Showing papers in "Journal of Applied Probability in 2013"
TL;DR: In this article, a functional central limit theorem for the nonlinear Hawkes process is obtained, and a functional law of the iterated logarithm is obtained for the same point process.
Abstract: The Hawkes process is a self-exciting point process with clustering effect whose intensity depends on its entire past history. It has wide applications in neuroscience, finance, and many other fields. In this paper we obtain a functional central limit theorem for the nonlinear Hawkes process. Under the same assumptions, we also obtain a Strassen's invariance principle, i.e. a functional law of the iterated logarithm.
92 citations
TL;DR: In this article, the sharpness of the dual bounds in the homogeneous case under general conditions was established under monotone densities and concave densities, and the corresponding optimal couplings were derived.
Abstract: Sharp tail bounds for the sum of d random variables with given marginal distributions and arbitrary dependence structure have been known since Makarov (1981) and Ruschendorf (1982) for d=2 and, in some examples, for d≥3. Based on a duality result, dual bounds have been introduced in Embrechts and Puccetti (2006b). In the homogeneous case,\break $F1=···=Fn, with monotone density, sharp tail bounds were recently found in Wang and Wang (2011). In this paper we establish the sharpness of the dual bounds in the homogeneous case under general conditions which include, in particular, the case of monotone densities and concave densities. We derive the corresponding optimal couplings and also give an effective method to calculate the sharp bounds.
73 citations
TL;DR: In this paper, Stein's method is applied to obtain bounds of the optimal order for the Wasserstein distance between the distribution of the scaled number of white balls drawn from a Polya-Eggenberger urn and its limiting beta distribution.
Abstract: Using a characterizing equation for the beta distribution, Stein's method is applied to obtain bounds of the optimal order for the Wasserstein distance between the distribution of the scaled number of white balls drawn from a Polya-Eggenberger urn and its limiting beta distribution. The bound is computed by making a direct comparison between characterizing operators of the target and the beta distribution, the former derived by extending Stein's density approach to discrete distributions. In addition, refinements are given to Dobler's (2012) result for the arcsine approximation for the fraction of time a simple random walk of even length spends positive, and so also to the distributions of its last return time to 0 and its first visit to its terminal point, by supplying explicit constants to the present Wasserstein bound and also demonstrating that its rate is of the optimal order.
64 citations
TL;DR: In this article, Chen and Wei introduced a generalized version of the Polya-Eggenberger urn model, where balls are added to the urn according to inspected colors, and exact expressions for the expectation and the variance of the number of white balls after n draws were derived.
Abstract: We study an urn model introduced in the paper of Chen and Wei (2005), where at each discrete time step m balls are drawn at random from the urn containing colors white and black. Balls are added to the urn according to the inspected colors, generalizing the well known Polya-Eggenberger urn model, case m = 1. We provide exact expressions for the expectation and the variance of the number of white balls after n draws, and determine the structure of higher moments. Moreover, we discuss extensions to more than two colors. Furthermore, we introduce and discuss a new urn model where the sampling of the m balls is carried out in a step-by-step fashion, and also introduce a generalized Friedman's urn model.
48 citations
TL;DR: In this paper, a new model for seed banks, where direct ancestors of individuals may have lived in the near as well as the very far past, is presented, and the ancestral process converges to a time-changed Kingman coalescent, while in the case of infinite mean, ancestral lineages might not merge at all with positive probability.
Abstract: We present a new model for seed banks, where direct ancestors of individuals may have lived in the near as well as the very far past. The classical Wright‒Fisher model, as well as a seed bank model with bounded age distribution considered in Kaj, Krone and Lascoux (2001) are special cases of our model. We discern three parameter regimes of the seed bank age distribution, which lead to substantially different behaviour in terms of genetic variability, in particular with respect to fixation of types and time to the most recent common ancestor. We prove that, for age distributions with finite mean, the ancestral process converges to a time-changed Kingman coalescent, while in the case of infinite mean, ancestral lineages might not merge at all with positive probability. Furthermore, we present a construction of the forward-in-time process in equilibrium. The mathematical methods are based on renewal theory, the urn process introduced in Kaj, Krone and Lascoux (2001) as well as on a paper by Hammond and Sheffield (2013).
46 citations
TL;DR: To model point processes with varying baseline intensity, SEPP models with time-varying background intensities (SEPPVB, for short) are introduced and it is shown that SEPPVB models are competitive with autoregressive conditional SE PP models (Engle and Russell 1998) for modeling ultra-high frequency data.
Abstract: Self-exciting point processes (SEPPs), or Hawkes processes, have found applications in a wide range of fields, such as epidemiology, seismology, neuroscience, engineering, and more recently financial econometrics and social interactions. In the traditional SEPP models, the baseline intensity is assumed to be a constant. This has restricted the application of SEPPs to situations where there is clearly a self-exciting phenomenon, but a constant baseline intensity is inappropriate. In this paper, to model point processes with varying baseline intensity, we introduce SEPP models with time-varying background intensities (SEPPVB, for short). We show that SEPPVB models are competitive with autoregressive conditional SEPP models (Engle and Russell 1998) for modeling ultra-high frequency data. We also develop asymptotic theory for maximum likelihood estimation based inference of parametric SEPP models, including SEPPVB. We illustrate applications to ultra-high frequency financial data analysis, and we compare performance with the autoregressive conditional duration models
36 citations
TL;DR: In this paper, a continuously monitored system is considered, which is subject to accumulating deterioration modelled as a gamma process, and a condition-based maintenance strategy is applied, with imperfect repair.
Abstract: A continuously monitored system is considered, which is subject to accumulating deterioration modelled as a gamma process. The system fails when its degradation level exceeds a limit threshold. At failure, a delayed replacement is performed. To shorten the down period, a condition-based maintenance strategy is applied, with imperfect repair. Mimicking virtual age models used for recurrent events, imperfect repair actions are assumed to lower the system degradation through a first-order arithmetic reduction of age model. Under these assumptions, Markov renewal equations are obtained for several reliability indicators. Numerical examples illustrate the behaviour of the system.
34 citations
TL;DR: In this article, the authors consider the two-dimensional risk model introduced in Avram, Palmowski and Pistorius (2008) with constant interest rate and derive integral-differential equations of the Laplace transforms, and asymptotic expressions for the finite-time ruin probabilities with respect to the joint ruin times T max(u 1,u 2) and T min(u 2,u 3) respectively.
Abstract: In this note we consider the two-dimensional risk model introduced in Avram, Palmowski and Pistorius (2008) with constant interest rate. We derive the integral-differential equations of the Laplace transforms, and asymptotic expressions for the finite-time ruin probabilities with respect to the joint ruin times T max(u 1,u 2) and T min(u 1,u 2) respectively.
33 citations
TL;DR: This paper derived conditional distributions for fractional analogies of prominent affine processes, including important examples like fractional Ornstein-Uhlenbeck or fractional Cox-Ingersoll-Ross processes.
Abstract: Conditional distributions for affine Markov processes are at the core of present (defaultable) bond pricing. There is, however, evidence that Markov processes may not be realistic models for short rates. Fractional Brownian motion (FBM) can be introduced by an integral representation with respect to standard Brownian motion. Using a simple prediction formula for the conditional expectation of an FBM and its Gaussianity, we derive the conditional distributions of FBM and related processes. We derive conditional distributions for fractional analogies of prominent affine processes, including important examples like fractional Ornstein–Uhlenbeck or fractional Cox–Ingersoll–Ross processes. As an application, we propose a fractional Vasicek bond market model and compare prices of zero-coupon bonds to those achieved in the classical Vasicek model.
33 citations
TL;DR: In this article, new conditions for the stability of V-geometrically ergodic Markov chains are introduced based on an extension of the standard perturbation theory formulated by Keller and Liverani.
Abstract: In this paper, new conditions for the stability of V-geometrically ergodic Markov chains are introduced. The results are based on an extension of the standard perturbation theory formulated by Keller and Liverani. The continuity and higher regularity properties are investigated. As an illustration, an asymptotic expansion of the invariant probability measure for an autoregressive model with independent and identically distributed noises (with a nonstandard probability density function) is obtained.
31 citations
TL;DR: In this article, the authors consider a multi asset financial market with stochastic volatility modeled by a Wishart process and study the problem of maximizing the expected utility of terminal wealth for power and logarithmic utility.
Abstract: We consider a multi asset financial market with stochastic volatility modeled by a Wishart process. This is an extension of the one-dimensional Heston model. Within this framework we study the problem of maximizing the expected utility of terminal wealth for power and logarithmic utility. We apply the usual stochastic control approach and obtain, explicitly, the optimal portfolio strategy and the value function in some parameter settings. In particular, we do this when the drift of the assets is a linear function of the volatility matrix. In this case the affine structure of the model can be exploited. In some cases we obtain a Feynman-Kac representation of the candidate value function. Though the approach we use is quite standard, the hard part is to identify when the solution of the Hamilton-Jacobi-Bellman equation is finite. This involves a couple of matrix analytic arguments. In a numerical study we discuss the influence of the investors' risk aversion on the hedging demand.
TL;DR: A matrix-analytic scheme is provided for numerically computing stationary expectations corresponding to long-run averages of additive functionals on level-dependent quasi-birth-and-death processes without explicitly computing the stationary distribution of the process, which yields an algorithm that is as quick as its counterparts for stationary distributions but requires far less computer storage.
Abstract: Stationary expectations corresponding to long-run averages of additive functionals on level-dependent quasi-birth-and-death processes are considered. Special cases include long-run average costs or rewards, moments and cumulants of steady-state queueing network performance measures, and many others. We provide a matrix-analytic scheme for numerically computing such stationary expectations without explicitly computing the stationary distribution of the process, which yields an algorithm that is as quick as its counterparts for stationary distributions but requires far less computer storage. Specific problems arising in the case of infinite state spaces are discussed and the application of the algorithm is demonstrated by a queueing network example.
TL;DR: In this article, the authors consider a large class of exponential random graph models and prove the existence of a region of parameter space corresponding to the emergent multipartite structure, separated by a phase transition from disordered graphs.
Abstract: We consider a large class of exponential random graph models and prove the existence of a region of parameter space corresponding to the emergent multipartite structure, separated by a phase transition from a region of disordered graphs. An essential feature is the formalism of graph limits as developed by Lovasz et al. for dense random graphs.
TL;DR: In this article, the authors considered a generalized telegraph process which follows an alternating renewal process and is subject to random jumps, and developed the distribution of the location of the particle at an arbitrary fixed time t, and study this distribution under the assumption of exponentially distributed alternating random times.
Abstract: We consider a generalized telegraph process which follows an alternating renewal process and is subject to random jumps. More specifically, consider a particle at the origin of the real line at time t=0. Then it goes along two alternating velocities with opposite directions, and performs a random jump toward the alternating direction at each velocity reversal. We develop the distribution of the location of the particle at an arbitrary fixed time t, and study this distribution under the assumption of exponentially distributed alternating random times. The cases of jumps having exponential distributions with constant rates and with linearly increasing rates are treated in detail.
TL;DR: In this article, necessary and sufficient conditions for embeddability are formulated and the probability square roots of the transition matrix are presented in analytic form for two-state Markov chains, and properties of row-normalized matrices are examined.
Abstract: When a discrete-time homogenous Markov chain is observed at time intervals that correspond to its time unit, then the transition probabilities of the chain can be estimated using known maximum likelihood estimators. In this paper we consider a situation when a Markov chain is observed on time intervals with length equal to twice the time unit of the Markov chain. The issue then arises of characterizing probability matrices whose square root(s) are also probability matrices. This characterization is referred to in the literature as the embedding problem for discrete time Markov chains. The probability matrix which has probability root(s) is called embeddable. In this paper for two-state Markov chains, necessary and sufficient conditions for embeddability are formulated and the probability square roots of the transition matrix are presented in analytic form. In finding conditions for the existence of probability square roots for (k x k) transition matrices, properties of row-normalized matrices are examined. Besides the existence of probability square roots, the uniqueness of these solutions is discussed: In the case of nonuniqueness, a procedure is introduced to identify a transition matrix that takes into account the specificity of the concrete context. In the case of nonexistence of a probability root, the concept of an approximate probability root is introduced as a solution of an optimization problem related to approximate nonnegative matrix factorization.
TL;DR: In this paper, a general homogeneous continuous-time Markov process with restarts is considered, where the process is forced to restart from a given distribution at time moments generated by an independent Poisson process.
Abstract: We consider a general homogeneous continuous-time Markov process with restarts. The process is forced to restart from a given distribution at time moments generated by an independent Poisson process. The motivation to study such processes comes from modeling human and animal mobility patterns, restart processes in communication protocols, and from application of restarting random walks in information retrieval. We provide a connection between the transition probability functions of the original Markov process and the modified process with restarts. We give closed-form expressions for the invariant probability measure of the modified process. When the process evolves on the Euclidean space, there is also a closed-form expression for the moments of the modified process. We show that the modified process is always positive Harris recurrent and exponentially ergodic with the index equal to (or greater than) the rate of restarts. Finally, we illustrate the general results by the standard and geometric Brownian motions.
TL;DR: A general set of sufficient conditions which are invariant to orthonormal transformation of the coordinate axes and which ensure that the limiting optimal acceptance rate is 0.234 are presented.
Abstract: Scaling of proposals for Metropolis algorithms is an important practical problem in Markov chain Monte Carlo implementation. Analyses of the random walk Metropolis for high-dimensional targets with specific functional forms have shown that in many cases the optimal scaling is achieved when the acceptance rate is approximately 0.234, but that there are exceptions. We present a general set of sufficient conditions which are invariant to orthonormal transformation of the coordinate axes and which ensure that the limiting optimal acceptance rate is 0.234. The criteria are shown to hold for the joint distribution of successive elements of a stationary pth-order multivariate Markov process.
TL;DR: In this paper, the authors study asymptotic consistency of law invariant convex risk measures and the corresponding risk averse stochastic programming problems for independent, identically distributed data.
Abstract: In this paper we study asymptotic consistency of law invariant convex risk measures and the corresponding risk averse stochastic programming problems for independent, identically distributed data. Under mild regularity conditions, we prove a law of large numbers and epiconvergence of the corresponding statistical estimators. This can be applied in a straightforward way to establish convergence with probability 1 of sample-based estimators of risk averse stochastic programming problems.
TL;DR: In this paper, the authors studied the two-sided exit problem of a time-homogeneous diffusion process with tax payments of loss-carry-forward type and obtained explicit formulae for the Laplace transforms associated with the problem.
Abstract: We study the two-sided exit problem of a time-homogeneous diffusion process with tax payments of loss-carry-forward type and obtain explicit formulae for the Laplace transforms associated with the two-sided exit problem. The expected present value of tax payments until default, the two-sided exit probabilities, and, hence, the nondefault probability with the default threshold equal to the lower bound are solved as immediate corollaries. A sufficient and necessary condition for the tax identity in ruin theory is discovered.
TL;DR: The authors survey the results of Bertoin and Uribe Bravo about Bernoulli bond percolation on certain large random trees with logarithmic height and provide a general criterion for the existence of giant percolations clusters in large trees.
Abstract: This paper is based on works presented at the 2012 Applied Probability Trust Lecture in Sheffield; its main purpose is to survey the recent asymptotic results of Bertoin (2012a) and Bertoin and Uribe Bravo (2012b) about Bernoulli bond percolation on certain large random trees with logarithmic height. We also provide a general criterion for the existence of giant percolation clusters in large trees, which answers a question raised by David Croydon.
TL;DR: In this paper, a characterization of continuous or discrete distributions with shape monotone of finite integer order t is presented as a mixture of a minimum of t independent uniform distributions, and a comparison of t-monotone distributions is made using the s-convex stochastic orders.
Abstract: This paper is concerned with the class of distributions, continuous or discrete, whose shape is monotone of finite integer order t. A characterization is presented as a mixture of a minimum of t independent uniform distributions. Then, a comparison of t-monotone distributions is made using the s-convex stochastic orders. A link is also pointed out with an alternative approach to monotonicity based on a stationary-excess operator. Finally, the monotonicity property is exploited to reinforce the classical Markov and Lyapunov inequalities. The results are illustrated by several applications to insurance.
TL;DR: In this paper, a mixture representation of the residual lifetime and the inactivity time of systems with such failure-dependent components are considered. But the authors do not consider the effect of the failure of a component on the performance of remaining components.
Abstract: Sequential order statistics can be used to describe the ordered lifetimes of components in a system, where the failure of a component may affect the performance of remaining components. In this paper mixture representations of the residual lifetime and the inactivity time of systems with such failure-dependent components are considered. Stochastic comparisons of differently structured systems are obtained and properties of the weights in the mixture representations are examined. Furthermore, corresponding representations of the residual lifetime and the inactivity time of a system given the additional information about a previous failure time are derived.
TL;DR: In this paper, the authors considered a branching population with independent and identically distributed (i.i.d.) life lengths (not necessarily exponential) and constant birth rates, and showed that the joint law of (Nt, T, X(T)), where X (T) is the jumping contour process of the tree truncated at time T, is equal to that of (M, -IM, Y'M) conditional on {M≠0}.
Abstract: We consider a branching population where individuals have independent and identically distributed (i.i.d.) life lengths (not necessarily exponential) and constant birth rates. We let Nt denote the population size at time t. We further assume that all individuals, at their birth times, are equipped with independent exponential clocks with parameter δ. We are interested in the genealogical tree stopped at the first time T when one of these clocks rings. This question has applications in epidemiology, population genetics, ecology, and queueing theory. We show that, conditional on {T<∞}, the joint law of (Nt, T, X(T)), where X(T)is the jumping contour process of the tree truncated at time T, is equal to that of (M, -IM, Y'M) conditional on {M≠0}. HereM+1 is the number of visits of 0, before some single, independent exponential clock e with parameter δ rings, by some specified Levy process Y without negative jumps reflected below its supremum; IM is the infimum of the path YM, which in turn is defined as Y killed at its last visit of 0 before e; and Y'M is the Vervaat transform of YM. This identity yields an explanation for the geometric distribution of NT (see Kitaev (1993) and Trapman and Bootsma (2009)) and has numerous other applications. In particular, conditional on {NT=n}, and also on {NT=n,T
TL;DR: In this paper, the authors make a connection between continuous time and lazy discrete time Markov chains through the comparison of cutoffs and mixing time in total variation distance, and provide a criterion on cutoffs using eigenvalues of the transition matrix.
Abstract: We make a connection between the continuous time and lazy discrete time Markov chains through the comparison of cutoffs and mixing time in total variation distance. For illustration, we consider finite birth and death chains and provide a criterion on cutoffs using eigenvalues of the transition matrix.
TL;DR: In this article, limit theorems for the generalized Feller game were studied and the weak law of large numbers for α = 1 was shown to hold for α > 0.
Abstract: In this paper we study limit theorems for the Feller game which is constructed from one-dimensional simple symmetric random walks, and corresponds to the St. Petersburg game. Motivated by a generalization of the St. Petersburg game which was investigated by Gut (2010), we generalize the Feller game by introducing the parameter α. We investigate limit distributions of the generalized Feller game corresponding to the results of Gut. Firstly, we give the weak law of large numbers for α=1. Moreover, for 0<α≤1, we have convergence in distribution to a stable law with index α. Finally, some limit theorems for a polynomial size and a geometric size deviation are given.
TL;DR: In this paper, the authors consider a stochastic SIR epidemic on a random graph with specified degree distribution, constructed using the configuration model, and investigate the "acquaintance vaccination" method for targeting individuals of high degree for vaccination.
Abstract: We consider a stochastic SIR (susceptible ? infective ? removed) epidemic on a random graph with specified degree distribution, constructed using the configuration model, and investigate the 'acquaintance vaccination' method for targeting individuals of high degree for vaccination. Branching process approximations are developed which yield a post-vaccination threshold parameter, and the asymptotic (large population) probability and final size of a major outbreak. We find that introducing an imperfect vaccine response into the present model for acquaintance vaccination leads to sibling dependence in the approximating branching processes, which may then require infinite type spaces for their analysis and are generally not amenable to numerical calculation. Thus, we propose and analyse an alternative model for acquaintance vaccination, which avoids these difficulties. The theory is illustrated by a brief numerical study, which suggests that the two models for acquaintance vaccination yield quantitatively very similar disease properties
TL;DR: In this article, the optimal timing for an asset sale for an agent with a long position in a momentum trade is studied, where the asset price is modelled as a geometric Brownian motion with a drift that initially exceeds the discount rate, but with the opposite relation after an unobservable and exponentially distributed time.
Abstract: There is an extensive academic literature that documents that stocks which have performed well in the past often continue to perform well over some holding period - so-called momentum. We study the optimal timing for an asset sale for an agent with a long position in a momentum trade. The asset price is modelled as a geometric Brownian motion with a drift that initially exceeds the discount rate, but with the opposite relation after an unobservable and exponentially distributed time. The problem of optimal selling of the asset is then formulated as an optimal stopping problem under incomplete information. Based on the observations of the asset, the agent wants to detect the unobservable change point as accurately as possible. Using filtering techniques and stochastic analysis, we reduce the problem to a one-dimensional optimal stopping problem, which we solve explicitly. We also show that the optimal boundary at which the investor should liquidate the trade depends monotonically on the model parameters.
TL;DR: In this article, the convergence rate of the distribution of the normalized maximum to the Gumbel extreme value distribution is proportional to 1/(log n) ½, where n is the number of nodes in the graph.
Abstract: We discuss tail behaviors, subexponentiality, and the extreme value distribution of logarithmic skew-normal random variables. With optimal normalized constants, the asymptotic expansion of the distribution of the normalized maximum of logarithmic skew-normal random variables is derived. We show that the convergence rate of the distribution of the normalized maximum to the Gumbel extreme value distribution is proportional to 1/(log n)½.
TL;DR: In this paper, the authors considered a drift parameter estimation problem when the state process is a reflected fractional Brownian motion (RFBM) with a nonzero drift parameter and the observation is the associated local time process and established its strong consistency and asymptotic normality.
Abstract: We consider a drift parameter estimation problem when the state process is a reflected fractional Brownian motion (RFBM) with a nonzero drift parameter and the observation is the associated local time process. The RFBM process arises as the key approximating process for queueing systems with long-range dependent and self-similar input processes, where the drift parameter carries the physical meaning of the surplus service rate and plays a central role in the heavy-traffic approximation theory for queueing systems. We study a statistical estimator based on the cumulative local time process and establish its strong consistency and asymptotic normality.
TL;DR: In this paper, a response adaptive design, described in terms of a two-color urn model, targeting fixed asymptotic allocations, is presented in an estimation problem context.
Abstract: We construct a response adaptive design, described in terms of a two-color urn model, targeting fixed asymptotic allocations. We prove asymptotic results for the process of colors generated by the urn and for the process of its compositions. An application of the proposed urn model is presented in an estimation problem context.