Showing papers on "Subordinator published in 2015"
TL;DR: In this article, the authors take under consideration subordinators and their inverse processes (hitting-times) and present the governing equations of such processes by means of convolution-type integro-differential operators similar to the fractional derivatives.
Abstract: This paper takes under consideration subordinators and their inverse processes (hitting-times). The governing equations of such processes are presented by means of convolution-type integro-differential operators similar to the fractional derivatives. Furthermore the concept of time-changed C0-semigroup is discussed in case the time-change is performed by means of the hitting-time of a subordinator. Such time-change gives rise to bounded linear operators governed by integro-differential time-operators. Because these operators are non-local the presence of long-range dependence is investigated.
102 citations
01 Oct 2015
TL;DR: In this article, the authors studied the asymptotic behavior of the time-changed stochas-tic process f X(t) = B( f S(t)), where B is a standard one-dimensional Brow-nian motion and f S is the (generalized) inverse of a subordinator.
Abstract: We study the asymptotic behaviour of the time-changed stochas- tic process f X(t) = B( f S(t)), where B is a standard one-dimensional Brow- nian motion and f S is the (generalized) inverse of a subordinator, i.e. the first-passage time process corresponding to an increasing Lprocess with Laplace exponent f. This type of processes plays an important role in statis- tical physics in the modeling of anomalous subdiffusive dynamics. The main result of the paper is the proof of the mixing property for the sequence of stationary increments of a subdiffusion process. We also investigate various martingale properties, derive a generalized Feynman-Kac formula, the laws of large numbers and of the iterated logarithm for f X.
72 citations
TL;DR: In this paper, the first-exit time of a tempered β-stable subordinator, also called inverse tempered stable (ITS) subordinator was investigated and the limiting form of the ITS density and its k-th order derivatives were derived as the space variable x → 0 +.
64 citations
TL;DR: In this article, the authors consider the time-changed Ornstein-Uhlenbeck process, in which time is replaced by an inverse subordinator of general infinite divisible distribution.
Abstract: The Ornstein–Uhlenbeck process is one of the most popular systems used for financial data description However, this process has also been examined in the context of many other phenomena In this paper we consider the so-called time-changed Ornstein–Uhlenbeck process, in which time is replaced by an inverse subordinator of general infinite divisible distribution Time-changed processes nowadays play an important role in various fields of mathematical physics, chemistry, and biology as well as in finance In this paper we examine the main characteristics of the time-changed Ornstein–Uhlenbeck process, such as the covariance function Moreover, we also prove the formula for a generalized fractional Fokker–Planck equation that describes the one-dimensional probability density function of the analyzed system For three cases of subordinators we show the special forms of obtained general formulas Furthermore, we mention how to simulate the trajectory of the Ornstein–Uhlenbeck process delayed by a general inverse subordinator
44 citations
TL;DR: In this article, a generalization of the space-fractional Poisson process by extending the difference operator acting on state space present in the associated difference-differential equations to a much more general form is introduced.
Abstract: This paper introduces a generalization of the so-called space-fractional Poisson process by extending the difference operator acting on state space present in the associated difference-differential equations to a much more general form. It turns out that this generalization can be put in relation to a specific subordination of a homogeneous Poisson process by means of a subordinator for which it is possible to express the characterizing L\'evy measure explicitly. Moreover, the law of this subordinator solves a one-sided first-order differential equation in which a particular convolution-type integral operator appears, called Prabhakar derivative. In the last section of the paper, a similar model is introduced in which the Prabhakar derivative also acts in time. In this case, too, the probability generating function of the corresponding process and the probability distribution are determined.
41 citations
TL;DR: The fractional wave equation governs a stochastic model for wave propagation, with deterministic time replaced by the inverse of a stable subordinator whose index is one-half the order of the fractional time derivative.
Abstract: A fractional wave equation replaces the second time derivative by a Caputo derivative of order between one and two. In this paper, we show that the fractional wave equation governs a stochastic model for wave propagation, with deterministic time replaced by the inverse of a stable subordinator whose index is one half the order of the fractional time derivative.
39 citations
04 Aug 2015
TL;DR: In this paper, the renewal counting number process N = N(t) is considered as a forward march over the non-negative integers with independent identically distributed waiting times, and the Laplace transform with respect to both variables x and t is applied.
Abstract: We consider the renewal counting number process N = N(t) as a forward march over the non-negative integers with independent identically distributed waiting times. We embed the values of the counting numbers N in a “pseudo-spatial” non-negative half-line x ≥ 0 and observe that for physical time likewise we have t ≥ 0. Thus we apply the Laplace transform with respect to both variables x and t. Applying then a modification of the Montroll-Weiss-Cox formalism of continuous time random walk we obtain the essential characteristics of a renewal process in the transform domain and, if we are lucky, also in the physical domain. The process t = t(N) of accumulation of waiting times is inverse to the counting number process, in honour of the Danish mathematician and telecommunication engineer A.K. Erlang we call it the Erlang process. It yields the probability of exactly n renewal events in the interval (0; t]. We apply our Laplace-Laplace formalism to the fractional Poisson process whose waiting times are of Mittag-Leffler type and to a renewal process whose waiting times are of Wright type. The process of Mittag-Leffler type includes as a limiting case the classical Poisson process, the process of Wright type represents the discretized stable subordinator and a re-scaled version of it was used in our method of parametric subordination of time-space fractional diffusion processes. Properly rescaling the counting number process N(t) and the Erlang process t(N) yields as diffusion limits the inverse stable and the stable subordinator, respectively.
33 citations
TL;DR: The governing equation of the Tempered Stable Subordinator is derived, which generalizes the space-fractional differential equation satisfied by the law of the α-stable subordinator itself and is expressed in terms of the shifted fractional derivative of order α ?
31 citations
TL;DR: In this paper, the authors provided a systematic study on effectively approximating the Gerber-Shiu functions, which is a hardly touched topic in the current literature, by incorporating the recently popular Fourier-cosine method.
Abstract: In this article, we provide a systematic study on effectively approximating the Gerber–Shiu functions, which is a hardly touched topic in the current literature, by incorporating the recently popular Fourier-cosine method. Fourier-cosine method has been a prevailing numerical method in option pricing theory since the work of Fang and Oosterlee (2009). Our approximant of Gerber–Shiu functions under Levy subordinator model has O ( n ) computational complexity in comparison with that of O ( n log n ) via the fast Fourier transform algorithm. Also, for Gerber–Shiu functions within our proposed refined Sobolev space, we introduce an explicit error bound, which seems to be absent from the literature. In contrast with our previous work (Chau et al., 2015), this error bound is more conservative without making heavy assumptions on the Fourier transform of the Gerber–Shiu function. The effectiveness of our result will be further demonstrated in the numerical studies.
27 citations
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TL;DR: In this paper, the transition probabilities of a class of subordinator motions with Laplace exponents that vary regularly at infinity with index one with respect to the Laplace exponent of the corresponding subordinator are established.
Abstract: In this article we study transition probabilities of a class of subordinate Brownian motions. Under mild assumptions on the Laplace exponent of the corresponding subordinator, sharp two sided estimates of the transition probability are established. This approach, in particular, covers subordinators with Laplace exponents that vary regularly at infinity with index one, e.g. \[
\phi(\lambda)=\frac{\lambda}{\log(1+\lambda)}-1 \quad \text{ or }\quad \phi(\lambda)=\frac{\lambda}{\log(1+\lambda^{\beta/2})},\ \beta\in (0,2)\, \] that correspond to subordinate Brownian motions with scaling order that is not necessarily strictly between 0 and 2. These estimates are applied to estimate Green function (potential) of subordinate Brownian motion. We also prove the equivalence of the lower scaling condition of the Laplace exponent and the near diagonal upper estimate of the transition estimate.
26 citations
TL;DR: In this paper, the authors studied the convergence rate of a Markov process to its invariant measure with a subordinator and the corresponding Bernstein function and showed that subordination can dramatically change the speed of convergence to equilibrium.
Abstract: We are interested in the rate of convergence of a subordinate Markov process to its invariant measure. Given a subordinator and the corresponding Bernstein function (Laplace exponent) we characterize the convergence rate of the subordinate Markov process; the key ingredients are the rate of convergence of the original process and the (inverse of the) Bernstein function. At a technical level, the crucial point is to bound three types of moments (sub-exponential, algebraic and logarithmic) for subordinators as time $t$ tends to infinity. At the end we discuss some concrete models and we show that subordination can dramatically change the speed of convergence to equilibrium.
TL;DR: In this paper, a fractional generalization of the well-known Gamma process is studied, in which the corresponding densities satisfy the same differential equation as the usual Gamma process, but with the shift operator replaced by its fractional version of order ν > 0.
Abstract: We introduce and study fractional generalizations of the well-known Gamma process, in the following sense: the corresponding densities are proved to satisfy the same differential equation as the usual Gamma process, but with the shift operator replaced by its fractional version of order ν > 0 In the case ν > 1, the solution corresponds to the density of a Gamma process time-changed by an independent stable subordinator of index 1/ν For ν less than one an analogous result holds, with the subordinator replaced by the inverse In this case the fractional Gamma process is proved to be a non-stationary version of the standard one, with power law behavior of the expected value Hence it can be considered a useful tool in modelling stochastic deterioration in the non-linear cases, a situation which often occurs in real data (see ie, [42] and the references therein)As a consequence of the previous results, the fractional generalizations of some Gamma subordinated processes (ie the Variance Gamma, the Geome
TL;DR: In this article, a micro-scale model is proposed for the evolution of the limit order book, where the flows of orders (claims) are described by doubly stochastic Poisson processes taking account of the stochastically character of intensities of bid and ask orders that determine the price discovery mechanism in financial markets.
TL;DR: The detailed structure of the scaling limits of such multidimensional Levy walks with power-law dependence between waiting times and jumps is obtained and it is concluded that the resulting limiting processes belong to sub-, quasi- and superdiffusion regimes.
TL;DR: In this paper, a generalized fractional version of the difference-differential equation governing the classical nonlinear birth process was studied, which was obtained by adding in the equation some extra terms, which made the expression of its solution much more complicated.
Abstract: We consider here generalized fractional versions of the difference-differential equation governing the classical nonlinear birth process. Orsingher and Polito (Bernoulli 16(3):858–881, 2010) defined a fractional birth process by replacing, in its governing equation, the first order time derivative with the Caputo fractional derivative of order υ ∈ (0, 1]. We study here a further generalization, obtained by adding in the equation some extra terms; as we shall see, this makes the expression of its solution much more complicated. Moreover we consider also the case υ ∈ (1, +∞ ), as well as υ ∈ (0, 1], using correspondingly two different definitions of fractional derivative: we apply the fractional Caputo derivative and the right-sided fractional Riemann–Liouville derivative on ℝ+, for υ ∈ (0, 1] and υ ∈ (1, +∞ ), respectively. For the two cases, we obtain the exact solutions and prove that they coincide with the distribution of some subordinated stochastic processes, whose random time argument is represented by a stable subordinator (for υ ∈ (1, +∞ )) or its inverse (for υ ∈ (0, 1]).
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TL;DR: In this paper, the authors study the Metropolis dynamics of the simplest mean-field spin glass model, the Random Energy Model, and show that this dynamics exhibits aging by showing that the properly rescaled time change process between the metropolis dynamics and a suitably chosen ''fast' Markov chain converges in distribution to a stable subordinator.
Abstract: We study the Metropolis dynamics of the simplest mean-field spin glass model, the Random Energy Model. We show that this dynamics exhibits aging by showing that the properly rescaled time change process between the Metropolis dynamics and a suitably chosen `fast' Markov chain converges in distribution to a stable subordinator. The rescaling might depend on the realization of the environment, but we show that its exponential growth rate is deterministic.
TL;DR: In this article, the notion of a Markovian embedding of a fragmenter in an exchangeable fragmentation process was introduced and a symmetrisation operation was investigated in a general study of pairs that coincide up to a junction time and then evolve independently.
Abstract: Some, but not all processes of the form Mt = exp(−ξt) for a pure-jump subordinator ξ with Laplace exponentarise as residual mass processes of particle 1 (tagged particle) in an exchangeable fragmentation processes. We introduce the notion of a Markovian embedding of M in an exchangeable fragmentation process and show that for each �, there is a unique binary dislocation measure ν such that M has a Markovian embedding in an associated exchangeable fragmentation process. The identification of the Laplace exponent � � of its tagged particle process Mgives rise to a symmetrisation operation � 7→� � , which we investigate in a general study of pairs (M,M � ) that coincide up to a junction time and then evolve independently. We call M a fragmenter and (M,M � ) a bifurcator. For alland α > 0, we can represent a fragmenter M as an interval R1 = (0, R 1 0 M � t dt) equipped with a purely atomic probability measure µ1 capturing the jump sizes of Mt after an α-self-similar time-change. We call (R1,µ1) an (α,�)-string of beads. We study binary tree growth processes that in the nth step sample a bead from µn and build (Rn+1,µn+1) by splitting the bead into a new string of beads, a rescaled independent copy of (R1,µ1) that we tie to the position of the sampled bead. We show that all such bead splitting processes converge almost surely to an α-self-similar CRT, in the Gromov-Hausdorff-Prohorov sense. AMS 2000 subject classifications: 60J80.
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TL;DR: In this article, the authors unify and extend a number of approaches related to constructing multivariate Variance-Gamma (V.G.) models for option pricing, and derive an overarching model by subordinating multivariate Brownian motion to a subordinator from the Thorin (1977) class of generalised Gamma convolution subordinators.
Abstract: We unify and extend a number of approaches related to constructing multivariate Variance-Gamma (V.G.) models for option pricing. An overarching model is derived by subordinating multivariate Brownian motion to a subordinator from the Thorin (1977) class of generalised Gamma convolution subordinators. A class of models due to Grigelionis (2007), which contains the well-known Madan-Seneta V.G. model, is of this type, but our multivariate generalization is considerably wider, allowing in particular for processes with infinite variation and a variety of dependencies between the underlying processes. Multivariate classes developed by Perez-Abreu and Stelzer (2012) and Semeraro (2008) and Guillaume (2013) are also submodels. The new models are shown to be invariant under Esscher transforms, and quite explicit expressions for canonical measures (and transition densities in some cases) are obtained, which permit applications such as option pricing using PIDEs or tree based methodologies. We illustrate with best-of and worst-of European and American options on two assets.
TL;DR: In this paper, the first-crossing-time problem for the iterated Poisson process is finally tackled in the cases of (i) a decreasing and constant boundary, where they provide some closed-form results, and (ii) a linearly increasing boundary.
Abstract: A compound Poisson process whose randomized time is an independent Poisson process is called compound Poisson process with Poisson subordinator. We provide its probability distribution, which is expressed in terms of the Bell polynomials, and investigate in detail both the special cases in which the compound Poisson process has exponential jumps and normal jumps. Then for the iterated Poisson process we discuss some properties and provide convergence results to a Poisson process. The first-crossing-time problem for the iterated Poisson process is finally tackled in the cases of (i) a decreasing and constant boundary, where we provide some closed-form results, and (ii) a linearly increasing boundary, where we propose an iterative procedure to compute the first-crossing-time density and survival functions.
TL;DR: In this article, the scaling limit of a simple random walk on Z d is shown to be either Brownian motion or the fractional Kinetics process, as conjectured in Ben Arous et al.
TL;DR: In this article, a multistable subordinator is introduced to generalize the stable subordinator to the case of time-varying stability index, which enables the convergence of a continuous-time random walk to the multifractional Poisson process.
TL;DR: Using inverse subordinators and Mittag-Leffler functions, this paper presented a new definition of a fractional Poisson process parametrized by points of the Euclidean space.
Abstract: Using inverse subordinators and Mittag-Leffler functions, we present a new definition of a fractional Poisson process parametrized by points of the Euclidean space $\mathbb{R}_+^2$
. Some properties are given and, in particular, we prove a long-range dependence property.
TL;DR: In this article, the authors presented new results on Hunt's hypothesis (H) for Levy processes and obtained an extended Kanda-Forst-Rao theorem for the case of Levy processes.
Abstract: In this paper, we present new results on Hunt’s hypothesis (H) for Levy processes. We start with a comparison result on Levy processes which implies that big jumps have no effect on the validity of (H). Based on this result and the Kanda-Forst-Rao theorem, we give examples of subordinators satisfying (H). Afterwards we give a new necessary and sufficient condition for (H) and obtain an extended Kanda-Forst-Rao theorem. By virtue of this theorem, we give a new class of Levy processes satisfying (H). Finally, we construct a type of subordinators that does not satisfy Rao’s condition.
TL;DR: Theorem 25 in Section 6 describes the asymptotic distribution of the last few terms in a finite i.i.d. size-biased permutation via a Poisson coupling with its few smallest order statistics.
Abstract: This paper focuses on the size-biased permutation of $n$ independent and identically distributed (i.i.d.) positive random variables. This is a finite dimensional analogue of the size-biased permutation of ranked jumps of a subordinator studied in Perman–Pitman–Yor (PPY) [ Probab. Theory Related Fields 92 (1992) 21–39], as well as a special form of induced order statistics [ Bull. Inst. Internat. Statist. 45 (1973) 295–300; Ann. Statist. 2 (1974) 1034–1039]. This intersection grants us different tools for deriving distributional properties. Their comparisons lead to new results, as well as simpler proofs of existing ones. Our main contribution, Theorem 25 in Section 6, describes the asymptotic distribution of the last few terms in a finite i.i.d. size-biased permutation via a Poisson coupling with its few smallest order statistics.
TL;DR: In this paper, a class of regenerative sets R, indexed by all measurable functions α : [ 0, 1] → [0, 1], has the law of the range of a special subordinator.
Abstract: We construct, on a single probability space, a class of regenerative sets R, indexed by all measurable functions α : [0, 1] → [0, 1]. For each function α, R, has the law of the range of a special subordinator. Constant functions correspond to stable subordinators. If α ≤ β, then R ⊂ R. Other examples of special subordinators are given in the lattice case.
TL;DR: In this article, the mathematical technique known as subordination is used to consider transient flow and obtain pressure distributions in a horizontal well under subdiffusion, which broadens the scope of existing solutions to consider a variety of bounded systems.
Abstract: Flow to a horizontal well under subdiffusion is considered. The mathematical technique known as subordination is shown to provide a succinct method to consider transient flow and obtain pressure distributions. Subordination broadens the scope of existing solutions to consider a variety of problems of interest, particularly bounded systems.
TL;DR: In this article, the authors introduce a class of stochastic volatility models (X t ) t ≥ 0 for which the absolute moments of the increments exhibit anomalous scaling: E ( ∣ X t + h − X t ∣ q ) scales as h q / 2 for q q ∗, but as h A ( q ) with A q ) q/2 for q > q ∆, for some threshold q ∐.
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TL;DR: The running infimum of a Levy process relative to its point of issue is known to have the same range that of the negative of a certain subordinator as mentioned in this paper, and conditioning the Levy process issued from a strictly positive value to stay positive may therefore be seen as implicitly conditioning its descending ladder heigh subordinator to remain in a strip.
Abstract: The running infimum of a Levy process relative to its point of issue is know to have the same range that of the negative of a certain subordinator. Conditioning a Levy process issued from a strictly positive value to stay positive may therefore be seen as implicitly conditioning its descending ladder heigh subordinator to remain in a strip. Motivated by this observation, we consider the general problem of conditioning a subordinator to remain in a strip. Thereafter we consider more general contexts in which subordinators embedded in the path decompositions of Markov processes are conditioned to remain in a strip.
TL;DR: In this paper, a normal inverse Gaussian Levy process in Hilbert space is defined and conditions for integrability and martingale properties are derived under various assumptions of the Levy process and subordinator.
Abstract: We investigate multivariate subordination of Levy processes which was first introduced by Barndorff-Nielsen et al. [O.E. Barndorff-Nielsen, F.E. Benth, and A. Veraart, Modelling electricity forward markets by ambit fields, J. Adv. Appl. Probab. (2010)], in a Hilbert space valued setting which has been introduced in Perez-Abreu and Rocha-Arteaga [V. Perez-Abreu and A. Rocha-Arteaga, Covariance-parameter Levy processes in the space of trace-class operators, Infin. Dimens. Anal. Quantum Probab. Related Top. 8(1) (2005), pp. 33–54]. The processes are explicitly characterized and conditions for integrability and martingale properties are derived under various assumptions of the Levy process and subordinator. As an application of our theory we construct explicitly some Hilbert space valued versions of Levy processes which are popular in the univariate and multivariate case. In particular, we define a normal inverse Gaussian Levy process in Hilbert space. The resulting process has the property that at each time ...
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TL;DR: In this article, the authors established small-ball probabilities for a class of time-changed processes, where the inner process is a self-similar process and the outer process is an independent continuous process, each with a certain small ball probability.
Abstract: This paper establishes small ball probabilities for a class of time-changed processes $X\circ E$, where $X$ is a self-similar process and $E$ is an independent continuous process, each with a certain small ball probability. In particular, examples of the outer process $X$ and the time change $E$ include an iterated fractional Brownian motion and the inverse of a general subordinator with infinite L\'evy measure, respectively. The small ball probabilities of such time-changed processes show power law decay, and the rate of decay does not depend on the small deviation order of the outer process $X$, but on the self-similarity index of $X$.