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Showing papers on "Subordinator published in 2022"


Journal ArticleDOI
11 Jan 2022
TL;DR: In this paper , the authors developed a theoretical framework for search with home returns in the case of subdiffusion and made a realistic description of restart by accounting for random walks with random stops.
Abstract: Stochastic resetting with home returns is widely found in various manifestations in life and nature. Using the solution to the home return problem in terms of the solution to the corresponding problem without home returns (Pal et al 2020 Phys. Rev. Res. 2 043174), we develop a theoretical framework for search with home returns in the case of subdiffusion. This makes a realistic description of restart by accounting for random walks with random stops. The model considers stochastic processes, arising from Brownian motion subordinated by an inverse infinitely divisible process (subordinator).

6 citations


Journal ArticleDOI
TL;DR: In this paper, the convergence behavior of α → 2 of the solutions to stochastic differential equations (SDEs) with Lipschitz and Holder drifts, and driven by α-stable Levy processes is investigated.

5 citations


Journal ArticleDOI
TL;DR: The proposed fractional self-exciting model for the risk of corporate default uses the inverse of an α -stable subordinator as a time-change to incorporate two particular features in the survival probability curves implied by the model.

4 citations


Journal ArticleDOI
TL;DR: In this paper , the time value of ruin in a pure jump Lévy risk model is estimated by the Fourier cosine method, and the uniform convergence rate is also derived.
Abstract: In this paper, we consider statistical estimation of the time value of ruin in a Lévy risk model. Suppose that the aggregate claims process of an insurance company is modeled by a pure jump Lévy subordinator, and we can observe the data set on the aggregate claims based on low-frequency sampling. The time value of ruin is estimated by the Fourier-cosine method, and the uniform convergence rate is also derived. Through a lot of simulation studies, we show that our estimators are very effective when the sample size is finite.

4 citations


Journal ArticleDOI
TL;DR: In this paper , a fractional self-exciting model for the risk of corporate default is proposed, where the inverse of an α-stable subordinator is used as a time-changed version of an intensity-based model.

4 citations


Journal ArticleDOI
TL;DR: In this paper , a Langevin system with a random diffusivity and a stable subordinator was considered and the analytic expressions of ergodicity breaking parameter and probability density function of the time-averaged mean-squared displacement were derived.
Abstract: A Brownian yet non-Gaussian phenomenon has recently been observed in many biological and active matter systems. The main idea of explaining this phenomenon is to introduce a random diffusivity for particles moving in inhomogeneous environment. This paper considers a Langevin system containing a random diffusivity and an $\ensuremath{\alpha}$-stable subordinator with $\ensuremath{\alpha}<1$. This model describes the particle's motion in complex media where both the long trapping events and random diffusivity exist. We derive the general expressions of ensemble- and time-averaged mean-squared displacements which only contain the values of the inverse subordinator and diffusivity. Further taking specific time-dependent diffusivity, we obtain the analytic expressions of ergodicity breaking parameter and probability density function of the time-averaged mean-squared displacement. The results imply the nonergodicity of the random diffusivity model with any kind of diffusivity, including the critical case where the model presents normal diffusion.

3 citations



Journal ArticleDOI
TL;DR: The authors identify just so as a newly emerging purpose subordinator and use it to represent a case of semantic specialization where the purpose meaning wins out over the conditional meaning, thus filling the niche of informal purpose subordinators and providing an alternative to its multifunctional and semantically ambiguous competitors.
Abstract: This article identifies just so as a newly emerging purpose subordinator. Using data from the Corpus of Contemporary American English and the Corpus of Historical American English , it traces its development and steady increase in frequency from its first attestation in the mid nineteenth century to the present day. Just so is shown to represent a case of semantic specialization where the purpose meaning wins out over the conditional meaning, thus filling the niche of an informal purpose subordinator and providing an alternative to its multifunctional and semantically ambiguous competitors so that and so . With increasing grammaticalization the just so purpose subordinator also exhibits signs of intersubjectification, being coopted for syntactically independent, interpersonal uses (e.g. just so we're clear ) and culminating in the emergence of a new discourse marker in the form of just so you know in the late twentieth and early twenty-first century. To account for the emergence of purpose just so , a constructional network approach is adopted, which considers the network links to other purpose subordinators, notably so that and so .

2 citations


Journal ArticleDOI
TL;DR: In this article , Dong, Goldschmidt and Martin showed that repeated application of independent single-block fragmentation operators based on mass partitions following a two-parameter Poisson-Dirichlet distribution with parameters (α, 1−α) to a mass partition having a Poisson−DirichLET distribution with parameter (α , θ+r) leads to a remarkable nested family of distributed mass partitions with parameters r = 0, 1, 2,⋯.
Abstract: Dong, Goldschmidt and Martin (2006) (DGM) showed that, for 0−α, the repeated application of independent single-block fragmentation operators based on mass partitions following a two-parameter Poisson–Dirichlet distribution with parameters (α,1−α) to a mass partition having a Poisson–Dirichlet distribution with parameters (α,θ) leads to a remarkable nested family of Poisson—Dirichlet distributed mass partitions with parameters (α,θ+r) for r=0,1,2,⋯. Furthermore, these generate a Markovian sequence of α-diversities following Mittag-Leffler distributions, whose ratios lead to independent Beta-distributed variables. These Markov chains are referred to as Mittag-Leffler Markov chains and arise in the broader literature involving Pólya urn and random tree/graph growth models. Here we obtain explicit descriptions of properties of these processes when conditioned on a mixed Poisson process when it equates to an integer n, which has interpretations in a species sampling context. This is equivalent to obtaining properties of the fragmentation operations of (DGM) when applied to mass partitions formed by the normalized jumps of a generalized gamma subordinator and its generalizations. We focus primarily on the case where n=0,1.

2 citations


Journal ArticleDOI
TL;DR: In this paper , the authors studied the probability that a bivariate subordinator (Y,Z) issued from 0 creeps through a graph in terms of its renewal function and the drifts of the components $Y$ and $Z$.
Abstract: A L\'evy process is said to creep through a curve if, at its first passage time across this curve, the process reaches it with positive probability. We first study this property for bivariate subordinators. Given the graph $\{(t,f(t)):t\ge0\}$ of any continuous, non increasing function $f$ such that $f(0)>0$, we give an expression of the probability that a bivariate subordinator $(Y,Z)$ issued from 0 creeps through this graph in terms of its renewal function and the drifts of the components $Y$ and $Z$. We apply this result to the creeping probability of any real L\'evy process through the graph of any continuous, non increasing function at a time where the process also reaches its past supremum. This probability involves the density of the renewal function of the bivariate upward ladder process as well as its drift coefficients. We also investigate the case of L\'evy processes conditioned to stay positive creeping at their last passage time below the graph of a function. Then we provide some examples and we give an application to the probability of creeping through fixed levels by stable Ornstein-Uhlenbeck processes. We also raise a couple of open questions along the text.

1 citations


Journal ArticleDOI
TL;DR: In this article , the authors studied complex spatial diffusion equations with time-fractional derivative and studied their stochastic solutions, in particular, the integral operator solution to the heat-type equation where the time derivative is replaced with the convolution-type generalization of the regularized Caputo derivative.
Abstract: Abstract We deal with complex spatial diffusion equations with time-fractional derivative and study their stochastic solutions. In particular, we complexify the integral operator solution to the heat-type equation where the time derivative is replaced with the convolution-type generalization of the regularized Caputo derivative. We prove that this operator is solution of a complex time-fractional heat equation with complex spatial variable. This approach leads to a wrapped Brownian motion on a circle time-changed by the inverse of the related subordinator. This time-changed Brownian motion is analyzed and, in particular, some results on its moments, as well as its construction as weak limit of continuous-time random walks, are obtained. The extension of our approach to the higher dimensional case is also provided.

Journal ArticleDOI
TL;DR: In this paper , a generalized fractional Ornstein-Uhlenbeck process driven by a Lévy subordinator and an independent sinusoidal-composite lévy process is proposed for volatility derivatives, which allows the characteristic function of average forward variance to be obtainable in semiclosed form.
Abstract: This paper proposes a novel analytical pricing–hedging framework for volatility derivatives which simultaneously takes into account rough volatility and volatility jumps. Directly targeting the instantaneous variance of a risky asset, our model consists of a generalized fractional Ornstein–Uhlenbeck process driven by a Lévy subordinator and an independent sinusoidal-composite Lévy process, and allows the characteristic function of average forward variance to be obtainable in semiclosed form, without having to invoke any geometric-mean approximations. Pricing–hedging formulae are proposed for a general class of power-type derivatives, in the spirit of numerical Fourier transform. A comparative empirical study is conducted on two independent recent data sets on Volatility Index options, before and during the COVID-19 pandemic, to demonstrate that the proposed framework is highly amenable to efficient model calibration under various choices of kernels. The price dynamics of the underlying asset can be readily considered and the possibility of studying rough volatility of volatility is given as well.

Journal ArticleDOI
TL;DR: In this article , a fractional time-changed stochastic risk model was considered for the infinite-horizon ruin probability, a specialized version of the Gerber-Shiu function.
Abstract: The paper deals with a fractional time-changed stochastic risk model, including stochastic premiums, dividends and also a stochastic initial surplus as a capital derived from a previous investment. The inverse of a ν-stable subordinator is used for the time-change. The submartingale property is assumed to guarantee the net-profit condition. The long-range dependence behavior is proven. The infinite-horizon ruin probability, a specialized version of the Gerber–Shiu function, is considered and investigated. In particular, we prove that the distribution function of the infinite-horizon ruin time satisfies an integral-differential equation. The case of the dividends paid according to a multi-layer dividend strategy is also considered.

Journal ArticleDOI
TL;DR: In this paper , a power-law vector random field whose finite-dimensional characteristic functions consist merely of a power function or the ratio of two power functions is proposed, which can be used as the building blocks to construct other Lévy processes (random fields).
Abstract: 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).

Journal ArticleDOI
TL;DR: In this article , the moments of the inverse gamma subordinator and the governing equations associated with Gamma subordinators and inverse processes are given in terms of higher transcendental functions, also known as Volterra functions.
Abstract: In this paper we deal with some open problems concerned with Gamma subordinators. In particular, we first provide a representation for the moments of the inverse gamma subordinator. Then, we focus on λ-potentials and we study the governing equations associated with Gamma subordinators and inverse processes. Such representations are given in terms of higher transcendental functions, also known as Volterra functions.

Journal ArticleDOI
TL;DR: In this article , the authors derived a probabilistic representation of the density of the anomalous process under the Feynman-Kac transform of non-Markov processes.

Journal ArticleDOI
TL;DR: In this paper , the authors studied the large-time and small-time asymptotic properties of the spectral heat content for time-changed stable processes, where the time change belongs to a large class of inverse subordinators.
Abstract: We study the large-time and small-time asymptotic behaviors of the spectral heat content for time-changed stable processes, where the time change belongs to a large class of inverse subordinators. For the large-time behavior, the spectral heat content decays polynomially with the decay rate determined by the Laplace exponent of the underlying subordinator, which is in sharp contrast to the exponential decay observed in the case when the time change is a subordinator. On the other hand, the small-time behavior exhibits three different decay regimes, where the decay rate is determined by both the Laplace exponent and the index of the stable process.

Journal ArticleDOI
01 Feb 2022-Lingua
TL;DR: The authors explored contextual factors around these two causal markers, incorporating the effective factors in multifactorial models to measure their weights in discriminating between these two markers, and examined the different weights borne by individual factors from the perspective of markedness correspondence strategies.

Journal ArticleDOI
TL;DR: In this paper , the spectral heat content for time-changed killed Brownian motions on open sets was investigated for either a subordinator or an inverse subordinator, with the underlying Laplace exponent being regularly varying at $$\infty $$ with index $$\beta \in (0,1)$$ .
Abstract: 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 $$\infty $$ with index $$\beta \in (0,1)$$ . 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 $$\beta \in (0,1)$$ . In contrast, in the case of subordinators, this universality holds only when $$\beta \in (\frac{1}{2}, 1)$$ .

Proceedings ArticleDOI
TL;DR: This paper studied syntactic, pragmatic and prosodic features of insubordinated clauses introduced by the adverbial subordinator potomu čto ‘because' and found that more than 30% of reason clauses in spoken discourse appear to be inattention.
Abstract: Based on data from the multimedia subcorpus of the Russian National Corpus, the paper addresses syntactic, pragmatic and prosodic features of insubordinated adverbial clauses introduced by the adverbial subordinator potomu čto ‘because’. The quantitative analysis showed that more than 30% of reason clauses in spoken discourse appear to be insubordinated. Qualitatively, we observed symptoms of insubordination at various levels. (1) Prosodically, insubordinated clauses are placed after discourse fragments that are articulated with falling pitch projecting no continuation and are separated from them by the prosodic break. (2) Pragmatically, they can have independent illocutionary force and can form separate turns in dialogues. (3) Grammatically, they allow right dislocation of the adverbial subordinator – otherwise blocked in adverbial clauses.

Posted ContentDOI
30 Apr 2022
TL;DR: In this paper , the authors proposed a significant improvement of stochastic clocks for the same objective but without decreasing the number of trades or changing the trading intensity, based on various choices of regulating kernels motivated from repeated averaging.
Abstract: Stochastic clocks represent a class of time change methods for incorporating trading activity into continuous-time financial models, with the ability to deal with typical asymmetrical and tail risks in financial returns. In this paper we propose a significant improvement of stochastic clocks for the same objective but without decreasing the number of trades or changing the trading intensity. Our methodology targets any L\'{e}vy subordinator, or more generally any process of nonnegative independent increments, and is based on various choices of regulating kernels motivated from repeated averaging. By way of a hyperparameter linked to the degree of regulation, arbitrarily large skewness and excess kurtosis of returns can be easily achieved. Generic-time Laplace transforms, characterizing triplets, and cumulants of the regulated clocks and subsequent mixed models are analyzed, serving purposes ranging from statistical estimation and option price calibration to simulation techniques. Under specified jump--diffusion processes and tempered stable processes, a robust moment-based estimation procedure with profile likelihood is developed and a comprehensive empirical study involving S\&P500 and Bitcoin daily returns is conducted to demonstrate a series of desirable effects of the proposed methods.

Journal ArticleDOI
TL;DR: In this paper , a stochastic model for the failure times of items subject to two external random shocks occurring as events in an underlying bivariate counting process is proposed, where the two kinds of shocks occur according to a bivariate space-fractional Poisson process.
Abstract: Abstract We 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.

Journal ArticleDOI
TL;DR: In this article , the generalized fractional Skellam process (GFSP) is considered by time-changing it with an independent inverse stable subordinator, and it is observed that the GFSP is a Skellham type version of the generalized fractional counting process (GFCP), which is a fractional variant of the GCP.
Abstract: 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.

Journal ArticleDOI
TL;DR: In this article , a Dirichlet distribution with expected values given by the option pricing model is used to test the consistency of risk neutral return distributions with a Brownian motion with drift time changed by a subordinator.
Abstract: At each maturity a discrete return distribution is inferred from option prices. Option pricing models imply a comparable theoretical distribution. As both the transformed data and the option pricing model deliver points on a simplex, the data is statistically modeled by a Dirichlet distribution with expected values given by the option pricing model. The resulting setup allows for maximum likelihood estimation of option pricing model parameters with standard errors that enable the testing of hypotheses. Hypothesis testing is then illustrated by testing for the consistency of risk neutral return distributions being those of a Brownian motion with drift time changed by a subordinator. Models mixing processes of independent increments with processes related to solutions of Ornstein–Uhlenbeck (OU) equations are also tested for the presence of the OU component. Solutions to OU equations may be viewed as processes of perpetual motion responding continuously to their past movements. The tests support the rejection of Brownian subordination and the presence of a perpetual motion component.

Journal ArticleDOI
TL;DR: In this paper , the authors assume the existence of some anticipative information in a market whose risky asset dynamics evolve according to a Brownian motion and a Poisson process, and derive the information drift of the mentioned processes and, both in the pure jump case and in the mixed one, compute the additional expected logarithmic utility.
Abstract: The anticipative information refers to some information about future events that may be disclosed in advance. This information may regard, for example, financial assets and their future trends. In our paper, we assume the existence of some anticipative information in a market whose risky asset dynamics evolve according to a Brownian motion and a Poisson process. Using Malliavin calculus and filtration enlargement techniques, we derive the information drift of the mentioned processes and, both in the pure jump case and in the mixed one, we compute the additional expected logarithmic utility. Many examples are shown, where the anticipative information is related to some conditions that the constituent processes or their running maximum may verify, in particular, we show new examples considering Bernoulli random variables.

Book ChapterDOI
01 Jan 2022
TL;DR: In this paper , it was shown that the mean return time in a renewal process with interarrival times is bounded by a factor of 2 when the Lévy exponent of the associated subordinator is a special Bernstein function.
Abstract: Lorden’s inequality asserts that the mean return time in a renewal process with (iid) interarrival times $$Y_1, Y_2,\ldots $$ , is bounded above by $$2\textbf{E}[Y_1]/\textbf{E}[Y_1^2]$$ . We establish this result in the context of regenerative sets, and remove the factor of 2 when the regenerative set enjoys a certain monotonicity property. This property occurs precisely when the Lévy exponent of the associated subordinator is a special Bernstein function. Several equivalent stochastic monotonicity properties of such a regenerative set are demonstrated.

Journal ArticleDOI
TL;DR: The authors examines verbal sequences in Arabic dialects which can correspond either to complex sentences with embedded clauses or to complex predicates with reduction of one or the other verb with a marker that is generally specific, but sometimes polyfunctional.
Abstract: Abstract This article examines verbal sequences in Arabic dialects which can correspond either to complex sentences with embedded clauses or to complex predicates with reduction of one or the other verb. The first part is devoted to complex sentences where completives and subordinates of purpose and consequence are introduced by a marker that is generally specific, but sometimes polyfunctional. The second part explores embedding without a subordinator (with distinct or identical subjects), as well as with cases of complex predicates, sometimes with reduction of V1 (cases of auxiliarization and pragmatization), sometimes of V2 (rare cases of serial verbs).

Journal ArticleDOI
TL;DR: In this paper , a class of multivariate tempered stable distributions and associated class of tempered stable Sato subordinators are used to build additive inhomogeneous processes by subordination of a multiparameter Brownian motion.
Abstract: Abstract 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.

Posted ContentDOI
01 Nov 2022
TL;DR: In this article , the authors considered a class of random fields, known as multiparameter L\'evy processes, where related multiparameter semigroups of operators and their generators are represented as pseudo-differential operators.
Abstract: Let $\mathbb{R}^N_+= [0,\infty)^N$. We here consider a class of random fields $(X_t)_{t\in \mathbb{R}^N_+}$ which are known as Multiparameter L\'evy processes. Related multiparameter semigroups of operators and their generators are represented as pseudo-differential operators. We also consider the composition of $(X_t)_{t\in \mathbb{R}^N_+}$ by means of the so-called subordinator fields and we provide a Phillips formula. We finally study the composition of $(X_t)_{t\in \mathbb{R}^N_+}$ by means of the so-called inverse random fields, which gives rise to interesting long range dependence properties. As a byproduct of our analysis, we study a model of anomalous diffusion in an anisotropic medium which extends the one treated in [8].

Posted ContentDOI
17 May 2022
TL;DR: In this article , the authors studied the large-time and small-time asymptotic behaviors of the spectral heat content for time-changed stable processes, where the time change belongs to a large class of inverse subordinators.
Abstract: We study the large-time and small-time asymptotic behaviors of the spectral heat content for time-changed stable processes, where the time change belongs to a large class of inverse subordinators. For the large-time behavior, the spectral heat content decays polynomially with the decay rate determined by the Laplace exponent of the underlying subordinator, which is in sharp contrast to the exponential decay observed in the case when the time change is a subordinator. On the other hand, the small-time behavior exhibits three different decay regimes, where the decay rate is determined by both the Laplace exponent and the index of the stable process.