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Models of generalized counting processes time-changed by a general inverse subordinator are considered, their distributions are characterized, and governing equations for them are presented. The equations are given in terms of the generalized fractional derivatives, namely, convolution-type derivatives with respect to Bernštein functions. Some particular examples are presented. [ABSTRACT FROM AUTHOR]

Models of generalized counting processes time-changed by a general inverse subordinator are considered, their distributions are characterized, and governing equations for them are presented. The equations are given in terms of the generalized fractional derivatives, namely, convolution-type derivatives with respect to Bernštein functions. Some particular examples are presented.

In Russian, the subordinator kak ‘how’ is both a manner question word and an eventive complementizer. The Russian linguistic tradition explains the colexification of the two functions in terms of a semantic shift from manner as characteristic of a situation to event description as a whole. Alternatively, a grammaticalization scenario from manner complements to event/propositional complements has been suggested: manner complements originally have a propositional frame, which is foregrounded concurrently with the loss of the manner meaning, giving rise to both eventive and propositional interpretations. This article is aimed at testing both hypotheses. We study several large Old Russian manuscripts, starting from the first available documents of the 11th century, and show that at the earliest documented period Old Russian kako/kakъ could be used in all types of complement clauses. It could introduce eventive, propositional, manner and irrealis purposive-like complements. Accordingly, the evolution of the subordinator kak in complementation involves a narrowing of its functional domain. We classify Old Russian texts based on the period and trace the gradual loss of particular functions during the centuries. Thus, we show that the Russian data supports the second grammaticalization scenario.

Based on the Girsanov theorem for a kind of Volterra-Gaussian process, which are the generalization of fractional Brownian motion, Liouville fractional Brownian motion, and fractional Ornstein-Uhlenbeck process, we establish the Harnack inequalities for a class of stochastic functional differential equations driven by a kind of Volterra-Gaussian processes with a subordinator by an approximation technique. Some known results have been generalized and improved. [ABSTRACT FROM AUTHOR]

This article focuses on adverbial clause construction in Dobbi language. Dobbi is a member of the Ethio-Semitic subgroup within the Gurage language family which is mostly spoken in the Southern Nations, Nationalities, and People's Region (SNNPR). Dobbi is one of the least researched and documented languages in the Gurage language family, which is why this study is being done. It is important to note that this study lacks any theoretical foundation, and its data analysis strictly adheres to a descriptive approach. The study relies on the participation of native speakers who were consulted for the collection of data through elicitation techniques. Adverbial clauses can also contribute to the modification of main clauses. Subordinators, acting as logical connectors, generally establish a link with a preceding clause. In the case of Dobbi, there exist two types of subordinators: (1) grammatical morphemes devoid of any lexical meaning and (2) grammatical morphemes carrying lexical content. The usage of a dependent clause verb entails attaching subordinate prefixes and suffixes, along with an agreement marker. Consequently, this research aims to provide an in-depth elaboration on the construction of temporal, spatial, comparative, concessive, and conditional adverbial clauses. [ABSTRACT FROM AUTHOR]

The concept and term polovedlejší věta ('semi-subordinate clause') is theoretically anchored in the formal-semantic level description of the language system. This description has been methodologically inspired by the distinctive features of the Prague Linguistic Circle phonology. Based on three features (subordinator, parenthesis, commenting), the semi-subordinate clause, by which the speaker comments on another clause, is delimited within this descriptive framework. Semi-subordinate clauses are unambiguously identifiable using a test of reversed relationship of clauses, cf. Jak vidím, jste soukromý detektiv 'As I see, you are a private detective' → Vidím, že jste soukromý detektiv 'I (can) see that you are a private detective.' The main section of the article is formed by a classification of semi-subordinate clauses in several parts. The first part is based on the classification according to subordinators: the centre of the system of semi-subordinate clauses is formed by clauses introduced by the relative pronominal adverb jak. Clauses with the relative consequential pronoun což and clauses with the relative introducing pronoun co are on the periphery. Clauses with conjunctions (jestli and its variants, and pokud and aby) lie between the centre and the periph ery. The second part differentiates between attitude and stylization clauses. In the conclusion, the term semisubordinate clause, reflecting a specific semi-category, is explained, and defended. [ABSTRACT FROM AUTHOR]

The existence of moments of first downward passage times of a spectrally negative Lévy process is governed by the general dynamics of the Lévy process, i.e. whether it is drifting to $+\infty$ , $-\infty$ , or oscillating. Whenever the Lévy process drifts to $+\infty$ , we prove that the $\kappa$ th moment of the first passage time (conditioned to be finite) exists if and only if the $(\kappa+1)$ th moment of the Lévy jump measure exists. This generalizes a result shown earlier by Delbaen for Cramér–Lundberg risk processes. Whenever the Lévy process drifts to $-\infty$ , we prove that all moments of the first passage time exist, while for an oscillating Lévy process we derive conditions for non-existence of the moments, and in particular we show that no integer moments exist. [ABSTRACT FROM AUTHOR]

The spectral heat content is investigated for time-changed killed Brownian motions on C 1 , 1 open sets, where the time change is given by either a subordinator or an inverse subordinator, with the underlying Laplace exponent being regularly varying at ∞ with index β ∈ (0 , 1) . In the case of inverse subordinators, the asymptotic limit of the spectral heat content in small time is shown to involve a probabilistic term depending only on β ∈ (0 , 1) . In contrast, in the case of subordinators, this universality holds only when β ∈ (1 2 , 1) . [ABSTRACT FROM AUTHOR]

This study traces the relationship between two erstwhile separate linguistic elements, just and so, within the framework of Functional Discourse Grammar (FDG). In keeping with FDG’s form-oriented function-to-form approach, the study proceeds semasiologically by, first, examining the uses of relatively independent forms (i.e. the focus particle just modifying so as a degree word and a manner proform), then turning to more tightly-knit structures (i.e. just so as a subordinator of purpose and condition), and finally, looking at the fixed expression just so used as a part of a pragmatic marker. Using data from the Corpus of Contemporary American English and the Corpus of Historical American English, we argue that the different meanings of just so raise a number of issues related to the analysis of modification in FDG, namely the status and function of the modifier just in the constructions under discussion and the concomitant representation of so. Furthermore, the analysis shows that FDG can model very precisely the interplay of semantic and pragmatic information in the stages when just so is still compositional, with just providing interpersonal (i.e. pragmatic) and so representational (i.e. semantic) information, as well as its development into the non-compositional and purely pragmatic discourse marker just so you know.

A new model which we called time driven model for option pricing is proposed. In this model, the price of underlying asset is driven by different random driving source in different time interval. Explicit option pricing formulas for the European call option are obtained by using the Mellin transform. Moreover, some numerical illustrations are given by computing European call option prices. [ABSTRACT FROM AUTHOR]

This paper introduces a power-law subordinator and a power-law Lévy process whose Laplace transform and characteristic function are simply made up of power functions or the ratio of power functions, respectively, and proposes a power-law vector random field whose finite-dimensional characteristic functions consist merely of a power function or the ratio of two power functions. They may or may not have first-order moment, and contain Linnik, variance Gamma, and Laplace Lévy processes (vector random fields) as special cases. For a second-order power-law vector random field, it is fully characterized by its mean vector function and its covariance matrix function, just like a Gaussian vector random field. An important feature of the power-law Lévy processes (random fields) is that they can be used as the building blocks to construct other Lévy processes (random fields), such as hyperbolic secant, cosine ratio, and sine ratio Lévy processes (random fields). [ABSTRACT FROM AUTHOR]

We investigate the Gerber-Shiu discounted penalty function for Markov-modulated Lévy risk processes with random incomes. Firstly, we consider the case when the downward and upward jumps (respectively, claims and random gains) are given by independent compound Poisson processes, with claim sizes with a general distribution function and gains in such a way that their distribution has a rational Laplace transform. Afterwards, we use the above results and weak convergence techniques to study the case when the claims are given by a subordinator and, subsequently, we establish results when the claims are governed by a pure spectrally positive Lévy jump process. Some numerical examples are presented in order to illustrate our results. [ABSTRACT FROM AUTHOR]

Being base on coupling by change of measure and an approximation technique, the Harnack inequalities for a class of stochastic functional differential equations driven by subordinate fractional Brownian motion with Hurst parameter 0 < H < 1/2 are established. By using a transformation formulas for fractional Brownian motion, the Harnack inequalities for stochastic functional differential equations driven by subordinate fractional Brownian motion with Hurst parameter 1/2 < H < 1 are established. [ABSTRACT FROM AUTHOR]

We consider the number of blocks involved in the last merger of a$\Lambda$-coalescent started with $n$ blocks. We give conditions under which,as $n \to \infty$, the sequence of these random variables a) is tight, b)converges in distribution to a finite random variable or c) converges toinfinity in probability. Our conditions are optimal for $\Lambda$-coalescentsthat have a dust component. For general $\Lambda$, we relate the three cases tothe existence, uniqueness and non-existence of quasi-invariant measures for thedynamics of the block-counting process, and in case b) investigate thetime-reversal of the block-counting process back from the time of the lastmerger.

We consider the number of blocks involved in the last merger of a$\Lambda$-coalescent started with $n$ blocks. We give conditions under which,as $n \to \infty$, the sequence of these random variables a) is tight, b)converges in distribution to a finite random variable or c) converges toinfinity in probability. Our conditions are optimal for $\Lambda$-coalescentsthat have a dust component. For general $\Lambda$, we relate the three cases tothe existence, uniqueness and non-existence of quasi-invariant measures for thedynamics of the block-counting process, and in case b) investigate thetime-reversal of the block-counting process back from the time of the lastmerger.

We study stochastic integrals driven by a general subordinator and establish a zero-one law for the finiteness of the resulting integral as well as moment estimates. As an application, we use these results to obtain structural properties of SPDEs driven by multiplicative pure jump noise, which include (1) a maximal inequality for a multiplicative stochastic convolution Z_t, (2) a small ball probability of Z_t, (3) the existence of invariant measures and accessibility to zero of SPDEs, and (4) a Galerkin approximation of solutions to SPDEs. [ABSTRACT FROM AUTHOR]

In this paper, we study some properties of the generalized Fokker–Planck equation induced by the time-changed fractional Ornstein–Uhlenbeck process. First of all, we exploit some sufficient conditions to show that a mild solution of such equation is actually a classical solution. Then, we discuss an isolation result for mild solutions. Finally, we prove the weak maximum principle for strong solutions of the aforementioned equation and then a uniqueness result. [ABSTRACT FROM AUTHOR]

In this paper, we study strong solutions of some non-local difference–differential equations linked to a class of birth–death processes arising as discrete approximations of Pearson diffusions by means of a spectral decomposition in terms of orthogonal polynomials and eigenfunctions of some non-local derivatives. Moreover, we give a stochastic representation of such solutions in terms of time-changed birth–death processes and study their invariant and their limit distribution. Finally, we describe the correlation structure of the aforementioned time-changed birth–death processes. [ABSTRACT FROM AUTHOR]

Let X = (X t) t ≥ 0 be a process behaving as a general increasing Lévy process (subordinator) prior to hitting a given unknown level m 0 , then behaving as another different subordinator once this threshold is crossed. This paper addresses the detection of this unknown threshold m 0 ∈ [ 0 , + ∞ ] from an observed trajectory of the process. These kind of model and issue are encountered in many areas such as reliability and quality control in degradation problems. More precisely, we construct, from a sample path and for each ε > 0 , a so-called detection level L ε by considering a CUSUM inspired procedure. Under mild assumptions, this level is such that, while m 0 is infinite (i.e. when no changes occur), its expectation E ∞ (L ε) tends to + ∞ as ε tends to 0, and the expected overshoot E m 0 ( [ L ε − m 0 ] +) , while the threshold m 0 is finite, is negligible compared to E ∞ (L ε) as ε tends to 0. Numerical illustrations are provided when the Lévy processes are gamma processes with different shape parameters. [ABSTRACT FROM AUTHOR]

In this paper, we develop a novel and rigorous approach to the Fokker–Planck equation, or Kolmogorov forward equation, for the Feynman–Kac transform of non-Markov anomalous processes. The equation describes the evolution of the density of the anomalous process Y t = X E t under the influence of potentials, where X is a strong Markov process on a Lusin space X that is in weak duality with another strong Markov process X ̂ on X and { E t , t ≥ 0 } is the inverse of a driftless subordinator S that is independent of X and has infinite Lévy measure. We derive a probabilistic representation of the density of the anomalous process under the Feynman–Kac transform by the dual Feynman–Kac transform in terms of the weak dual process X ̂ t and the inverse subordinator { E t ; t ≥ 0 }. We then establish the regularity of the density function, and show that it is the unique mild solution as well as the unique weak solution of a non-local Fokker–Planck equation that involves the dual generator of X and the potential measure of the subordinator S. During the course of the study, we are naturally led to extend the notation of Riemann–Liouville integral to measures that are locally finite on [ 0 , ∞). [ABSTRACT FROM AUTHOR]

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Reaction-diffusion equations are one of the most common mathematical models in the natural sciences and are used to model systems that combine reactions with diffusive motion. However, rather than normal diffusion, anomalous subdiffusion is observed in many systems and is especially prevalent in cell biology. What are the reaction-subdiffusion equations describing a system that involves first-order reactions and subdiffusive motion? In this paper, we derive fractional reaction-subdiffusion equations describing an arbitrary number of molecular species that react at first-order rates and move subdiffusively. Importantly, different species may have different diffusivities and drifts, which contrasts previous approaches to this question that assume that each species has the same movement dynamics. We derive the equations by combining results on timedependent fractional Fokker--Planck equations with methods of analyzing stochastically switching evolution equations. Furthermore, we construct the stochastic description of individual molecules whose deterministic concentrations follow these reaction-subdiffusion equations. This stochastic description involves subordinating a diffusion process whose dynamics are controlled by a subordinated jump process. We illustrate our results in several examples and show that solutions of the reactionsubdiffusion equations agree with stochastic simulations of individual molecules. [ABSTRACT FROM AUTHOR]

We develop a new approach to the study of the Feynman--Kac transform for non-Markov anomalous process Yt = XE using methods from stochastic analysis, where X is a strong Markov process on a Lusin space X and Et, t ≥ 0} is the inverse of a driftless subordinator S that is independent of X and has infinite Lévy measure. For a bounded function κ X and f in a suitable functional space over X, we establish regularity of ... and show that it is the unique mild solution to a time fractional equation with initial value f. When X is a symmetric Markov process on X, we further show that u is the unique weak solution to that time fractional equation. The main results are applied to compute the probability distribution of several random quantities of anomalous subdiffusion Y where X is a one-dimensional Brownian motion, including the first passage time, occupation time, and stochastic areas of Y. [ABSTRACT FROM AUTHOR]

In this paper, we discuss estimates on the transition densities of subordinators, which are global in time. We establish sharp two-sided estimates on the transition densities of subordinators whose Lévy measures are absolutely continuous and decaying in mixed polynomial orders. Under a weaker assumption on Lévy measures, we also obtain precise asymptotic behaviors of the transition densities at infinity. Our results cover geometric stable subordinators, Gamma subordinators and much more. [ABSTRACT FROM AUTHOR]

In this paper we focus on strong solutions of some heat-like problems with a non-local derivative in time induced by a Bernstein function and an elliptic operator given by the generator or the Fokker–Planck operator of a Pearson diffusion, covering a large class of important stochastic processes. Such kind of time-non-local equations naturally arise in the treatment of particle motion in heterogeneous media. In particular, we use spectral decomposition results for the usual Pearson diffusions to exploit explicit solutions of the aforementioned equations. Moreover, we provide stochastic representation of such solutions in terms of time-changed Pearson diffusions. Finally, we exploit some further properties of these processes, such as limit distributions and long/short-range dependence. [ABSTRACT FROM AUTHOR]

Hunt's hypothesis (H) and the related Getoor's conjecture is one of the most important problems in the basic theory of Markov processes. In this paper, we investigate the invariance of Hunt's hypothesis (H) for Markov processes under two classes of transformations, which are change of measure and subordination. Our first theorem shows that for two standard processes (Xt) and (Yt), if (Xt) satisfies (H) and (Yt) is locally absolutely continuous with respect to (Xt), then (Yt) satisfies (H). Our second theorem shows that a standard process (Xt) satisfies (H) if and only if (X τ t ) satisfies (H) for some (and hence any) subordinator (τt) which is independent of (Xt) and has a positive drift coefficient. Applications of the two theorems are given. [ABSTRACT FROM AUTHOR]

The degradations of engineering components can be dependent. This study considers that a d-dimensional subordinator constructed based on Lévy copula can be used to model the dependency. Three procedures to estimate model parameters of the process constructed based on Clayton Lévy copula with the gamma processes as marginals are investigated through simulation. The first one assumes that jumps of the processes are observable; the second one uses the increments of degradation as samples; and the third one is based on the concept of jump tail dependence coefficient. An application is presented by estimating model parameter and calculating the system reliability. [ABSTRACT FROM AUTHOR]

In the literature, one can find several applications of the time-fractional heat equation, particularly in the context of time-changed stochastic processes. Stochastic representation results for such an equation can be used to provide a Monte Carlo simulation method, upon proving that the solution is actually unique. In the classical case, however, this is not true if we do not consider any additional assumption, showing, thus, that the Monte Carlo simulation method identifies only a particular solution. In this paper, we consider the problem of the uniqueness of the solutions of the time-fractional heat equation with initial data. Precisely, under suitable assumptions about the regularity of the initial datum, we prove that such an equation admits an infinity of classical solutions. The proof mimics the construction of the Tychonoff solutions of the classical heat equation. As a consequence, one has to add some addtional conditions to the time-fractional Cauchy problem to ensure the uniqueness of the solution.

The syntactic structures used by adolescents can be calculated by measuring various markers that have the scope of developing with age. The use of such markers can be influenced by the type and modality of discourse on the individual uses. The present study was aimed at exploring the syntactic development in 10–16-year-old Indian adolescents using markers such as sentences, clauses, subordinators, coordinators, T-units and the subordination index (SI) using written expository texts. The study followed a cross-sectional study design following a non-random convenient sampling procedure. A total number of 180 typically-developing adolescents divided into six groups participated in the study. Phase 1 of the study comprised the preparation of the stimuli; Phase 2 included the data collection; Phase 3 focused on the data analysis; and Phase 4 involved the statistical analysis performed on the obtained data. One-way analysis of variance (ANOVA) revealed a main level of significance at p < 0.05 for all variables (except SI) indicating an overall change in the development across the six age groups. Bonferroni's post hoc analysis indicated poor significance between the groups in almost all the six variables. Understanding the syntactic markers in the language of adolescents is essential to determine the academic and communicative effectiveness of these individuals. [ABSTRACT FROM AUTHOR]

Exchangeable coalescents with dust are studied. The rate of convergence as the sample size tends to infinity of the scaled block counting process to the frequency of singleton process is determined. This rate is expressed in terms of a certain Bernstein function. The proofs are based on Taylor expansions of the infinitesimal generators and semigroups and involve a particular concentration inequality arising in the context of Karlin's infinite urn model. The rate of convergence is calculated for several examples of coalescents. [ABSTRACT FROM AUTHOR]

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.

Free regular convolution semigroups describe the distribution of free subordinators, while Bondesson class convolution semigroups correspond to classical subordinators with completely monotone Lévy density. We show that these two classes of convolution semigroups are in bijection with the class of complete Bernstein functions, and we establish an integral identity linking the two semigroups. We provide several explicit examples that illustrate this result. [ABSTRACT FROM AUTHOR]

In this paper, we study estimates on tail probabilities of several classes of subordinators under mild assumptions on the tails of their Lévy measures. As an application of that result, we obtain two-sided estimates for fundamental solutions of general homogeneous time fractional equations including those with Dirichlet boundary conditions. [ABSTRACT FROM AUTHOR]

We consider the variational structure of a time-fractional second-order mean field games (MFG) system. The MFG system consists of time-fractional Fokker–Planck and Hamilton–Jacobi–Bellman equations. In such a situation, the individual agent follows a non-Markovian dynamics given by a subdiffusion process. Hence, the results of this paper extend the theory of variational MFG to the subdiffusive situation. [ABSTRACT FROM AUTHOR]

We define a time-changed fractional Ornstein-Uhlenbeck process by composing a fractional Ornstein-Uhlenbeck process with the inverse of a subordinator. Properties of the moments of such process are investigated and the existence of the density is shown. We also provide a generalized Fokker-Planck equation for the density of the process. [ABSTRACT FROM AUTHOR]

We consider a subordinate Brownian motion X with Gaussian components when the scaling order of purely discontinuous part is between 0 and 2 including 2. In this paper we establish sharp two-sided bounds for transition density of X in ℝ d and C1,1-open sets. As a corollary, we obtain a sharp Green function estimates. [ABSTRACT FROM AUTHOR]

This study of two syntactic patterns (quasi + predicative future participles and quasi introducing actantial clauses) delineates the process through which they evolve, centering around juridical texts and drawing also on translated texts. The question of the Greek genesis of each of these two constructions is examined with special consideration of the distinct constraints for Lat. quasi vis-àvis those for Gr. ὡς, leading to disparate answers for each.

The aim of this paper is to describe the specific features of quia introducing complement clauses in Biblical Latin, compared with quod, and to investigate its particular link with ὅτι. To this end, I shall explore the contrasting ways of Jerome’s using quia in his Biblical translation, depending on whether the source language is Hebrew (Old Testament) or Greek (Gospels), and I shall compare the use of quia in the Vulgate and in the previous Vetus Latina.

This paper aims to study the asymptotic behaviour of the fundamental solutions (heat kernels) of non-local (partial and pseudo differential) equations with fractional operators in time and space. In particular, we obtain exact asymptotic formulas for the heat kernels of time-changed Brownian motions and Cauchy processes. As an application, we obtain exact asymptotic formulas for the fundamental solutions to the n-dimensional fractional heat equations in both time and space ∂ β ∂ t β u (t , x) = − (− Δ x) γ u (t , x) , β , γ ∈ (0 , 1). $$\begin{array}{} \displaystyle \frac{\partial^\beta}{\partial t^\beta}u(t,x) = -(-\Delta_x)^\gamma u(t,x), \quad \beta,\gamma\in(0,1). \end{array}$$ [ABSTRACT FROM AUTHOR]

We consider a network of infinite-server queues where the input process is a Cox process of the following form: The arrival rate is a vector-valued linear transform of a multivariate generalized (i.e., being driven by a subordinator rather than a compound Poisson process) shot-noise process. We first derive some distributional properties of the multivariate generalized shot-noise process. Then these are exploited to obtain the joint transform of the numbers of customers, at various time epochs, in a single infinite-server queue fed by the above-mentioned Cox process. We also obtain transforms pertaining to the joint stationary arrival rate and queue length processes (thus facilitating the analysis of the corresponding departure process), as well as their means and covariance structure. Finally, we extend to the setting of a network of infinite-server queues. [ABSTRACT FROM AUTHOR]

We develop a multivariate Lévy model and apply the bivariate model for the pricing of quanto options that captures three characteristics observed in real-world markets 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 Lévy process, exponential tilting subordinator. We refer to this model as a multivariate exponential tilting process. We then compare using a time series of daily log-returns and market prices of European-style quanto options the relative performance of the exponential tilting process to that of the Black-Scholes and the normal tempered stable process. We find that, due to more flexibility on capturing the information of tails and skewness, the proposed modeling process is superior to the other two processes for fitting market distribution and pricing quanto options. [ABSTRACT FROM AUTHOR]