process is a multivariate Levy process constructed by weakly subordinating Brownian motion, possibly with correlated components with an alpha-gamma subordinator. It generalises the variance-alpha-gamma process of Semeraro constructed by traditional subordination. We compare three calibration methods for the weak variance-alpha-gamma process, method of moments, maximum likelihood estimation (MLE) and digital moment estimation (DME). We derive a condition for Fourier invertibility needed to apply MLE and show in our simulations that MLE produces a better fit when this condition holds, while DME produces a better fit when it is violated. We also find that the weak variance-alpha-gamma process exhibits a wider range of dependence and produces a significantly better fit than the variance-alpha-gamma process on an S&P500-FTSE100 data set, and that DME produces the best fit in this situation.

Estimating the input of a L\'evy-driven queue by Poisson sampling of the workload process

Abstract: This paper aims at semi-parametrically estimating the input process to a L\'evy-driven queue by sampling the workload process at Poisson times. We construct a method-of-moments based estimator for the L\'evy process' characteristic exponent. This method exploits the known distribution of the workload sampled at an exponential time, thus taking into account the dependence between subsequent samples. Verifiable conditions for consistency and asymptotic normality are provided, along with explicit expressions for the asymptotic variance. The method requires an intermediate estimation step of estimating a constant (related to both the input distribution and the sampling rate); this constant also features in the asymptotic analysis. For subordinator L\'evy input, a partial MLE is constructed for the intermediate step and we show that it satisfies the consistency and asymptotic normality conditions. For general spectrally-positive L\'evy input a biased estimator is proposed that only uses workload observations above some threshold; the bias can be made arbitrarily small by appropriately choosing the threshold.

Maximum likelihood estimation for stochastic Lotka–Volterra model with jumps

Abstract: In this paper, we consider the stochastic Lotka–Volterra model with additive jump noises. We show some desired properties of the solution such as existence and uniqueness of positive strong solution, unique stationary distribution, and exponential ergodicity. After that, we investigate the maximum likelihood estimation for the drift coefficients based on continuous time observations. The likelihood function and explicit estimator are derived by using semimartingale theory. In addition, consistency and asymptotic normality of the estimator are proved. Finally, computer simulations are presented to illustrate our results.

Random time-changes and asymptotic results for a class of continuous-time Markov chains on integers with alternating rates

Abstract: We consider continuous-time Markov chains on integers which allow transitions to adjacent states only, with alternating rates. We give explicit formulas for probability generating functions, and also for means, variances and state probabilities of the random variables of the process. Moreover we study independent random time-changes with the inverse of the stable subordinator, the stable subordinator and the tempered stable subodinator. We also present some asymptotic results in the fashion of large deviations. These results give some generalizations of those presented in Di Crescenzo A., Macci C., Martinucci B. (2014).

Subordinators which are infinitely divisible w.r.t. time: Construction, properties, and simulation of max-stable sequences and infinitely divisible laws

Abstract: The concept of a L\'evy subordinator is generalized to a family of non-decreasing stochastic processes, which are parameterized in terms of two Bernstein functions. Whereas the independent increments property is only maintained in the L\'evy subordinator special case, the considered family is always strongly infinitely divisible with respect to time, meaning that a path can be represented in distribution as a finite sum with arbitrarily many summands of independent and identically distributed paths of another process. Besides distributional properties of the process, we present two applications to the design of accurate and efficient simulation algorithms. First, each member of the considered family corresponds uniquely to an exchangeable max-stable sequence of random variables, and we demonstrate how the associated extreme-value copula can be simulated exactly and efficiently from its Pickands dependence measure. Second, we show how one obtains different series and integral representations for infinitely divisible probability laws by varying the parameterizing pair of Bernstein functions, without changing the law of one-dimensional margins of the process. As a particular example, we present an exact simulation algorithm for compound Poisson distributions from the Bondesson class, for which the generalized inverse of the distribution function of the associated Stieltjes measure can be evaluated accurately.

On the infinite divisibility of distributions of some inverse subordinators

Abstract: We consider the infinite divisibility of distributions of some well-known inverse subordinators. Using a tail probability bound, we establish that distributions of many of the inverse subordinators used in the literature are not infinitely divisible. We further show that the distribution of a renewal process time-changed by an inverse stable subordinator is not infinitely divisible, which in particular implies that the distribution of the fractional Poisson process is not infinitely divisible.

Say that she rail; why then I'll tell her plain she sings as sweetly as a nightingale: A corpus-based study of the development of say as a conditional subordinator in English

Abstract: in Norwegian ............................................................................................................. iii Acknowledgements ................................................................................................................... iv List of tables ............................................................................................................................. vii List of figures ......................................................................................................................... viii Abbreviations and conventions ................................................................................................. ix

The time to absorption in Λ-coalescents

Abstract: We present a law of large numbers and a central limit theorem for the time to absorption of Λ-coalescents with dust started from n blocks, as n→∞. The proofs rely on an approximation of the logarithm of the block-counting process by means of a drifted subordinator.

Limiting behaviour of the stationary search cost distribution driven by a generalized gamma process

Abstract: Consider a list of labeled objects that are organized in a heap. At each time, object $j$ is selected with probability $p_j$ and moved to the top of the heap. This procedure defines a Markov chain on the set of permutations which is referred to in the literature as Move-to-Front rule. The present contribution focuses on the stationary search cost, namely the position of the requested item in the heap when the Markov chain is in equilibrium. We consider the scenario where the number of objects is infinite and the probabilities $p_j$’s are defined as the normalization of the increments of a subordinator. In this setting, we provide an exact formula for the moments of any order of the stationary search cost distribution. We illustrate the new findings in the case of a generalized gamma subordinator and deal with an extension to the two–parameter Poisson–Dirichlet process, also known as Pitman–Yor process.

Simulation of Student–Lévy processes using series representations

Abstract: Levy processes have become very popular in many applications in finance, physics and beyond. The Student–Levy process is one interesting special case where increments are heavy-tailed and, for 1-increments, Student t distributed. Although theoretically available, there is a lack of path simulation techniques in the literature due to its complicated form. In this paper we address this issue using series representations with the inverse Levy measure method and the rejection method and prove upper bounds for the mean squared approximation error. In the numerical section we discuss a numerical inversion scheme to find the inverse Levy measure efficiently. We extend the existing numerical inverse Levy measure method to incorporate explosive Levy tail measures. Monte Carlo studies verify the error bounds and the effectiveness of the simulation routine. As a side result we obtain series representations of the so called inverse gamma subordinator which are used to generate paths in this model.

Extinction time of a CB-processes with competition in a L\'evy random environment

Abstract: In this paper, we are interested on the extinction time of continuous state branching processes with competition in a L\'evy random environment. In particular we prove, under the so-called Grey's condition together with the assumption that the L\'evy random environment does not drift towards infinity, that for any starting point the process gets extinct in finite time a.s. Moreover if we replace the condition on the L\'evy random environment by a technical integrability condition on the competition mechanism, then the process also gets extinct in finite time a.s. and it comes down from infinity. Then the logistic case in a Brownian random environment is treated. Our arguments are base on a Lamperti-type representation where the driven process turns out to be a perturbed Feller diffusion by an independent spectrally positive L\'evy process. If the independent random perturbation is a subordinator then the process converges to a specified distribution; otherwise, it goes extinct a.s. In the latter case and following a similar approach to Lambert (Ann. Appl. Probab., 15(2):1506-1535, 2005.), we provide the expectation and the Laplace transform of the absorption time, as a functional of the solution to a Riccati differential equation.

Term structure of interest rates with stickiness: a subdiffusion approach

Abstract: In this paper, we propose a new class of term structure of interest rate models which is built on the subdiffusion processes. We assume that the spot rate is a function of a time changed diffusion process belonging to a symmetric pricing semigroup for which its spectral representation is known. The time change process is taken to be an inverse Levy subordinator in order to capture the stickiness feature observed in the short-term interest rates. We derive the analytical formulas for both bond and bond option prices based on eigenfunction expansion method. We also numerically implement a specific subdiffusive model by testing the sensitivities of bond and bond option prices with respect to the parameters of time change process.

Time-changed Poisson processes of order $k$

Abstract: In this article, we study the Poisson process of order k (PPoK) time-changed with an independent Levy subordinator and its inverse, which we call respectively, as TCPPoK-I and TCPPoK-II, through various distributional properties, long-range dependence and limit theorems for the PPoK and the TCPPoK-I. Further, we study the governing difference-differential equations of the TCPPoK-I for the case inverse Gaussian subordinator. Similarly, we study the distributional properties, asymptotic moments and the governing difference-differential equation of TCPPoK-II. As an application to ruin theory, we give a governing differential equation of ruin probability in insurance ruin using these processes. Finally, we present some simulated sample paths of both the processes.