Showing papers in "Advances in Applied Probability in 2003"
TL;DR: In this article, the authors studied the first passage times to flat boundaries for a double exponential jump diffusion process, which consists of a continuous part driven by a Brownian motion and a jump part with jump sizes having double exponential distribution.
Abstract: This paper studies the first passage times to flat boundaries for a double exponential jump diffusion process, which consists of a continuous part driven by a Brownian motion and a jump part with jump sizes having a double exponential distribution. Explicit solutions of the Laplace transforms, of both the distribution of the first passage times and the joint distribution of the process and its running maxima, are obtained. Because of the overshoot problems associated with general jump diffusion processes, the double exponential jump diffusion process offers a rare case in which analytical solutions for the first passage times are feasible. In addition, it leads to several interesting probabilistic results. Numerical examples are also given. The finance applications include pricing barrier and lookback options.
458 citations
TL;DR: In this article, shot noise Cox processes constitute a large class of Cox and Poisson cluster processes in ℝ d, including Neyman-Scott, Poisson-gamma and shot noise G Cox processes.
Abstract: Shot noise Cox processes constitute a large class of Cox and Poisson cluster processes in ℝ d , including Neyman-Scott, Poisson-gamma and shot noise G Cox processes It is demonstrated that, due to the structure of such models, a number of useful and general results can easily be established The focus is on the probabilistic aspects with a view to statistical applications, particularly results for summary statistics, reduced Palm distributions, simulation with or without edge effects, conditional simulation of the intensity function and local and spatial Markov properties
173 citations
TL;DR: In this paper, Taylor series expansions of stationary characteristics of general-state-space Markov chains are explicitly calculated and a lower bound for the radius of convergence of the Taylor series is established.
Abstract: We study Taylor series expansions of stationary characteristics of general-state-space Markov chains. The elements of the Taylor series are explicitly calculated and a lower bound for the radius of convergence of the Taylor series is established. The analysis provided in this paper applies to the case where the stationary characteristic is given through an unbounded sample performance function such as the second moment of the stationary waiting time in a queueing system.
69 citations
TL;DR: In this article, an explicit integral expression for the joint distribution of the number and the respective positions of the sides of the typical cell of a two-dimensional Poisson-Voronoi tessellation is given.
Abstract: In this paper, we give an explicit integral expression for the joint distribution of the number and the respective positions of the sides of the typical cell 𝒞 of a two-dimensional Poisson-Voronoi tessellation. We deduce from it precise formulae for the distributions of the principal geometric characteristics of 𝒞 (area, perimeter, area of the fundamental domain). We also adapt the method to the Crofton cell and the empirical (or typical) cell of a Poisson line process.
66 citations
TL;DR: In this article, the authors demonstrate that the high volatility of share prices can nevertheless be used in building a model that leads to a particular cross-sectional size distribution, which in turn is modeled as a birth-death process.
Abstract: The inability to predict the future growth rates and earnings of growth stocks (such as biotechnology and internet stocks) leads to the high volatility of share prices and difficulty in applying the traditional valuation methods. This paper attempts to demonstrate that the high volatility of share prices can nevertheless be used in building a model that leads to a particular cross-sectional size distribution. The model focuses on both transient and steady-state behavior of the market capitalization of the stock, which in turn is modeled as a birth–death process. Numerical illustrations of the cross-sectional size distribution are also presented.
65 citations
TL;DR: In this paper, the state of a failure-prone system is modeled as a continuous-time Markov process with a finite state space, where the observation process is stochastically related to the state process which is unobservable, except for the failure state.
Abstract: We consider a failure-prone system which operates in continuous time and is subject to condition monitoring at discrete time epochs. It is assumed that the state of the system evolves as a continuous-time Markov process with a finite state space. The observation process is stochastically related to the state process which is unobservable, except for the failure state. Combining the failure information and the information obtained from condition monitoring, and using the change of measure approach, we derive a general recursive filter, and, as special cases, we obtain recursive formulae for the state estimation and other quantities of interest. Up-dated parameter estimates are obtained using the EM algorithm. Some practical prediction problems are discussed and an illustrative example is given using a real dataset.
64 citations
TL;DR: For supercritical multitype Markov branching processes in continuous time, the authors investigated the evolution of types along those lineages that survive up to some time t and established almost-sure convergence theorems for both time and population averages of ancestral types.
Abstract: For supercritical multitype Markov branching processes in continuous time, we investigate the evolution of types along those lineages that survive up to some time t. We establish almost-sure convergence theorems for both time and population averages of ancestral types (conditioned on non-extinction), and identify the mutation process describing the type evolution along typical lineages. An important tool is a representation of the family tree in terms of a suitable size-biased tree with trunk. As a by-product, this representation allows a ‘conceptual proof’ (in the sense of [19]) of the continuous-time version of the Kesten-Stigum theorem.
63 citations
TL;DR: In this paper, the distribution of the coalescence time for two individuals picked at random (uniformly) in the current generation of a branching process founded t units of time ago, in both the discrete and continuous (time and state-space) settings, is investigated.
Abstract: We investigate the distribution of the coalescence time (most recent common ancestor) for two individuals picked at random (uniformly) in the current generation of a branching process founded t units of time ago, in both the discrete and continuous (time and state-space) settings. We obtain limiting distributions as t→∞ in the subcritical case. In the continuous setting, these distributions are specified for quadratic branching mechanisms (corresponding to Brownian motion and Brownian motion with positive drift), and we also extend our results for two individuals to the joint distribution of coalescence times for any finite number of individuals sampled in the current generation.
58 citations
TL;DR: In this article, the authors give an explicit expression for the distribution of the number of sides of the typical cell of a two-dimensional Poisson-Voronoi tessellation.
Abstract: In this paper, we give an explicit expression for the distribution of the number of sides (or equivalently vertices) of the typical cell of a two-dimensional Poisson-Voronoi tessellation. We use this formula to give a table of numerical values of the distribution function.
52 citations
TL;DR: In this article, a continuum growth model is introduced, where the state at time t, S t, is a subset of a connected union of randomly sized Euclidean balls, which emerge from outbursts at their center points.
Abstract: A continuum growth model is introduced. The state at time t, S t , is a subset of
ℝ d and consists of a connected union of randomly sized Euclidean balls, which emerge from outbursts at their centre points. An outburst occurs somewhere in S t after an exponentially distributed time with expected value
|S t |-1 and the location of the outburst is uniformly distributed over S t . The main result is that, if the distribution of the radii of the outburst balls has bounded support, then S t grows linearly and S t /t has a nonrandom shape as t → ∞. Due to rotational invariance the asymptotic shape must be a Euclidean ball.
50 citations
TL;DR: In this article, the authors studied the asymptotic properties of singularly perturbed Markov chains in discrete time and proved that a suitably scaled sequence of occupation measures converges to a switching diffusion.
Abstract: This work is devoted to asymptotic properties of singularly perturbed Markov chains in discrete time. The motivation stems from applications in discrete-time control and optimization problems, manufacturing and production planning, stochastic networks, and communication systems, in which finite-state Markov chains are used to model large-scale and complex systems. To reduce the complexity of the underlying system, the states in each recurrent class are aggregated into a single state. Although the aggregated process may not be Markovian, its continuous-time interpolation converges to a continuous-time Markov chain whose generator is a function determined by the invariant measures of the recurrent states. Sequences of occupation measures are defined. A mean square estimate on a sequence of unscaled occupation measures is obtained. Furthermore, it is proved that a suitably scaled sequence of occupation measures converges to a switching diffusion.
TL;DR: In this article, the authors proposed to model the human frailty as a weighted Levy process, and derived the population hazard and survival functions for the LUvy process with respect to the power variance function.
Abstract: Generalizing the standard frailty models of survival analysis, we propose to model frailty as a weighted Levy process. Hence, the frailty of an individual is not a fixed quantity, but develops over time. Formulae for the population hazard and survival functions are derived. The power variance function LUvy process is a prominent example. In many cases, notably for compound Poisson processes, quasi-stationary distributions of survivors may arise. Quasi-stationarity implies limiting population hazard rates that are constant, in spite of the continual increase of the individual hazards. A brief discussion is given of the biological relevance of this finding.
TL;DR: It is shown that the locally scaled versions of these locally scaled point processes are again Markov and that locally the Papangelou conditional intensity of the new process behaves like that of a global scaling of the homogeneous process.
Abstract: A new class of models for inhomogeneous spatial point processes is introduced. These locally scaled point processes are modifications of homogeneous template point processes, having the property that regions with different intensities differ only by a scale factor. This is achieved by replacing volume measures used in the density with locally scaled analogues defined by a location-dependent scaling function. The new approach is particularly appealing for modelling inhomogeneous Markov point processes. Distance-interaction and shot noise weighted Markov point processes are discussed in detail. It is shown that the locally scaled versions are again Markov and that locally the Papangelou conditional intensity of the new process behaves like that of a global scaling of the homogeneous process. Approximations are suggested that simplify calculation of the density, for example, in simulation. For sequential point processes, an alternative and simpler definition of local scaling is proposed.
TL;DR: In this paper, the partial sums process of the AR(1) process with random coefficient was studied and it was shown that the covariance function of {X t, t ∈ 𝕫} decays hyperbolically with exponent between 0 and 1.
Abstract: We discuss long-memory properties and the partial sums process of the AR(1) process {X t , t ∈ 𝕫} with random coefficient {a t , t ∈ 𝕫} taking independent values A j ∈ [0,1] on consecutive intervals of a stationary renewal process with a power-law interrenewal distribution. In the case when the distribution of generic A j has either an atom at the point a=1 or a beta-type probability density in a neighborhood of a=1, we show that the covariance function of {X t } decays hyperbolically with exponent between 0 and 1, and that a suitably normalized partial sums process of {X t } weakly converges to a stable Levy process.
TL;DR: In this article, the authors give different proofs of Dufresne's results and present extensions of them for the processes {A, t > 0}. But they do not consider diffusion processes on hyperbolic spaces.
Abstract: Denote by a the probability law of AG) = fo exp(2(Bs + ps)) ds for a Brownian motion {Bs, s > 0}. It is well known that a is of interest in a number of domains, e.g. mathematical finance, diffusion processes in random environments, stochastic analysis on hyperbolic spaces and so on, but that it has complicated expressions. Recently, Dufresne obtained some remarkably simple expressions for a0) and a) , as well as an equally remarkable relationship between aGO) and a(v) for two different drifts # and v. In this paper, hinging on previous results about atC), we give different proofs of Dufresne's results and present extensions of them for the processes {A ), t > 0}.
TL;DR: In this paper, the authors analyse the limit behavior of a stochastic structured metapopulation model as the number of its patches goes to infinity and show that the limit of every convergent subsequence satisfies an infinite system of ordinary differential equations.
Abstract: We analyse the limit behaviour of a stochastic structured metapopulation model as the number of its patches goes to infinity. The sequence of probability measures associated with the random process, whose components are the proportions of patches with different number of individuals, is tight. The limit of every convergent subsequence satisfies an infinite system of ordinary differential equations. The existence and the uniqueness of the solution are shown by semigroup methods, so that the whole random process converges weakly to the solution of the system.
TL;DR: In this article, the limits for sequences of tessellations that are generated by consecutive application of superposition or iteration are investigated and necessary conditions are formulated for those TESsellations which may occur as limits.
Abstract: In order to increase the variety of feasible models for random stationary tessellations (mosaics), two operations acting on tessellations are studied: superposition and iteration (the latter is also referred to as nesting). The superposition of two planar tessellations is the superposition of the edges of the cells of both tessellations. The iteration of tessellations means that one tessellation is chosen as a ‘frame’ tessellation. The single cells of this ‘frame’ are simultaneously and independently subdivided by cut-outs of tessellations of an independent and identically distributed sequence of tessellations. In the present paper, we investigate the limits for sequences of tessellations that are generated by consecutive application of superposition or iteration respectively. Sequences of (renormalised) superpositions of stationary planar tessellations converge weakly to Poisson line tessellations. For consecutive iteration the notion of stability of distributions is adapted and necessary conditions are formulated for those tessellations which may occur as limits.
TL;DR: In this article, the authors considered the iteration of random tessellations in ℝ d and derived sufficient conditions for stationarity and isotropy of iterated TESsellations.
Abstract: The iteration of random tessellations in ℝ d is considered, where each cell of an initial tessellation is further subdivided into smaller cells by so-called component tessellations. Sufficient conditions for stationarity and isotropy of iterated tessellations are given. Formulae are derived for the intensities of their facet processes, and for the expected intrinsic volumes of their typical facets. Particular emphasis is put on two special cases: superposition and nesting of tessellations. Bernoulli thinning of iterated tessellations is also considered.
TL;DR: For arbitrary stationary sequences of random variables satisfying a mild mixing condition, distributional approximations are established for functionals of clusters of exceedances over a high threshold in this article.
Abstract: For arbitrary stationary sequences of random variables satisfying a mild mixing condition, distributional approximations are established for functionals of clusters of exceedances over a high threshold. The approximations are in terms of the distribution of the process conditionally on the event that the first variable exceeds the threshold. This conditional distribution is shown to converge to a nontrivial limit if the finite-dimensional distributions of the process are in the domain of attraction of a multivariate extreme-value distribution. In this case, therefore, limit distributions are obtained for functionals of clusters of extremes, thereby generalizing results for higher-order stationary Markov chains by Yun (2000).
TL;DR: In this paper, a spectral theory for stationary random closed sets is developed and provided with a sound mathematical basis, where the Bartlett spectrum and the power spectrum can be used as second order characteristics in frequency space.
Abstract: A spectral theory for stationary random closed sets is developed and provided with a sound mathematical basis. Definition and proof of existence of the Bartlett spectrum of a stationary random closed set as well as the proof of a Wiener-Khintchine theorem for the power spectrum are used to two ends: First, well known second order characteristics like the covariance can be estimated faster than usual via frequency space. Second, the Bartlett spectrum and the power spectrum can be used as second order characteristics in frequency space. Examples show, that in some cases information about the random closed set is easier to obtain from these characteristics in frequency space than from their real world counterparts.
TL;DR: In this article, the Hartman-Watson Ansatz of the integral of geometric Brownian motion over finite time intervals is derived and an apparently new integral representation is derived, and its interrelations with the integral representations for these laws originating by Yor and by Dufresne are established.
Abstract: This paper studies the law of any real powers of the integral of geometric Brownian motion over finite time intervals. As its main results, an apparently new integral representation is derived and its interrelations with the integral representations for these laws originating by Yor and by Dufresne are established. In fact, our representation is found to furnish what seems to be a natural bridge between these other two representations. Our results are obtained by enhancing the Hartman-Watson Ansatz of Yor, based on Bessel processes and the Laplace transform, by complex analytic techniques. Systematizing this idea in order to overcome the limits of Yor's theory seems to be the main methodological contribution of the paper.
TL;DR: In this paper, the authors studied the convergence rate of the generalized Pareto distribution to 0 of F u ( x )-G γ (( x + u -α( u ))/σ( u )), where α and σ are two appropriate normalizing functions.
Abstract: Let F be a distribution function in the domain of attraction of an extreme-value distribution H γ . If F u is the distribution function of the excesses over u and G γ the distribution function of the generalized Pareto distribution, then it is well known that F u ( x ) converges to G γ ( x /σ( u )) as u tends to the end point of F , where σ is an appropriate normalizing function. We study the rate of (uniform) convergence to 0 of F u ( x )- G γ (( x + u -α( u ))/σ( u )), where α and σ are two appropriate normalizing functions.
TL;DR: In this paper, the authors define a class of tessellation models based on perturbing or deforming standard Tessellations, such as the Voronoi torsion, and show how distributions over this class of Torsion can be used to define prior distributions.
Abstract: We define a class of tessellation models based on perturbing or deforming standard tessellations such as the Voronoi tessellation. We show how distributions over this class of ‘deformed’ tessellations can be used to define prior distributions for models based on tessellations, and how inference for such models can be carried out using Markov chain Monte Carlo methods; stability properties of the algorithms are investigated. Our approach applies not only to fixed dimension problems, but also to variable dimension problems, in which the number of cells in the tessellation is unknown. We illustrate our methods with two real examples. The first relates to reconstructing animal territories, represented by the individual cells of a tessellation, from observation of an inhomogeneous Poisson point process. The second example involves the analysis of an image of a cross-section through a sample of metal, with the tessellation modelling the micro-crystalline structure of the metal.
TL;DR: In this article, the authors consider random recursive fractals and prove fine results about their local behaviour. And they use the large deviation principle for a class of general branching processes which extends the known large deviation estimates for the supercritical Galton-Watson process.
Abstract: We consider random recursive fractals and prove fine results about their local behaviour. We show that for a class of random recursive fractals the usual multifractal spectrum is trivial in that all points have the same local dimension. However, by examining the local behaviour of the measure at typical points in the set, we establish the size of fine fluctuations in the measure. The results are proved using a large deviation principle for a class of general branching processes which extends the known large deviation estimates for the supercritical Galton-Watson process.
TL;DR: In this paper, it was shown that the representation of general symmetric solutions is a mixture of certain infinitely divisible distributions, and that the existence of nontrivial symmetric solution is exactly determined under the condition that E ∑ j = 1 ∞ T j 2 log+ Tj 2 < ∞.
Abstract: Let T = (T 1, T 2,…) be a sequence of real random variables with ∑ j=1 ∞ 1 |T j |>0 < ∞ almost surely. We consider the following equation for distributions μ: W ≅ ∑ j=1 ∞ T j W j , where W, W 1, W 2,… have distribution μ and T, W 1, W 2,… are independent. We show that the representation of general solutions is a mixture of certain infinitely divisible distributions. This result can be applied to investigate the existence of symmetric solutions for T j ≥ 0: essentially under the condition that E ∑ j=1 ∞ T j 2 log+ T j 2 < ∞, the existence of nontrivial symmetric solutions is exactly determined, revealing a connection with the existence of positive solutions of a related fixed-point equation. Furthermore, we derive results about a special class of canonical symmetric solutions including statements about Lebesgue density and moments.
TL;DR: It is shown that the probabilities of observing the various types of configurations can be expressed in terms of the first contact distribution function of Z, and an important prerequisite result concerning deterministic dilation areas is established.
Abstract: Estimation methods for the directional measure of a stationary planar random set Z, based only on discretized realizations of Z, are discussed. Properties of the discretized set that can be derived by comparing neighbouring grid points are used. Larger grid configurations of more than two grid points are considered. It is shown that the probabilities of observing the various types of configurations can be expressed in terms of the first contact distribution function of Z (with a finite structuring element). An important prerequisite result concerning deterministic dilation areas is also established. The inference on the mean normal measure based on 2 x 2 configurations is discussed in detail.
TL;DR: In this article, the cubed intercept length is replaced by a moment of the distance between two points in the section profile, which can be computed as a moments of the set covariance function, which in turn is computable using the fast Fourier transform.
Abstract: Our aim is to estimate the volume-weighted mean of the volumes of three-dimensional ‘particles’ (compact, not-necessarily-convex subsets) from plane sections of the particle population. The standard stereological technique is to place test lines in the plane section, and measure cubed intercept lengths with the two-dimensional particle profiles. This paper discusses more efficient estimators obtained by integrating over all possible placements of the test line. We prove that these estimators have smaller variance than the line transect estimators, and indeed are related to them by the Rao-Blackwell process. In the improved estimators, the cubed intercept length is replaced by a moment of the distance between two points in the section profile. This can be computed as a moment of the set covariance function, which in turn is computable using the fast Fourier transform. We also derive an isoperimetric-type inequality between the improved estimator and the area-weighted 3/2th moment of the profile areas. Finally, we present two practical applications to particles of silicon carbide and to synaptic boutons in brain tissue. We estimate the variance of the technique and the gain in efficiency over line transect techniques; the efficiency improvement appears to be as much as one order of magnitude.
TL;DR: In this article, the authors derived conditions on the internal wear process under which the resulting time-to-failure model will be of the simple collapsible form when the usage accumulation history is available.
Abstract: In this paper we derive conditions on the internal wear process under which the resulting time-to-failure model will be of the simple collapsible form (Oakes, 1995, Duchesne and Lawless, 2000) when the usage accumulation history is available. We suppose that failure occurs when internal wear crosses a certain threshold and/or a traumatic event causes the item to fail (Cox, 1999 and Bagdonavi cius and Nikulin, 2001). We model the innitesimal increment in internal wear as a function of time, accumulated internal wear and usage history, and we derive conditions on this function to get a collapsible model for the distribution of time-to-failure given the usage history. We reach the conclusion that collapsible models form the subset of accelerated failure time models with time-varying covariates (Robins and Tsiatis, 1992) for which the time transformation function satises certain simple properties.
TL;DR: Under mild assumptions, it is proved that the workload distribution is asymptotically equivalent to that in a somewhat ‘dual’ reduced system, multiplied by a certain prefactor.
Abstract: We determine the exact large-buffer asymptotics for a mixture of light-tailed and heavytailed input flows. Earlier studies have found a ‘reduced-load equivalence’ in situations where the peak rate of the heavy-tailed flows plus the mean rate of the light-tailed flows is larger than the service rate. In that case, the workload is asymptotically equivalent to that in a reduced system, which consists of a certain ‘dominant’ subset of the heavytailed flows, with the service rate reduced by the mean rate of all other flows. In the present paper, we focus on the opposite case where the peak rate of the heavy-tailed flows plus the mean rate of the light-tailed flows is smaller than the service rate. Under mild assumptions, we prove that the workload distribution is asymptotically equivalent to that in a somewhat ‘dual’ reduced system, multiplied by a certain prefactor. The reduced system now consists of only the light-tailed flows, with the service rate reduced by the peak rate of the heavy-tailed flows. The prefactor represents the probability that the heavy-tailed flows have sent at their peak rate for more than a certain amount of time, which may be interpreted as the ‘time to overflow’ for the light-tailed flows in the reduced system. The results provide crucial insight into the typical overflow scenario.
TL;DR: In this article, a preorder of linear dependence is defined through inclusion of the zonoid of a d-dimensional random vector, which is used as a tool for measuring linear dependence among its components.
Abstract: The zonoid of a d-dimensional random vector is used as a tool for measuring linear dependence among its components. A preorder of linear dependence is defined through inclusion of the zonoids. The zonoid of a random vector does not characterize its distribution, but it does characterize the size-biased distribution of its compositional variables. This fact will allow a characterization of our linear dependence order in terms of a linear-convex order for the size-biased compositional variables. In dimension 2 the linear dependence preorder will be shown to be weaker than the concordance order. Some examples related to the Marshall-Olkin distribution and to a copula model will be presented, and a class of measures of linear dependence will be proposed.