Showing papers on "Subordinator published in 2008"

A multivariate Variance Gamma model for financial application

Abstract: In this paper we subordinate a multivariate Brownian motion with independent components by a multivariate gamma subordinator. The resulting process is a generalization of the bivariate variance gamma process proposed by Madan and Seneta [7], mentioned in Cont and Tankov [4] and calibrated in Luciano and Schoutens [5] as a price process. Our main contribution here is to introduce a multivariate subordinator with gamma margins. We investigate the process, determine its Levy triplet and analyze its dependence structure. At the end we propose an exponential Levy price model.

Universality of the REM for Dynamics of Mean-Field Spin Glasses

Abstract: We consider a version of Glauber dynamics for a p-spin Sherrington– Kirkpatrick model of a spin glass that can be seen as a time change of simple random walk on the N-dimensional hypercube. We show that, for all p ≥ 3 and all inverse temperatures β > 0, there exists a constant γ β ,p > 0, such that for all exponential time scales, exp(γ N), with γ < γ β ,p , the properly rescaled clock process (time-change process) converges to an α-stable subordinator where α = γ/β 2 < 1. Moreover, the dynamics exhibits aging at these time scales with a time-time correlation function converging to the arcsine law of this α-stable subordinator. In other words, up to rescaling, on these time scales (that are shorter than the equilibration time of the system) the dynamics of p-spin models ages in the same way as the REM, and by extension Bouchaud’s REM-like trap model, confirming the latter as a universal aging mechanism for a wide range of systems. The SK model (the case p = 2) seems to belong to a different universality class.

Moments of convex distribution functions and completely alternating sequences

Abstract: We solve the moment problem for convex distribution functions on [0, 1] in terms of completely alternating sequences. This complements a recent solution of this problem by Diaconis and Freedman, and relates this work to the Levy-Khintchine formula for the Laplace transform of a subordinator, and to regenerative composition structures.

On suprema of Lévy processes and application in risk theory

Abstract: Let $\wh{; ; ; X}; ; ; =C-Y$ where $Y$ is a general one-dimensional Levy process and $C$ an independent subordinator. Consider the times when a new supremum of $\wh{; ; ; X}; ; ; $ is reached by a jump of the subordinator $C$. We give a necessary and sufficient condition in order for such times to be discrete. When this is the case and $\wh{; ; ; X}; ; ; $ drifts to $-\infty$, we decompose the absolute supremum of $\wh{; ; ; X}; ; ; $ at these times, and derive a Pollaczek-Hinchin-type formula for the distribution function of the supremum.

Distributions of linear functionals of two parameter Poisson-Dirichlet random measures.

Abstract: The present paper provides exact expressions for the probability distributions of linear functionals of the two-parameter Poisson–Dirichlet process PD(α, θ). We obtain distributional results yielding exact forms for density functions of these functionals. Moreover, several interesting integral identities are obtained by exploiting a correspondence between the mean of a Poisson–Dirichlet process and the mean of a suitable Dirichlet process. Finally, some distributional characterizations in terms of mixture representations are proved. The usefulness of the results contained in the paper is demonstrated by means of some illustrative examples. Indeed, our formulae are relevant to occupation time phenomena connected with Brownian motion and more general Bessel processes, as well as to models arising in Bayesian nonparametric statistics.

Queues with delays in two-state strategies and Lévy input

Abstract: We consider a reflected Levy process without negative jumps, starting at the origin. When the reflected process first upcrosses level K, a timer is activated. After D time units, the timer expires and the Levy exponent of the Levy process is changed. As soon as the process hits zero again, the Levy exponent reverses to the original function. If the process has reached the origin before the timer expires then the Levy exponent does not change. Using martingale techniques, we analyze the steady-state distribution of the resulting process, reflected at the origin. We pay special attention to the cases of deterministic and exponential timers, and to the following three special Levy processes: (i) a compound Poisson process plus negative drift (corresponding to an M/G/1 queue), (ii) Brownian motion, and (iii) a Levy process that is a subordinator until the timer expires.

Stochastic solutions of a class of Higher order Cauchy problems in $\rd$

Abstract: We study solutions of a class of higher order partial differential equations in bounded domains. These partial differential equations appeared first time in the papers of Allouba and Zheng \cite{allouba1}, Baeumer, Meerschaert and Nane \cite{bmn-07}, Meerschaert, Nane and Vellaisamy \cite{MNV}, and Nane \cite{nane-h}. We express the solutions by subordinating a killed Markov process by a hitting time of a stable subordinator of index $0<\beta <1$, or by the absolute value of a symmetric $\alpha$-stable process with $0<\alpha\leq 2$, independent of the Markov process. In some special cases we represent the solutions by running composition of $k$ independent Brownian motions, called $k$-iterated Brownian motion for an integer $k\geq 2$. We make use of a connection between fractional-time diffusions and higher order partial differential equations established first by Allouba and Zheng \cite{allouba1} and later extended in several directions by Baeumer, Meerschaert and Nane \cite{bmn-07}.

Pricing Forward Start Options in Models Based on (Time-Changed) Levy Processes

Abstract: Options depending on the forward skew are very popular. One such option is the forward starting call option - the basic building block of a cliquet option. Widely applied models to account for the forward skew dynamics to price such options include the Heston model, the Heston-Hull-White model and the Bates model. Within these models solutions for options including forward start features are available using (semi) analytical formulas. Today exponential (subordinated) Levy models being increasingly popular for modelling the asset dynamics. While the simple exponential Levy models imply the same forward volatily surface for all future times the subordinated models do not. Depending on the subordinator the dynamic of the forward volatility surface and therefore stochastic volatility can be modelled. Analytical pricing formulas based on the characteristic function and Fourier transform methods are available for the class of these models. We extend the applicability of analytical pricing to options including forward start features. To this end we derive the forward characteristic functions which can be used in Fourier transform based methods. As examples we consider the Variance Gamma model and the NIG model subordinated by a Gamma Ornstein Uhlenbeck process and respectively by an Cox-Ingersoll-Ross process. We check our analytical results by applying Monte Carlo methods. These results can for instance be applied to calibration of the forward volatility surface.

On multifractality and time subordination for continuous functions

Abstract: We show that if $Z$ is "homogeneously multifractal" (in a sense we precisely define), then $Z$ is the composition of a monofractal function $g$ with a time subordinator $f$ (i.e. $f$ is the integral of a positive Borel measure supported by $\zu$). When the initial function $Z$ is given, the monofractality exponent of the associated function $g$ is uniquely determined. We study in details a classical example of multifractal functions $Z$, for which we exhibit the associated functions $g$ and $f$. This provides new insights into the understanding of multifractal behaviors of functions.

Small deviation of subordinated processes over compact sets

Abstract: Let A = ` A(t) ́ t­0 be a subordinator. Given a compact set K ⊂ [0,∞) we prove two-sided estimates for the covering numbers of the random set {A(t) : t ∈ K} which depend on the Laplace exponent Φ of A and on the covering numbers of K. This extends former results in the case K = [0, 1]. Using this we find the behavior of the small deviation probabilities for subordinated processes ` WH ` A(t) ́ ́ t∈K , where WH is a fractional Brownian motion with Hurst index 0 < H < 1. The results are valid in the quenched as well as in the annealed case. In particular, those questions are investigated for Gamma processes. Here some surprising new phenomena appear. As application of the general results we find the behavior of log P (supt∈K |Zα(t)| < ε) as ε→ 0 for the α-stable Lévy motion Zα. For example, if K is a self-similar set with Hausdorff dimension D > 0, then this behavior is of order −ε−αD in complete accordance with the Gaussian case α = 2. 2000 AMS Mathematics Subject Classification: Primary: 60G51; Secondary: 60G15, 60G52, 28A80, 60G18.

Aspects of Wireless Network Modeling based on Poisson Point Processes

Abstract: In these notes we discuss stochastic models for a simplified wireless network that consists of a collection of spatially distributed stations equipped with emitters and/or receivers for transmission over a common communication channel. The modeling approach is based on using Poisson point processes for the spatial locations as well as for other signaling characteristics of the network nodes. Throughout we work from the premises that the transmission of signals is syncronized and slotted in time so that in a fixed time slot each sender attempts to emit the equivalent of one symbol. The signal power is affected by random fading and attenuation proportional to the traveled distance. Using simple Poisson models and superpositioning the effect of all stations in one slot it is possible to obtain some insights into the balance between node density and node interference. For traffic sessions over a fixed or random number of slots and under assumptions of Rayleigh fading, we propose a modeling scenario based on the Levy gamma subordinator process and its relation to complex Gaussian waweforms. These models can be extended to sessions which are Poisson in both space and time, and have short-tailed or heavy-tailed random session duration times. It is possible also to include lognormal fading. This is a mechanism supposed to act on the slow time scale of sessions, which is in contrast to Rayleigh fading that generate random variation on the fast scale of slots. For the traffic session models it is possible to perform a scaling approximation to analyze the fluctuations that build up in the interference field.

On Ruin Probability for a Risk Process Perturbed by a Lévy Process with no Negative Jumps

Abstract: We study a risk process where the claim sizes and the interarrival times are phase-type distributed. The risk process is perturbed by a Levy process with no negative jumps. We derive the ruin probability, and the distribution of deficit at ruin, by constructing an unperturbed risk process with the same ruin probability. In this process, the claim sizes are also phase-type, and the interarrival times have some general distribution. The interarrival times and the claims are dependent. The model is analyzed via the dual queueing system, which we show to be of the Markov arrival process type.

Analysis of stochastic fluid queues driven by local-time processes

Abstract: We consider a stochastic fluid queue served by a constant rate server and driven by a process which is the local time of a reflected Levy process. Such a stochastic system can be used as a model in a priority service system, especially when the time scales involved are fast. The input (local time) in our model is typically (but not necessarily) singular with respect to the Lebesgue measure, a situation which, in view of the nonsmooth or bursty nature of several types of Internet traffic, is nowadays quite realistic. We first discuss how to rigorously construct the (necessarily) unique stationary version of the system under some natural stability conditions. We then consider the distribution of performance steady-state characteristics, namely, the buffer content, the idle period, and the busy period. These derivations are much based on the fact that the inverse of the local time of a Markov process is a Levy process (a subordinator), hence making the theory of Levy processes applicable. Another important ingredient in our approach is the use of Palm calculus for stationary random point processes and measures.

Pricing of the Time-Change Risks

Abstract: We develop a discrete-time real endowment economy featuring Epstein-Zin recursive utility and a Levy time-change subordinator, which represents a clock that connects business time to calendar time. This setup provides a convenient equilibrium framework for pricing non-Gaussian risks, where the solutions for financial prices are available up to integral operations in general, or in closed-form for tempered stable shocks. The non-Gaussianity of fundamentals due to time-deformation induces compensations for higher order moments and co-moments of consumption and dividend growth rates of the assets. Forecastability of the time change leads to predictability of the endowment streams and therefore to time-variation in financial prices and risk premia on assets. In numerical calibrations, we quantitatively analyze the compensations for different types of systematic risk.

A Multivariate Variance Gamma Model for Financial Applications

Abstract: In this paper we subordinate a multivariate Brownian motion with independent components by a multivariate gamma subordinator. The resulting process is a generalization of the bivariate variance gamma process proposed by Madan and Seneta [7], mentioned in Cont and Tankov [4] and calibrated in Luciano and Schoutens [5] as a price process. Our main contribution here is to introduce a multivariate subordinator with gamma margins. We investigate the process, determine its Levy triplet and analyze its dependence structure. At the end we propose an exponential Levy price model.

Lévy Processes in Asset Pricing

Abstract: After mentioning empirical motivation of Levy processes in asset pricing, an overview of properties related to Levy processes is given. Certain difficulties in applying Levy processes, such as the volatility clustering effect and those in distinguishing the tail behavior of asset returns, are also discussed. Keywords: leptokurtic distribution; jump diffusion; subordinator; infinite activity Levy processes; volatility clustering effect; tail distribution

Edgeworth expansions for subordinators via differential calculus for linear operators

Abstract: We apply a dierential calculus for linear operators, together with moduli of smooth- ness techniques, in order to obtain Edgeworth expansions for E`(Z(t)) i E`(Z), where (Z(t); t ‚ 1) is a standardized subordinator, Z is a standard normal ran- dom variable and the degree of smoothness ofgoes from infinite dierentiability to bounded variation. The main achievement of the method is to provide explicit upper bounds for the remainders, thus getting rid o the 'big or little o' terms. Other features are the relative simplicity of the proofs, the property of monotonic convergence for E`(Z(t)) under simple sucient conditions on ` and, in the lattice case, the obtention of explicit lower and upper Berry-Esseen bounds of the same order of magnitude which are asymptotically sharp.

On Lévy-driven vacation models with correlated busy periods and service interruptions

Abstract: This paper considers queues with server vacations, but departs from the traditional setting in two ways: (i) the queueing model is driven by Levy processes rather than just compound Poisson processes; (ii) the vacation lengths depend on the length of the server's preceding busy period. Regarding the former point: the Levy process active during the busy period is assumed to have no negative jumps, whereas the Levy process active during the vacation is a subordinator. Regarding the latter point: where in a previous study (Boxma et al. in Probab. Eng. Inf. Sci. 22:537---555, 2008) the durations of the vacations were positively correlated with the length of the preceding busy period, we now introduce a dependence structure that may give rise to both positive and negative correlations. We analyze the steady-state workload of the resulting queueing (or: storage) system, by first considering the queue at embedded epochs (viz. the beginnings of busy periods). We show that this embedded process does not always have a proper stationary distribution, due to the fact that there may occur an infinite number of busy-vacation cycles in a finite time interval; we specify conditions under which the embedded process is recurrent. Fortunately, irrespective of whether the embedded process has a stationary distribution, the steady-state workload of the continuous-time storage process can be determined. In addition, a number of ramifications are presented. The theory is illustrated by several examples.

The falling appart of the tagged fragment and the asymptotic disintegration of the Brownian height fragmentation

Abstract: We present a further analysis of the fragmentation at heights of the normalized Brownian excursion. Specifically we study a representation for the mass of a tagged fragment in terms of a Doob transformation of the 1/2-stable subordinator and use it to study its jumps; this accounts for a description of how a typical fragment falls apart. These results carry over to the height fragmentation of the stable tree. Additionally, the sizes of the fragments in the Brownian fragmentation when it is about to reduce to dust are described in a limit theorem.

Some recent developments in the theory of fragmentations.

Abstract: The main subject of this PHD thesis is the study of various quantities related to fragmentation processes. These processes are designed to modelize a unit mass object which fragments with time. This work is composed of four chapters. The aim of the first one is to study the Hausdorff dimension of the set of locations having exactly an exponential decay. In the second chapter, we construct a self-similar Markov process which generalizes the classical fragmentation by allowing in particular the size of the descendants to be bigger than the one of their parents. Then we show some Limit Theorems using the theory of self-similar Markov processes. In the third chapter, we are interested by the statistical estimation of the Levy measure of the classical subordinator associated to the fragmentation. More precisely, we observe the fragments only when their size reach a size smaller than a given threshold. Finally, in the fourth chapter, we study the energy cost of a succession of fragmentations.

On the first passage time for Brownian motion subordinated by a Levy process

Abstract: This paper considers the class of L\'evy processes that can be written as a Brownian motion time changed by an independent L\'evy subordinator. Examples in this class include the variance gamma model, the normal inverse Gaussian model, and other processes popular in financial modeling. The question addressed is the precise relation between the standard first passage time and an alternative notion, which we call first passage of the second kind, as suggested by Hurd (2007) and others. We are able to prove that standard first passage time is the almost sure limit of iterations of first passage of the second kind. Many different problems arising in financial mathematics are posed as first passage problems, and motivated by this fact, we are lead to consider the implications of the approximation scheme for fast numerical methods for computing first passage. We find that the generic form of the iteration can be competitive with other numerical techniques. In the particular case of the VG model, the scheme can be further refined to give very fast algorithms.

Fractional Cauchy problems on bounded domains

Abstract: Fractional Cauchy problems replace the usual first-order time derivative by a fractional derivative. This paper develops classical solutions and stochastic analogues for fractional Cauchy problems in a bounded domain $D\subset\mathbb{R}^d$ with Dirichlet boundary conditions. Stochastic solutions are constructed via an inverse stable subordinator whose scaling index corresponds to the order of the fractional time derivative. Dirichlet problems corresponding to iterated Brownian motion in a bounded domain are then solved by establishing a correspondence with the case of a half-derivative in time.