Showing papers on "Subordinator published in 2011"

The Fractional Poisson Process and the Inverse Stable Subordinator

Abstract: The fractional Poisson process is a renewal process with Mittag-Leffler waiting times. Its distributions solve a time-fractional analogue of the Kolmogorov forward equation for a Poisson process. This paper shows that a traditional Poisson process, with the time variable replaced by an independent inverse stable subordinator, is also a fractional Poisson process. This result unifies the two main approaches in the stochastic theory of time-fractional diffusion equations. The equivalence extends to a broad class of renewal processes that include models for tempered fractional diffusion, and distributed-order (e.g., ultraslow) fractional diffusion. The paper also {discusses the relation between} the fractional Poisson process and Brownian time.

Fokker-Planck-Kolmogorov equations associated with time-changed fractional Brownian motion

Abstract: In this paper Fokker-Planck-Kolmogorov type equations associated with stochastic differential equations driven by a time-changed fractional Brownian motion are derived. Two equivalent forms are suggested. The time-change process considered is the first hitting time process for either a stable subordinator or a mixture of stable subordinators. A family of operators arising in the representation of the Fokker-Plank-Kolmogorov equations is shown to have the semigroup property.

Convergence to fractional kinetics for random walks associated with unbounded conductances

Abstract: We consider a random walk among unbounded random conductances whose distribution has infinite expectation and polynomial tail. We prove that the scaling limit of this process is a Fractional-Kinetics process—that is the time change of a d-dimensional Brownian motion by the inverse of an independent α-stable subordinator. We further show that the same process appears in the scaling limit of the non-symmetric Bouchaud’s trap model.

The Fractional Poisson Process and the Inverse Stable Subordinator

Abstract: The fractional Poisson process is a renewal process with Mittag-Leffler waiting times. Its distributions solve a time-fractional analogue of the Kolmogorov forward equation for a Poisson process. This paper shows that a traditional Poisson process, with the time variable replaced by an independent inverse stable subordinator, is also a fractional Poisson process. This result unifies the two main approaches in the stochastic theory of time-fractional diffusion equations. The equivalence extends to a broad class of renewal processes that include models for tempered fractional diffusion, and distributed-order (e.g., ultraslow) fractional diffusion. The paper also {discusses the relation between} the fractional Poisson process and Brownian time.

Reparameterizing Marshall–Olkin copulas with applications to sampling

Abstract: It is shown that exchangeable Marshall–Olkin survival copulas coincide with a parametric family of copulas studied in [J.-F. Mai and M. Scherer, Levy-Frailty copulas, J. Multivariate Anal. 100 (2009), pp. 1567–1585]. This observation implies an alternative probabilistic interpretation in many cases and allows the transfer of known results from one family to the other. For instance, using the classical construction of [A.W. Marshall and I. Olkin, A multivariate exponential distribution, J. Am. Stat. Assoc. 62 (1967), pp. 30–44], sampling an n-dimensional Marshall–Olkin copula requires 2 n −1 exponentially distributed random variables, which is inefficient in large dimensions. Applying the alternative construction, sampling an exchangeable n-dimensional copula boils down to generating n independent exponentially distributed random variables and one path of a certain Levy subordinator, which is highly efficient in many cases. Furthermore, the alternative model and sampling methodology is generalized to high-...

On time-changed Gaussian processes and their associated Fokker-Planck-Kolmogorov equations

Abstract: This paper establishes Fokker-Planck-Kolmogorov type equations for time-changed Gaussian processes. Examples include those equations for a time-changed fractional Brownian motion with time-dependent Hurst parameter and for a time-changed Ornstein-Uhlenbeck process. The time-change process considered is the inverse of either a stable subordinator or a mixture of independent stable subordinators.

Calibration of the Subdiffusive Arithmetic Brownian Motion with Tempered Stable Waiting-Times

Abstract: In the classical analysis many models used to real data description are based on the standard Brownian diffusion-type processes. However, some real data exhibit characteristic periods of constant values. In such cases the popular systems seem not to be applicable. Therefore we propose an alternative approach, based on the combination of the popular Brownian motion with drift (called also the arithmetic Brownian motion) and tempered stable subordinator. The probability density function of the proposed model can be described by a Fokker-Planck type equation and therefore it has many similar properties as the popular Brownian motion with drift. In this paper we propose the estimation procedure for the considered tempered stable subdiffusive arithmetic Brownian motion and calibrate the analyzed process to the real financial data.

Potential theory of subordinate Brownian motions with Gaussian components

Abstract: In this paper we study a subordinate Brownian motion with a Gaussian component and a rather general discontinuous part. The assumption on the subordinator is that its Laplace exponent is a complete Bernstein function with a Levy density satisfying a certain growth condition near zero. The main result is a boundary Harnack principle with explicit boundary decay rate for non-negative harmonic functions of the process in $C^{1,1}$ open sets. As a consequence of the boundary Harnack principle, we establish sharp two-sided estimates on the Green function of the subordinate Brownian motion in any bounded $C^{1,1}$ open set $D$ and identify the Martin boundary of $D$ with respect to the subordinate Brownian motion with the Euclidean boundary.

Anomalous diffusion models: different types of subordinator distribution

Abstract: Subordinated processes play an important role in modeling anomalous diffusion-type behavior. In such models the observed constant time periods are described by the subordinator distribution. Therefore, on the basis of the observed time series, it is possible to conclude on the main properties of the subordinator. In this paper we analyze the anomalous diffusion models with three types of subordinator distribution: \alpha-stable, tempered stable and gamma. We present similarities and differences between the analyzed processes and point at their main properties (like the behavior of moments or the mean squared displacement).

The Time at which a Lévy Process Creeps

Abstract: We show that if a Levy process creeps then the renewal function of the bivariate ascending ladder process satisfies certain continuity and differentiability properties. Then a left derivative of the renewal function is shown to be proportional to the distribution function of the time at which the process creeps over a given level, where the constant of proportionality is the reciprocal of the (positive) drift of the ascending ladder height process. This allows us to add the term due to creeping in the recent quintuple law of Doney and Kyprianou (2006). As an application, we derive a Laplace transform identity which generalises the second factorization identity. We also relate Doney and Kyprianou's extension of Vigon's equation amicale inversee to creeping. Some results concerning the ladder process, including the second factorization identity, continue to hold for a general bivariate subordinator, and are given in this generality.

On Λ-coalescents with dust component

Abstract: We consider the Λ-coalescent processes with a positive frequency of singleton clusters. The class in focus covers, for instance, the beta(a, b)-coalescents with a > 1. We show that some large-sample properties of these processes can be derived by coupling the coalescent with an increasing Levy process (subordinator), and by exploiting parallels with the theory of regenerative composition structures. In particular, we discuss the limit distributions of the absorption time and the number of collisions.

Estimation of the expected discounted penalty function for Lévy insurance risks

Abstract: We consider a generalized risk process which consists of a subordinator plus a spectrally negative Levy process. Our interest is to estimate the expected discounted penalty function (EDPF) from a set of data which is practical in the insurance framework. We construct an empirical type estimator of the Laplace transform of the EDPF and obtain it by a regularized Laplace inversion. The asymptotic behavior of the estimator under a high frequency assumption is investigated.

Probabilistic Representation and Fall-Off of Bound States of Relativistic Schr\"odinger Operators with Spin 1/2

Abstract: A Feynman-Kac type formula of relativistic Schr\"odinger operators with unbounded vector potential and spin 1/2 is given in terms of a three-component process consisting of Brownian motion, a Poisson process and a subordinator. This formula is obtained for unbounded magnetic fields and magnetic field with zeros. From this formula an energy comparison inequality is derived. Spatial decay of bound states is established separately for growing and decaying potentials by using martingale methods.

On the density of exponential functionals of L\'evy processes

Abstract: In this paper, we study the existence of the density associated to the exponential functional of the Levy process $\xi$, \[ I_{\ee_q}:=\int_0^{\ee_q} e^{\xi_s} \, \mathrm{d}s, \] where $\ee_q$ is an independent exponential r.v. with parameter $q\geq 0$. In the case when $\xi$ is the negative of a subordinator, we prove that the density of $I_{\ee_q}$, here denoted by $k$, satisfies an integral equation that generalizes the one found by Carmona et al. \cite{Carmona97}. Finally when $q=0$, we describe explicitly the asymptotic behaviour at 0 of the density $k$ when $\xi$ is the negative of a subordinator and at $\infty$ when $\xi$ is a spectrally positive Levy process that drifts to $+\infty$.

The student subordinator model with dependence for risky asset returns

Abstract: A new, tractable model of the stock price due to Heyde (1999) (see also Heyde and Leonenko, 2005) is elaborated here and used for asset price movement The model is driven by a Brownian motion, which has a “fractal clock” rather than a calendar clock We incorporate the Student's t-distribution, and a special dependence structure is introduced through the construction of this fractal time The Student model described has desired features supported by real financial data

A Weak Approximation of Stochastic Differential Equations with Jumps Through Tempered Polynomial Optimization

Abstract: We present an optimization approach to the weak approximation of a general class of stochastic differential equations with jumps, in particular, when value functions with compact support are considered. Our approach employs a mathematical programming technique yielding upper and lower bounds of the expectation, without Monte Carlo sample paths simulations, based upon the exponential tempering of bounding polynomial functions to avoid their explosion at infinity. The resulting tempered polynomial optimization problems can be transformed into a solvable polynomial programming after a minor approximation. The exponential tempering widens the class of stochastic differential equations for which our methodology is well defined. The analysis is supported by numerical results on the tail probability of a stable subordinator and the survival probability of Ornstein-Uhlenbeck processes driven by a stable subordinator, both of which can be formulated with value functions with compact support and are not applicable ...

The time at which a L\'evy process creeps

Abstract: We show that if a Levy process creeps then, as a function of $u$, the renewal function $V(t,u)$ of the bivariate ascending ladder process $(L^{-1},H)$ is absolutely continuous on $[0,\infty)$ and left differentiable on $(0,\infty)$, and the left derivative at $u$ is proportional to the (improper) distribution function of the time at which the process creeps over level $u$, where the constant of proportionality is $\rmd_H^{-1}$, the reciprocal of the (positive) drift of $H$. This yields the (missing) term due to creeping in the recent quintuple law of Doney and Kyprianou (2006). As an application, we derive a Laplace transform identity which generalises the second factorization identity. We also relate Doney and Kyprianou's extension of Vigon's equation amicale inversee to creeping. Some results concerning the ladder process of $X$, including the second factorization identity, continue to hold for a general bivariate subordinator, and are given in this generality.

Conditional $\alpha$-diversity for exchangeable Gibbs partitions driven by the stable subordinator

Abstract: Asymptotic behaviour of conditional $\alpha$ diversity for the two-parameter Poisson-Dirichlet partition model and for the normalized generalized Gamma model has been recently investigated in Favaro et al. (2009, 2011) with a view to possible applications in Bayesian treatment of species richness estimation. Here we generalize those results to the larger class of mixed Poisson-Kingman species sampling models driven by the stable subordinator (Pitman, 2003).

Applications of Lévy processes in credit and volatility modelling

Abstract: Market participants are faced with the problem of finding a good trade-off between the model adequacy and its tractability. The aim of this thesis is to develop tractable and intuitive models in the credit and equity areas and to assess their performance according to different criteria of interest for practitioners. Although sometimes criticized for its inability to reproduce quoted option prices which manifests itself in what is commonly referred to as the volatility smile, the Black-Scholes model and its implied volatility are widespread and their success is due to their very simple and intuitive concept. The first contribution of this thesis consists of the introduction of two kinds of alternative implied volatility, namely the implied Levy space volatility and the implied Levy time volatility, and of the investigation of the resulting skew adjustment. Moreover, we show that under the Levy parameter settings, the model performs systematically better than the classical Black-Scholes model as regards the risk of the commonly used delta hedging strategy. This is illustrated by looking at the daily hedging error and P&L distributions and by noting that, for the Levy models under investigation, the empirical variance is smaller and, for a wide range of in the money options, the empirical acceptability indices are higher than in the Black-Scholes setting. The second part of this thesis illustrates the impact of calibration risk under the He-ston model. In particular, we show that different plausible calibration procedures lead to different optimal parameter sets and hence to significantly different prices for a wide range of exotic products, emphasizing the necessity to take into account some additional safety margin for the pricing of these structured products, as it has been recommended in a recent directive of the Basel committee. The third contribution consists of the extension of the aVG model where the constraints on the subordinator parameters are relaxed such that the calibration does not require the existence of a liquid multivariate derivatives market which is nowadays pretty rare since the marginal characteristic functions become dependent on the whole parameter set. Moreover, the stocks log-return volatility, which is an indicator of the trading volume, becomes dependent on both the idiosyncratic and common subordinator settings, making the generalized model more coherent with the empirical evidence of the presence of both an idiosyncratic and a common component in the business clock. Furthermore we emphasize the presence of model risk inside a particular class of multivariate models by pricing standard multivariate options. Finally, we compare the exponential Levy model with the classical Gaussian copula model for the pricing of CDO-squared tranches, using several approximations of the recursive approach, namely a full Monte Carlo approximation and a Monte Carlo approximation which rests either on the multivariate Normal approximation of the joint inner CDO loss distribution or on the multivariate Poisson approximation of the joint number of defaults affecting the inner CDOs. The numerical study shows in particular that the multivariate Poisson approximation method outperforms the multivariate Normal approximation for CDOs-squared made up of inner equity tranches.

The class of distributions associated with the generalized Pollaczek-Khinchine formula

Abstract: The goal is to identify the class of distributions to which the distribution of the maximum of a L\'evy process with no negative jumps and negative mean (equivalently, the stationary distribution of the reflected process) belongs. An explicit new distributional identity is obtained for the case where the L\'evy process is an independent sum of a Brownian motion and a general subordinator (nondecreasing L\'evy process) in terms of a geometrically distributed sum of independent random variables. This generalizes both the distributional form of the standard Pollaczeck-Khinchine formula for stationary workload distribution in the M/G/1 queue and the exponential stationary distribution of a reflected Brownian motion.

First-Exit Times of an Inverse Gaussian Process

Abstract: The first-exit time process of an inverse Gaussian L\'evy process is considered. The one-dimensional distribution functions of the process are obtained. They are not infinitely divisible and the tail probabilities decay exponentially. These distribution functions can also be viewed as distribution functions of supremum of the Brownian motion with drift. The density function is shown to solve a fractional PDE and the result is also generalized to tempered stable subordinators. The subordination of this process to the Brownian motion is considered and the underlying PDE of the subordinated process is obtained. The infinite divisibility of the first-exit time of a $\beta$-stable subordinator is also discussed.

Valuation of collateralized debt obligations in a multivariate subordinator model

Abstract: The paper develops valuation of multi-name credit derivatives, such as collateralized debt obligations (CDOs), based on a novel multivariate subordinator model of dependent default (failure) times. The model can account for high degree of dependence among defaults of multiple firms in a credit portfolio and, in particular, exhibits positive probabilities of simultaneous defaults of multiple firms. The paper proposes an efficient simulation algorithm for fast and accurate valuation of CDOs with large number of firms.

On the Limit Distributions of Continuous-State Branching Processes with Immigration

Abstract: We consider the class of continuous-state branching processes with immigration (CBI-processes), introduced by Kawazu and Watanabe (1971) and their limit distributions as time tends to infinity. We determine the Levy-Khintchine triplet of the limit distribution and give an explicit description in terms of the characteristic triplets of the Levy subordinator and the spectrally positive Levy process, which describe the immigration resp. branching mechanism of the CBI-process. This representation allows us to describe the support of the limit distribution and characterise its absolute continuity and asymptotic behavior at the boundary of the support, generalizing several known results on self-decomposable distributions.

Parameter Estimation of Multivariate Lévy Processes

Abstract: In this thesis, we apply Levy copulas to describe the dependence structure of multivariate Levy processes and build some Levy copula-based models. Parameter estimation of the models is the main part of this work. The estimation procedure is based on maximum likelihood principles. For compound Poisson processes (CPP) which have finite Levy measure, we decompose the mass on the axes and outside of the axes. This decomposition for a bivariate CPP generates three independent components and shows either the jumps only in one component, or the bivariate jumps in both components. The likelihood function can be derived based on these independent parts. We also suggest a new simulation algorithm for a bivariate CPP. We apply our method to model Danish fire insurance data and estimate the parameters of the model. The extension of the method for Levy Processes with infinite Levy measure is discussed in the second part. More precisely we take a bivariate stable Levy Process and truncate all the small jumps. We base the statistical analysis on the resulting CPP. The Fisher information matrix is also calculated and the asymptotic normality of the estimators is proved as the number of jumps tends to infinity. In this model this may happen either for the observation period going to infinity, or the truncation point going to 0 for a fixed observation period. A simulation study investigates the loss of efficiency because of the truncation. Finally, a new estimation procedure is introduced in the last chapter. The main idea of this approach, which we call two-step method, is similar to IFM (inference functions for margins) for multivariate distribution functions. First, the parameters of the marginal processes are estimated. Then, given the estimates from the first step, we estimate in a second step only the dependence structure parameters. This method is applied to a bivariate α-stable Clayton subordinator with different or common marginal parameters. For the latter, the Godambe information matrix and asymptotic covariance matrix are analytically calculated. Moreover, the asymptotic normality of the estimators is proved as the time span goes to infinity or the truncation point goes to zero. A simulation study compares the quality of all three estimation methods: the two-step estimates, the MLEs of a full model and the MLEs based on joint jumps only.

On harmonic functions of symmetric Levy processes

Abstract: We consider some classes of Levy processes for which the estimate of Krylov and Safonov (as in [BL02]) fails and thus it is not possible to use the standard iteration technique to obtain a-priori Holder continuity estimates of harmonic functions. Despite the faliure of this method, we obtain some a-priori regularity estimates of harmonic functions for these processes. Moreover, we extend results from [SSV06] and obtain asymptotic behavior of the Green function and the Levy density for a large class of subordinate Brownian motions, where the Laplace exponent of the corresponding subordinator is a slowly varying function.

Numerical Methods for Lévy Processes: Lattice Methods and the Density, the Subordinator and the Time Copula

Abstract: Evidence from the financial markets suggests that empirical returns distributions, both historical and implied, do not arise from diffusion processes. A growing literature models the returns process as a Lévy process, finding a number of explicit formulae for the values of some derivatives in special cases. Practical use of these models has been hindered by a relative paucity of numerical methods that can be used when explicit solutions are not present. This paper investigates lattice methods that can be used when the returns process is Lévy. We relate the transition density function of a Lévy process to its representation as a time-changed Brownian motion and to its time-copula, leading to alternative derivations of the lattice. We apply the lattice to models based on the variance-gamma, NIG and Meixner processes, contrasting the numerical difficulties in each case. We discuss implications for implied pricing, concluding that current methods based, directly or indirectly, on low order branching, are unlikely to be capable of calibrating to market prices. ∗We gratefully acknowledge the help and support of Manfred Gilli and the hospitality of the Department of Econometrics, University of Geneva. We would like to thank Grace Kuan and Stewart Hodges for their comments and advice. The paper has benefited from comments by Lynda McCarthy, Peter Carr, Philip Schönbucher, Steve Heston, Mark Broadie, Chris Rogers, Rupert Brotherton-Ratcliffe and Alessio Sancetta, and from participants at the 8th CAP workshop, New York, and QMF 2002, Sydney..