Showing papers in "Advances in Applied Probability in 1996"

Anticipative portfolio optimization

Abstract: We study a classical stochastic control problem arising in financial economics: to maximize expected logarithmic utility from terminal wealth and/or consumption. The novel feature of our work is that the portfoilo is allowed to anticipate the future, i.e. the terminal values of the prices, or of the driving Brownian motion, are known to the investor, either exactly or with some uncertainty. Results on the finiteness of the value of the control problem are obtained in various setups, using techniques from the so-called enlargement of filtrations. When the value of the problem is finite, we compute it explicitly and exhibit an optimal portfolio in closed form.

The quasi-stationary distribution of the closed endemic sis model

Abstract: The quasi-stationary distribution of the closed stochastic SIS model changes drastically as the basic reproduction ratio Ro passes the deterministic threshold value 1. Approximations are denved that describe these changes. The quasi-stationary distnbution is approximated by a geometric distribution (discrete!) for R,, distinctly below 1 and by a normal distribution (continuous! ) for Rn distinctly above 1. Uniformity of the approximation with respect to Ro allows one to study the transition between these two extreme distributions. We also study the time to extinction and the invasion and persistence thresholds of the model.

On a Voronoi Aggregative Process Related to a Bivariate Poisson Process

Abstract: We consider two independent homogeneous Poisson processes Π 0 and Π 1 in the plane with intensities λ 0 and λ 1 , respectively. We study additive functionals of the set of Π 0 -particles within a typical Voronoi Π 1 -cell. We find the first and the second moments of these variables as well as upper and lower bounds on their distribution functions, implying an exponential asymptotic behavior of their tails. Explicit formulae are given for the number and the sum of distances from Π 0 -particles to the nucleus within a typical Voronoi Π 1 -cell.

Spectral analysis of M/G/ 1 and G / M /1 type Markov chains

Abstract: When analyzing the equilibrium behavior of M/G/1 type Markov chains by transform methods, restrictive hypotheses are often made to avoid technical problems that arise in applying results from complex analysis and linear algebra. It is shown that such restrictive assumptions are unnecessary, and an analysis of these chains using generating functions is given under only the natural hypotheses that first moments (or second moments in the null recurrent case) exist. The key to the analysis is the identification of an important subspace of the space of bounded solutions of the system of homogeneous vector-valued Wiener–Hopf equations associated with the chain. In particular, the linear equations in the boundary probabilities obtained from the transform method are shown to correspond to a spectral basis of the shift operator on this subspace. Necessary and sufficient conditions under which the chain is ergodic, null recurrent or transient are derived in terms of properties of the matrix-valued generating functions determined by transitions of the Markov chain. In the transient case, the Martin exit boundary is identified and shown to be associated with certain eigenvalues and vectors of one of these generating functions. An equilibrium analysis of the class of G/M/1 type Markov chains by similar methods is also presented.

Random discrete distributions invariant under size-biased permutation

Abstract: Invariance of a random discrete distribution under size-biased permutation is equivalent to a conjunction of symmetry conditions on its finite-dimensional distributions. This is applied to characterize residual allocation models with independent factors that are invariant under size-biased permutation. Apart from some exceptional cases and minor modifications, such models form a two-parameter family of generalized Dirichlet distributions.

Perpetuities with thin tails

Abstract: We investigate the behaviour of P(R ≥ r) and P(R ≤ - r) as r → ∞ for the random variable R:= Formula math M k , where ((Q k , M k )) k ∈ N is an independent, identically distributed sequence with P(-1 ≤ M ≤ 1) = 1. Random variables of this type appear in insurance mathematics, as solutions of stochastic difference equations, in the analysis of probabilistic algorithms and elsewhere. Exponential and Poissonian tail behaviour can arise.

Queues with marked customers

Abstract: Queueing systems with distinguished arrivals are described on the basis of Markov arrival processes with marked transitions. Customers are distinguished by their types of arrival. Usually, the queues observed by customers of different types are different, especially for queueing systems with bursty arrival processes. We study queueing systems from the points of view of customers of different types. A detailed analysis of the fundamental period, queue lengths and waiting times at the epochs of arrivals is given. The results obtained are the generalizations of the results of the MAP/G/1 queue.

Some asymptotic results for transient random walks

Abstract: We consider a real-valued random walk S which drifts to -x and is such that E(exp OS) 0, but for which Cramdr's condition fails. We investigate the asymptotic tail behaviour of the distributions of the all time maximum, the upwards and downwards first passage times and the last passage times. As an application, we obtain new limit theorems for certain conditional laws. TRANSIENT RANDOM WALKS

Large deviations for discrete and continuous percolation

Abstract: Motivated by a statistical application, we consider continuum percolation in two or more dimensions, restricted to a large finite box, when above the critical point. We derive surface order large deviation estimates for the volume of the largest cluster and for its intersection with the boundary of the box. We also give some natural extensions to known, analogous results on lattice percolation.

Portfolio Choice and the Bayesian Kelly Criterion

Abstract: We derive optimal gambling and investment policies for cases in which the underlying stochastic process has parameter values that are unobserved random variables. For the objective of maximizing logarithmic utility when the underlying stochastic process is a simple random walk in a random environment, we show that a state-dependent control is optimal, which is a generalization of the celebrated Kelly strategy: the optimal strategy is to bet a fraction of current wealth equal to a linear function of the posterior mean increment. To approximate more general stochastic processes, we consider a continuous-time analog involving Brownian motion. To analyze the continuous-time problem, we study the diffusion limit of random walks in a random environment. We prove that they converge weakly to a Kiefer process, or tied-down Brownian sheet. We then find conditions under which the discrete-time process converges to a diffusion, and analyze the resulting process. We analyze in detail the case of the natural conjugate prior, where the success probability has a beta distribution, and show that the resulting limit diffusion can be viewed as a rescaled Brownian motion. These results allow explicit computation of the optimal control policies for the continuous-time gambling and investment problems without resorting to continuous-time stochastic-control procedures. Moreover they also allow an explicit quantitative evaluation of the financial value of randomness, the financial gain of perfect information and the financial cost of learning in the Bayesian problem.

The M/G/1 queue with negative customers

Abstract: We derive expressions for the generating function of the equilibrium queue length probability distribution in a single server queue with general service times and independent Poisson arrival streams of both ordinary, positive customers and negative customers which eliminate a positive customer if present. For the case of first come first served queueing discipline for the positive customers, we compare the killing strategies in which either the last customer in the queue or the one in service is removed by a negative customer. We then consider preemptive-restart with resampling last come first served queueing discipline for the positive customers, combined with the elimination of the customer in service by a negative customer- the case of elimination of the last customer yields an analysis similar to first come first served discipline for positive customers. The results show different generating functions in contrast to the case where service times are exponentially distributed. This is also reflected in the stability conditions. Incidently, this leads to a full study of the preemptive-restart with resampling last come first served case without negative customers. Finally, approaches to solving the Fredholm integral equation of the first kind which arises, for instance, in the first case are considered as well as an alternative iterative solution method.

Central limit theorem for a class of random measures associated with germ-grain models

Abstract: We introduce a family of stationary random measures in the Euclidean space generated by so-called germ-grain models. The germ-grain model is defined as the union of i.i.d. compact random sets (grains) shifted by points (germs) of a point process. This model gives rise to random measures defined by the sum of contributions of non-overlapping parts of the individual grains. The corresponding moment measures are calculated and an ergodic theorem is presented. The main result is the central limit theorem for the introduced random measures, which is valid for rather general independently marked germ-grain models, including those with non-Poisson distribution of germs and non-convex grains. The technique is based on a central limit theorem for β-mixing random fields. It is shown that this construction of random measures includes those random measures obtained by the so-called positive extensions of intrinsic volumes. In the Poisson case it is possible to prove a central limit theorem under weaker assumptions by using approximations by m-dependent random fields. Applications to statistics of the Boolean model are also discussed. They include a standard way to derive limit theorems for estimators of the model parameters.

Rates of convergence for random approximations of convex sets

Abstract: The Hausdorff distance between a compact convex set K CRd and random sets K c lRd iS studied. Basic inequalities are derived for the case of K being a convex subset of K. If applied to special sequences of such random sets, these inequalities yield rates of almost sure convergence. With the help of duality considerations these results are extended to the case of K being the intersection of a random family of halfspaces containing K

Stationary distributions under mutation- selection balance: structure and properties

Abstract: A general model for the evolution of the frequency distribution of types in a population under mutation and selection is derived and investigated. The approach is sufficiently general to subsume classical models with a finite number of alleles, as well as models with a continuum of possible alleles as used in quantitative genetics. The dynamics of the corresponding probability distributions is governed by an integrodifferential equation in the Banach space of Borel measures on a locally compact space. Existence and uniqueness of the solutions of the initial value problem is proved using basic semigroup theory. A complete characterization of the structure of stationary distributions is presented. Then, existence and uniqueness of stationary distributions is proved under mild conditions by applying operator theoretic generalizations of Perron-Frobenius theory. For an extension of Kingman's original house-of-cards model, a classification of possible stationary distributions is obtained. PROBABILITY MEASURES; PERRON-FROBENIUS THEORY; INTEGRO-DIFFERENTIAL EQUATIONS; POSITIVE OPERATORS; CONTINUUM-OF-ALLELES

Asymptotic distribution theory for hoare's selection algorithm

Abstract: We investigate the asymptotic behaviour of the distribution of the number of comparisons needed by a quicksort-style selection algorithm that finds the lth smallest in a set of n numbers. Letting n tend to infinity and considering the values = 1, * , n simultaneously we obtain a limiting stochastic process. This process admits various interpretations: it arises in connection with a representation of real numbers induced by nested random partitions and also in connection with expected path lengths of a random walk in a random environment on a binary tree. STOCHASTIC ALGORITHMS; ASYMPTOTIC DISTRIBUTION; ORDER STATISTICS; RANDOM BINARY NUMBERS; RANDOM WALK IN A RANDOM ENVIRONMENT

General conditions for bounded relative error in simulations of highly reliable Markovian systems

Abstract: We establish a necessary condition for any importance sampling scheme to give bounded relative error when estimating a performance measure of a highly reliable Markovian system. Also, a class of importance sampling methods is defined for which we prove a necessary and sufficient condition for bounded relative error for the performance measure estimator. This class of probability measures includes all of the currently existing failure biasing methods in the literature. Similar conditions for derivative estimators are established.

Convergence of stochastic algorithms: from the kushner-clark theorem to the lyapounov functional method

Abstract: In the first part of this paper a global Kushner-Clark theorem about the convergence of stochastic algorithms is proved: we show that, under some natural assumptions, one can 'read' from the trajectories of its ODE whether or not an algorithm converges. The classical stochastic optimization results are included in this theorem. In the second part, the above smoothness assumption on the mean vector field of the algorithm is relaxed using a new approach based on a path-dependent Lyapounov functional. Several applications, for non-smooth mean vector fields and/or bounded Lyapounov function settings, are derived. Examples and simulations are provided that illustrate and enlighten the field of application of the theoretical results.

An operational calculus for probability distributions via laplace transforms

Abstract: In this paper we investigate operators that map one or more probability distributions on the positive real line into another via their Laplace-Stieltjes transforms. Our goal is to make it easier to construct new transforms by manipulating known transforms. We envision the results here assisting modelling in conjunction with numerical transform inversion software. We primarily focus on operators related to infinitely divisible distributions and Levy processes, drawing upon Feller (1971). We give many concrete examples of infinitely divisible distributions. We consider a cumulant-moment-transfer operator that allows us to relate the cumulants of one distribution to the moments of another. We consider a power-mixture operator corresponding to an independently stopped Levy process. The special case of exponential power mixtures is a continuous analog of geometric random sums. We introduce a further special case which is remarkably tractable, exponential mixtures of inverse Gaussian distributions (EMIGs). EMIGs arise naturally as approximations for busy periods in queues. We show that the steady-state waiting time in an M/G/1 queue is the difference of two EMIGs when the service-time distribution is an EMIG. We consider several transforms related to first-passage times, e.g. for the M/M/1 queue, reflected Brownian motion and Levy processes. Some of the associated probability density functions involve Bessel functions and theta functions. We describe properties of the operators, including how

Markov chain models, time series analysis and extreme value theory

Abstract: Markov chain processes are becoming increasingly popular as a means of modelling various phenomena in different disciplines. For example, a new approach to the investigation of the electrical activity of molecular structures known as ion channels is to analyse raw digitized current recordings using Markov chain models. An outstanding question which arises with the application of such models is how to determine the number of states required for the Markov chain to characterize the observed process. In this paper we derive a realization theorem showing that observations on a finite state Markov chain embedded in continuous noise can be synthesized as values obtained from an autoregressive moving-average data generating mechanism. We then use this realization result to motivate the construction of a procedure for identifying the state dimension of the hidden Markov chain. The identification technique is based on a new approach to the estimation of the order of an autoregressive moving-average process. Conditions for the method to produce strongly consistent estimates of the state dimension are given. The asymptotic distribution of the statistic underlying the identification process is also presented and shown to yield critical values commensurate with the requirements for strong consistency.

Markov connected component fields

Abstract: A new class of Gibbsian models with potentials associated with the connected components or homogeneous parts of images is introduced. For these models the neighbourhood of a pixel is not fixed as for Markov random fields, but is given by the components which are adjacent to the pixel. The relationship to Markov random fields and marked point processes is explored and spatial Markov properties are established. Extensions to infinite lattices are also studied, and statistical inference problems including geostatistical applications and statistical image analysis are discussed. Finally, simulation studies are presented which show that the models may be appropriate for a variety of interesting patterns, including images exhibiting intermediate degrees of spatial continuity and images of objects against background.

Equivalent necessary and sufficient conditions on noise sequences for stochastic approximation algorithms

Abstract: We consider stochastic approximation algorithms on a general Hilbert space, and study four conditions on noise sequences for their analysis: Kushner and Clark's condition, Chen's condition, a decomposition condition, and Kulkarni and Horn's condition. We discuss various properties of these conditions. In our main result we show that the four conditions are all equivalent, and are both necessary and sufficient for convergence of stochastic approximation algorithms under appropriate assumptions.

On evaluations and asymptotic approximations of first-passage-time probabilities

Abstract: The series expansion for the solution of the integral equation for the first-passagetime probability density function, obtained by resorting to the fixed point theorem, is used to achieve approximate evaluations for which error bounds are indicated. A different use of the fixed point theorem is then made to determine lower and upper bounds for asymptotic approximations, and to examine their range of validity. FIRST-PASSAGE-TIME; DIFFUSION PROCESSES; ORSTEIN-UHLENBECK PROCESSES;

Large deviations in the piecewise linear approximation of Gaussian processes with stationary increments

Abstract: We consider the piecewise linear interpolation of Gaussian processes with continuous sample paths and stationary increments. The interrelation between the smoothness of the incremental variance function, d(t-s) = E[(X(t)-X(s)) 2 ], and the interpolation errors in mean square and uniform metrics is studied. The method of investigation can also be applied to the analysis of different methods of interpolation. It is based on some limit results for large deviations of a sequence of Gaussian non-stationary processes and related point processes. Non-stationarity in our case means mainly the local stationary condition for the sequence of correlation functions r n (t, s), n = 1, 2,..., which has to hold uniformly in n. Finally, we discuss some examples and an application to the calculation of the distribution function of the maximum of a continuous Gaussian process with a given precision.

Markov properties of cluster processes

Abstract: We show that a Poisson cluster point process is a nearest-neighbour Markov point process [2] if the clusters have uniformly bounded diameter. It is typically not a finite-range Markov point process in the sense of Ripley and Kelly [12]. Furthermore, when the parent Poisson process is replaced by a Markov or nearest-neighbour Markov point process, the resulting cluster process is also nearest-neighbour Markov, provided all clusters are non-empty. In particular, the nearest-neighbour Markov property is preserved when points of the process are independently randomly translated, but not when they are randomly thinned.

Gamma type results and other related properties of poisson processes

Abstract: Families of Poisson processes defined on general state spaces and with the intensity measure scaled by a positive parameter are investigated. In particular, mean value relations with respect to the scale parameter are established and used to derive various Gamma-type results for certain geometric characteristics determined by finite subprocesses. In particular, we deduce Miles' complementary theorem. Applications of the results within stochastic geometry and particularly for random tessellations are discussed.

First crossing of basic counting processes with lower non-linear boundaries: A unified approach through pseudopolynomials (I)

Abstract: The paper is concerned with the distribution of the level N of the first crossing of a counting process trajectory with a lower boundary. Compound and simple Poisson or binomial processes, gamma renewal processes, and finally birth processes are considered. In the simple Poisson case, expressing the exact distribution of N requires the use of a classical family of Abel-Gontcharoff polynomials. For other cases convenient extensions of these polynomials into pseudopolynomials with a similar structure are necessary. Such extensions being applicable to other fields of applied probability, the central part of the present paper has been devoted to the building of these pseudopolynomials in a rather general framework.

Homogeneous rectangular tessellations

Abstract: A rectangular tessellation is a covering of the plane by non-overlapping rectangles. A basic theory for general homogeneous random rectangular tessellations is developed, and it is shown that many first-order mean values may be expressed in terms of just three basic quantities. Corresponding values for independent superpositions of two or more such tessellations are derived. The most interesting homogeneous rectangular tessellations are those with only T-vertices (i.e. no X-vertices). Gilbert's (1967) isotropic model adapted to this two-orthogonal-orientations case, although simply specified, appears theoretically intractable, due to a complex ‘blocking' effect. However, the approximating penetration model, also introduced by Gilbert, is found to be both tractable and informative about the true model. A multi-stage method for simulating the model is developed, and the distributions of important characteristics estimated.

The range of a simple random walk on ℤ

Abstract: Let θ(a) be the first time when the range (R n ; n ≥ 0) is equal to a, R n being equal to the difference of the maximum and the minimum, taken at time n, of a simple random walk on Z. We compute the g.f. of θ(a); this allows us to compute the distributions of θ(a) and R n . We also investigate the asymptotic behaviour of θ(n), n going to infinity.

Confidence bounds for the adjustment coefficient

Abstract: Let ψ(u) be the probability of eventual ruin in the classical Sparre Andersen model of risk theory if the initial risk reserve is u. For a large class of such models ψ(u) behaves asymptotically like a multiple of exp (-Ru) where R is the adjustment coefficient; R depends on the premium income rate, the claim size distribution and the distribution of the time between claim arrivals. Estimation of R has been considered by many authors. In the present paper we deal with confidence bounds for R. A variety of methods is used, including jackknife estimation of asymptotic variances and the bootstrap. We show that, under certain assumptions, these procedures result in interval estimates that have asymptotically the correct coverage probabilities. We also give the results of a simulation study that compares the different techniques in some particular cases.

Throughput properties of a queueing network with distributed dynamic routing and flow control

Abstract: A queueing network with arbitrary topology, state dependent routing and flow control is considered. Customers may enter the network at any queue and they are routed through it until they reach certain queues from which they may leave the system. The routing is based on local state information. The service rate of a server is controlled based on local state information as well. A distributed policy for routing and service rate control is identified that achieves maximum throughput. The policy can be implemented without knowledge of the arrival and service rates. The importance of flow control is demonstrated by showing that, in certain networks, if the servers cannot be forced to idle, then no maximum throughput policy exists when the arrival rates are not known. Also a model for exchange of state information among neighboring nodes is presented and the network is studied when the routing is based on delayed state information. A distributed policy is shown to achieve maximum throughput in the case of delayed state information. Finally, some implications for deterministic flow networks are discussed.