Showing papers in "Journal of Applied Probability in 2014"
TL;DR: In this paper, the authors introduce a new class of Monte Carlo methods, called exact estimation algorithms, which provide unbiased estimators for equilibrium expectations associated with real-valued functionals defined on a Markov chain.
Abstract: We introduce a new class of Monte Carlo methods, which we call exact estimation algorithms. Such algorithms provide unbiased estimators for equilibrium expectations associated with real-valued functionals defined on a Markov chain. We provide easily implemented algorithms for the class of positive Harris recurrent Markov chains, and for chains that are contracting on average. We further argue that exact estimation in the Markov chain setting provides a significant theoretical relaxation relative to exact simulation methods.
115 citations
TL;DR: In this paper, a renewal risk model in which the surplus process of the insurance company is modelled by a compound fractional Poisson process is presented, and some results for the ruin probabilities are presented in various assumptions on the distribution of the claim sizes.
Abstract: We study a renewal risk model in which the surplus process of the insurance company is modelled by a compound fractional Poisson process. We establish the long-range dependence property of this non-stationary process. Some results for the ruin probabilities are presented in various assumptions on the distribution of the claim sizes.
60 citations
TL;DR: In this paper, a stochastic Cox-Ingersoll-Ross process with Hawkes jumps is proposed, which is a generalization of the classical Cox-Inersoll Ross process and the classical Hawkes process with exponential exciting function.
Abstract: In this paper we propose a stochastic process, which is a Cox-Ingersoll-Ross process with Hawkes jumps. It can be seen as a generalization of the classical Cox-Ingersoll-Ross process and the classical Hawkes process with exponential exciting function. Our model is a special case of the affine point processes. We obtain Laplace transforms and limit theorems, including the law of large numbers, central limit theorems, and large deviations.
59 citations
TL;DR: In this paper, two fractional versions of a family of nonnegative integer-valued processes are considered and it is shown that their probability mass functions solve fractional Kolmogorov forward equations.
Abstract: We consider two fractional versions of a family of nonnegative integer-valued processes. We prove that their probability mass functions solve fractional Kolmogorov forward equations, and we show the overdispersion of these processes. As particular examples in this family, we can define fractional versions of some processes in the literature as the Polya-Aeppli process, the Poisson inverse Gaussian process, and the negative binomial process. We also define and study some more general fractional versions with two fractional parameters
48 citations
TL;DR: In this article, the authors studied the one-dimensional random motion X = X(t), t ≥ 0, which takes two different velocities with two different alternating intensities, and obtained closed-form formulae for the density functions of X and for the moments of any order.
Abstract: We study the one-dimensional random motion X = X(t), t ≥ 0, which takes two different velocities with two different alternating intensities. The closed-form formulae for the density functions of X and for the moments of any order, as well as the distributions of the first passage times, are obtained. The limit behaviour of the moments is analysed under nonstandard Kac's scaling.
47 citations
TL;DR: An insurance ruin model with adaptive premium rate, thereafter refered to as restructuring/refraction, in which classical ruin and bankruptcy are distinguished is introduced, which is mainly focused on the time a refracted L\'evy risk process spends in the red zone.
Abstract: In this paper we introduce an insurance ruin model with an adaptive premium rate, henceforth referred to as restructuring/refraction, in which classical ruin and bankruptcy are distinguished. In this model the premium rate is increased as soon as the wealth process falls into the red zone and is brought back to its regular level when the wealth process recovers. The analysis is focused mainly on the time a refracted Levy risk process spends in the red zone (analogous to the duration of the negative surplus). Building on results from [11] and [16], we identify the distribution of various functionals related to occupation times of refracted spectrally negative Levy processes. For example, these results are used to compute both the probability of bankruptcy and the probability of Parisian ruin in this model with restructuring.
40 citations
TL;DR: In this article, a sufficient and necessary condition to have uniqueness of the location of the maximum of a stochastic process over an interval is provided. But this condition is not applicable to the case of a Brownian motion with variable drift.
Abstract: In this short article we will provide a sufficient and necessary condition to have uniqueness of the location of the maximum of a stochastic process over an interval. The result will also express the mean value of the location in terms of the derivative of the expectation of the maximum of a linear perturbation of the underlying process. As an application, we will consider a Brownian motion with variable drift. The ideas behind the method of proof will also be useful to study the location of the maximum, over the real line, of a two-sided Brownian motion minus a parabola and of a stationary process minus a parabola.
36 citations
TL;DR: In this article, a random multigraph G * with given vertex degrees d 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 14, 15, 16, 17, 18, 20, 21, 22, 23, 24, 25, 26, 28, 30, 31, 34, 35, 36, 38, 39, 40, 41, 42
Abstract: Consider a random multigraph G * with given vertex degrees d 1,…, d n , constructed by the configuration model. We give a new proof of the fact that, asymptotically for a sequence of such multigraphs with the number of edges the probability that the multigraph is simple stays away from 0 if and only if The new proof uses the method of moments, which makes it possible to use it in some applications concerning convergence in distribution. Corresponding results for bipartite graphs are included.
31 citations
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TL;DR: In this paper, the forward and backward tail chains of a Markovian forward tail chain were analyzed and shown to mutually determine each other through a kind of adjoint relation, and the resulting class of limiting processes was analyzed in detail.
Abstract: The extremes of a univariate Markov chain with regulary varying stationary marginal distribution and asymptotically linear behavior are known to exhibit a multiplicative random walk structure called the tail chain. In this paper, we extend this fact to Markov chains with multivariate regularly varying marginal distribution in R^{d}. We analyze both the forward and the backward tail process and show that they mutually determine each other through a kind of adjoint relation. In a broader setting, it will be seen that even for non-Markovian underlying processes a Markovian forward tail chain always implies that the backward tail chain is Markovian as well. We analyze the resulting class of limiting processes in detail. Applications of the theory yield the asymptotic distribution of both the past and the future of univariate and multivariate stochastic difference equations conditioned on an extreme event.
30 citations
TL;DR: In this paper, the authors studied the long-time behavior of a Markov process evolving in N and conditioned not to hit 0 and showed that the process admits a unique quasistationary distribution (in particular, the distribution admits a limit when time goes to ∞).
Abstract: We study the long-time behaviour of a Markov process evolving in N and conditioned not to hit 0. Assuming that the process comes back quickly from ∞, we prove that the process admits a unique quasistationary distribution (in particular, the distribution of the conditioned process admits a limit when time goes to ∞). Moreover, we prove that the distribution of the process converges exponentially fast in the total variation norm to its quasistationary distribution and we provide a bound for the rate of convergence. As a first application of our result, we bring a new insight on the speed of convergence to the quasistationary distribution for birth-and-death processes: we prove that starting from any initial distribution the conditional probability converges in law to a unique distribution ρ supported in N * if and only if the process has a unique quasistationary distribution. Moreover, ρ is this unique quasistationary distribution and the convergence is shown to be exponentially fast in the total variation norm. Also, considering the lack of results on quasistationary distributions for nonirreducible processes on countable spaces, we show, as a second application of our result, the existence and uniqueness of a quasistationary distribution for a class of possibly nonirreducible processes.
29 citations
TL;DR: In this article, it was shown that the conditional first and last passage times are jointly asymptotically Gaussian and completely dependent for the γ-reflected process.
Abstract: Define a γ-reflected process Wγ(t) = YH(t) - γinfs∈[0,t]YH(s), t ≥ 0, with input process {YH(t), t ≥ 0}, which is a fractional Brownian motion with Hurst index H ∈ (0, 1) and a negative linear trend. In risk theory Rγ(u) = u - Wγ(t), t ≥ 0, is referred to as the risk process with tax payments of a loss-carry-forward type. For various risk processes, numerous results are known for the approximation of the first and last passage times to 0 (ruin times) when the initial reserve u goes to ∞. In this paper we show that, for the γ-reflected process, the conditional (standardized) first and last passage times are jointly asymptotically Gaussian and completely dependent. An important contribution of this paper is that it links ruin problems with extremes of nonhomogeneous Gaussian random fields defined by YH, which we also investigate.
TL;DR: In this article, a system of polya-type urns with balls of two colors is considered, and the reinforcement of each urn depends both on the content of the same urn and on the average content of all the balls.
Abstract: We consider a system of urns of Polya-type, with balls of two colors; the reinforcement of each urn depends both on the content of the same urn and on the average content of all urns. We show that the urns synchronize almost surely, in the sense that the fraction of balls of a given color converges almost surely, as the time goes to infinity, to the same limit for all urns. A normal approximation for a large number of urns is also obtained.
TL;DR: In this article, the authors show that geometric ergodicity provides a simple and versatile framework to establish measure concentration for a broad class of non-contracting chains, including Markov trees and hidden Markov chains.
Abstract: We observe that the technique of Markov contraction can be used to establish measure concentration for a broad class of noncontracting chains. In particular, geometric ergodicity provides a simple and versatile framework. This leads to a short, elementary proof of a general concentration inequality for Markov and hidden Markov chains, which supersedes some of the known results and easily extends to other processes such as Markov trees. As applications, we provide a Dvoretzky-Kiefer-Wolfowitz-type inequality and a uniform Chernoff bound. All of our bounds are dimension-free and hold for countably infinite state spaces.
TL;DR: In this article, a one-sided Markov additive process with an upper and a lower barrier is considered, where each can be either reflecting or terminating, and the corresponding potential measures for both defective and non-defective processes are identified.
Abstract: Consider a one-sided Markov additive process with an upper and a lower barrier, where each can be either reflecting or terminating. For both defective and nondefective processes, and all possible scenarios, we identify the corresponding potential measures, which help to generalize a number of results for one-sided Levy processes. The resulting rather neat formulae have various applications in risk and queueing theories, and, in particular, they lead to quasistationary distributions of the corresponding processes.
TL;DR: In this paper, the authors considered some nonstandard renewal risk models with some dependent claim sizes and stochastic return, where an insurance company is allowed to invest her/his wealth in financial assets, and the price process of the investment portfolio is described as a geometric Levy process.
Abstract: In this paper we consider some nonstandard renewal risk models with some dependent claim sizes and stochastic return, where an insurance company is allowed to invest her/his wealth in financial assets, and the price process of the investment portfolio is described as a geometric Levy process. When the claim size distribution belongs to some classes of heavy-tailed distributions and a constraint is imposed on the Levy process in terms of its Laplace exponent, we obtain some asymptotic formulae for the tail probability of discounted aggregate claims and ruin probabilities holding uniformly for some finite or infinite time horizons.
TL;DR: In this article, the authors derived the explicit formula for the joint Laplace transform of the Wishart process and its time integral, which extends the original approach of Bru (1991), and compared their methodology with the alternative results given by the variation-of-constants method, linearization of the matrix Riccati ordinary differential equation, and the Runge-Kutta algorithm.
Abstract: We derive the explicit formula for the joint Laplace transform of the Wishart process and its time integral, which extends the original approach of Bru (1991). We compare our methodology with the alternative results given by the variation-of-constants method, the linearization of the matrix Riccati ordinary differential equation, and the Runge-Kutta algorithm. The new formula turns out to be fast and accurate.
TL;DR: The asymptotic tail behaviour of the aggregated risk is derived under weak conditions on the marginal tails and the dependence structure of a vector of positive risks and an application concerning log-normal risks with stochastic volatility is presented.
Abstract: In this paper we work in the framework of a k-dimensional vector of loglinear risks. Under weak conditions on the marginal tails and the dependence structure of a vector of positive risks we derive the asymptotic tail behaviour of the aggregated risk and present an application concerning log-normal risks
TL;DR: A survey of related combinatorial constructions and the large-sample behavior of the functionals which characterize in some way the speed of coalescence can be found in this paper, where the authors focus on the Λ-coalescents model the evolution of a coalescing system in which any number of components randomly sampled from the whole may merge into larger blocks.
Abstract: Λ-coalescents model the evolution of a coalescing system in which any number of components randomly sampled from the whole may merge into larger blocks. This survey focuses on related combinatorial constructions and the large-sample behaviour of the functionals which characterize in some way the speed of coalescence.
TL;DR: In this article, the smallest possible value of P(X1 + + Xn < s) over all possible dependence structures, denoted by mn;F(s), was derived for any s 2 R with an error of at most n 1=6 for general continuous distributions.
Abstract: Suppose X1; ; Xn are random variables with the same known marginal distribution F but unknown dependence structure. In this paper, we study the smallest possible value of P(X1 + + Xn < s) over all possible dependence structures, denoted by mn;F(s). We show that mn;F(ns) ! 0 for s no more than the mean of F under weak assumptions. We also derive a limit of mn;F(ns) for any s2 R with an error of at most n 1=6 for general continuous distributions. An application of our result in risk management confirms that the worst-case Value-at-Risk is asymptotically equivalent to the worst-case Expected Shortfall for risk aggregation with dependence uncertainty. In the last part of this paper we present a dual presentation of the theory of complete mixability and give dual proofs of theorems in the literature on this concept.
TL;DR: In this article, the distribution of the coalescence time for two individuals picked at random (uniformly) in the current generation of a continuous-time Bienayme-Galton-Watson process was 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 continuous-time Bienayme-Galton-Watson process founded t units of time ago. We also obtain limiting distributions as t ?8 in the subcritical case. We extend our results for two individuals to the joint distribution of coalescence times for any finite number of individuals sampled in the current generation.
TL;DR: In this article, the authors investigated optimal forms of dynamic reinsurance polices among a class of general reinsurance strategies in a dynamic control framework, where the original surplus process of an insurance portfolio is assumed to follow a Markov jump process with state-dependent income.
Abstract: In this paper we investigate optimal forms of dynamic reinsurance polices among a class of general reinsurance strategies. The original surplus process of an insurance portfolio is assumed to follow a Markov jump process with state-dependent income. We assume that the insurer uses a dynamic reinsurance policy to minimize the probability of absolute ruin, where the traditional ruin can be viewed as a special case of absolute ruin. In terms of approximation theory of stochastic process, the controlled diffusion model with a general reinsurance policy is established strictly. In such a risk model, absolute ruin is said to occur when the drift coefficient of the surplus process turns negative, when the insurer has no profitability any more. Under the expected value premium principle, we rigorously prove that a dynamic excess-of-loss reinsurance is the optimal form of reinsurance among a class of general reinsurance strategies in a dynamic control framework. Moreover, by solving the Hamilton-Jacobi-Bellman equation, we derive both the explicit expression of the optimal dynamic excess-of-loss reinsurance strategy and the closed-form solution to the absolute ruin probability under the optimal reinsurance strategy. We also illustrate these explicit solutions using numerical examples.
TL;DR: In this article, the authors established a connection between epidemic models on random networks with general infection times considered in Barbour and Reinert (2013) and first passage percolation, using techniques developed in Bhamidi, van der Hofstad and Hooghiemstra (2012), when each vertex has infinite contagious periods.
Abstract: We establish a connection between epidemic models on random networks with general infection times considered in Barbour and Reinert (2013) and first passage percolation. Using techniques developed in Bhamidi, van der Hofstad and Hooghiemstra (2012), when each vertex has infinite contagious periods, we extend results on the epidemic curve in Barbour and Reinert (2013) from bounded degree graphs to general sparse random graphs with degrees having finite second moments as n → ∞, with an appropriate X 2 log + X condition. We also study the epidemic trail between the source and typical vertices in the graph.
TL;DR: This paper provides tools for the study of the Dirichlet random walk in R d by compute explicitly the distribution of the random variable W using a form of Stieltjes transform of W instead of the Laplace transform, replacing the Bessel functions with hypergeometric functions.
Abstract: This paper provides tools for the study of the Dirichlet random walk in R d . We compute explicitly, for a number of cases, the distribution of the random variable W using a form of Stieltjes transform of W instead of the Laplace transform, replacing the Bessel functions with hypergeometric functions. This enables us to simplify some existing results, in particular, some of the proofs by Le Caer (2010), (2011). We extend our results to the study of the limits of the Dirichlet random walk when the number of added terms goes to ∞, interpreting the results in terms of an integral by a Dirichlet process. We introduce the ideas of Dirichlet semigroups and Dirichlet infinite divisibility and characterize these infinite divisible distributions in the sense of Dirichlet when they are concentrated on the unit sphere of R d .
TL;DR: In this paper, the authors studied the properties of the multivariate skew normal distribution as an approximation to the distribution of the sum of n independent, identically distributed random vectors, and established conditions ensuring that the uniform distance between the two distribution functions converges to 0 at a rate of n -2/3.
Abstract: We study the properties of the multivariate skew normal distribution as an approximation to the distribution of the sum of n independent, identically distributed random vectors. More precisely, we establish conditions ensuring that the uniform distance between the two distribution functions converges to 0 at a rate of n -2/3. The advantage over the corresponding normal approximation is particularly relevant when the summands are skewed and n is small, as illustrated for the special case of exponentially distributed random variables. Applications to some well-known multivariate distributions are also discussed.
TL;DR: In this paper, the tail order of copulas and hidden regular variation (HRV) on subcones generated by order statistics were studied. And the relation between the tail-order of a copula and the tail indexes for MRV and HRV was established.
Abstract: We study the relations between the tail order of copulas and hidden regular variation (HRV) on subcones generated by order statistics. Multivariate regular variation (MRV) and HRV deal with extremal dependence of random vectors with Pareto-like univariate margins. Alternatively, if one uses a copula to model the dependence structure of a random vector then the upper exponent and tail order functions can be used to capture the extremal dependence structure. After defining upper exponent functions on a series of subcones, we establish the relation between the tail order of a copula and the tail indexes for MRV and HRV. We show that upper exponent functions of a copula and intensity measures of MRV/HRV can be represented by each other, and the upper exponent function on subcones can be expressed by a Pickands-type integral representation. Finally, a mixture model is given with the mixing random vector leading to the finite-directional measure in a product-measure representation of HRV intensity measures.
TL;DR: In this paper, the authors extend the results of Bairamov and Arnold (2008) on the residual life lengths of the live components in an (n - k + 1)-out-of-n system, and also present a new result concerning the multivariate stochastic ordering of live components.
Abstract: Suppose that a system consists of n independent and identically distributed components and that the life lengths of the n components are Xi, i = 1,..., n. For k ? {1,..., n - 1}, let X1(k),...,Xn-k(k) be the residual life lengths of the live components following the kth failure in the system. In this paper we extend various stochastic ordering results presented in Bairamov and Arnold (2008) on the residual life lengths of the live components in an (n - k + 1)-out-of-n system, and also present a new result concerning the multivariate stochastic ordering of live components in the two-sample situation. Finally, we also characterize exponential distributions under a weaker condition than those introduced in Bairamov and Arnold (2008) and show that some special ageing properties of the original residual life lengths get preserved by residual life lengths
TL;DR: In this paper, the optimal stopping times of the exercise are shown to be the first times at which the price of the underlying asset exits some regions restricted by certain boundaries depending on the running values of the associated maximum and maximum drawdown processes.
Abstract: We study optimal stopping problems related to the pricing of perpetual American options in an extension of the Black-Merton-Scholes model in which the dividend and volatility rates of the underlying risky asset depend on the running values of its maximum and maximum drawdown. The optimal stopping times of the exercise are shown to be the first times at which the price of the underlying asset exits some regions restricted by certain boundaries depending on the running values of the associated maximum and maximum drawdown processes. We obtain closed-form solutions to the equivalent free-boundary problems for the value functions with smooth fit at the optimal stopping boundaries and normal reflection at the edges of the state space of the resulting three-dimensional Markov process. We derive first-order nonlinear ordinary differential equations for the optimal exercise boundaries of the perpetual American standard options.
TL;DR: In this article, a single-step network consisting of n links and the links are subject to failure is considered and the residual lifetimes of the networks under different scenarios on the states and the number of failed links of the network.
Abstract: This paper is an investigation into the reliability and stochastic properties of three-state networks. We consider a single-step network consisting of n links and we assume that the links are subject to failure. We assume that the network can be in three states, up (K = 2), partial performance (K = 1), and down (K = 0). Using the concept of the two-dimensional signature, we study the residual lifetimes of the networks under different scenarios on the states and the number of failed links of the network. In the process of doing so, we define variants of the concept of the dynamic signature in a bivariate setting. Then, we obtain signature based mixture representations of the reliability of the residual lifetimes of the network states under the condition that the network is in state K = 2 (or K = 1) and exactly k links in the network have failed. We prove preservation theorems showing that stochastic orderings and dependence between the elements of the dynamic signatures (which relies on the network structure) are preserved by the residual lifetimes of the states of the network (which relies on the network ageing). Various illustrative examples are also provided.
TL;DR: The algorithm is described in full generality and applied to the problem of computing the probability that a heavy-tailed random walk exceeds a high threshold, and its performance is illustrated numerically and compared to existing importance sampling algorithms.
Abstract: In this paper a method based on a Markov chain Monte Carlo (MCMC) algorithm is proposed to compute the probability of a rare event. The conditional distribution of the underlying process given that the rare event occurs has the probability of the rare event as its normalizing constant. Using the MCMC methodology, a Markov chain is simulated, with the aforementioned conditional distribution as its invariant distribution, and information about the normalizing constant is extracted from its trajectory. The algorithm is described in full generality and applied to the problem of computing the probability that a heavy-tailed random walk exceeds a high threshold. An unbiased estimator of the reciprocal probability is constructed whose normalized variance vanishes asymptotically. The algorithm is extended to random sums and its performance is illustrated numerically and compared to existing importance sampling algorithms.
TL;DR: This short note investigates convergence of adaptive MCMC algorithms, i.e. algorithms which modify the Markov chain update probabilities on the fly, and shows that if the Containment condition is not satisfied, then the algorithm will perform very poorly.
Abstract: This short note investigates convergence of adaptive Markov chain Monte Carlo algorithms, i.e. algorithms which modify the Markov chain update probabilities on the fly. We focus on the containment condition introduced Roberts and Rosenthal (2007). We show that if the containment condition is not
satisfied, then the algorithm will perform very poorly. Specifically, with positive probability, the adaptive algorithm will be asymptotically less efficient then any nonadaptive ergodic MCMC algorithm. We call such algorithms AdapFail, and conclude that they should not be used.