Showing papers in "Journal of Applied Probability in 2011"
TL;DR: The statistical estimation and goodness-of-fit are derived for multivariate Hawkes processes with possibly dependent marks and two data sets from finance are analyzed.
Abstract: A Hawkes process is also known under the name of a self-exciting point process and has numerous applications throughout science and engineering. We derive the statistical estimation (maximum likelihood estimation) and goodness-of-fit (mainly graphical) for multivariate Hawkes processes with possibly dependent marks. As an application, we analyze two data sets from finance.
268 citations
TL;DR: In this paper, the authors study new classes of extreme shock models and, based on the obtained results and model interpretations, extend these results to several specific combined shock models, and derive the corresponding survival probabilities and discuss some meaningful interpretations and examples.
Abstract: In extreme shock models, only the impact of the current, possibly fatal shock is usually taken into account, whereas in cumulative shock models, the impact of the preceding shocks is accumulated as well. A shock model which combines these two types is called a `combined shock model'. In this paper we study new classes of extreme shock models and, based on the obtained results and model interpretations, we extend these results to several specific combined shock models. For systems subject to nonhomogeneous Poisson processes of shocks, we derive the corresponding survival probabilities and discuss some meaningful interpretations and examples.
122 citations
TL;DR: In this paper, the Parisian ruin probability, which arises when the surplus process stays below 0 longer than a fixed amount of time ζ > 0, is analyzed. And the authors derive an expression for the Parisians ruin probability in terms of quantities that can be calculated explicitly in many models.
Abstract: In this paper we analyze the so-called Parisian ruin probability, which arises when the surplus process stays below 0 longer than a fixed amount of time ζ > 0. We focus on a general spectrally negative Levy insurance risk process. For this class of processes, we derive an expression for the ruin probability in terms of quantities that can be calculated explicitly in many models. We find its Cramer-type and convolution-equivalent asymptotics when reserves tend to ∞. Finally, we analyze some explicit examples.
86 citations
TL;DR: In this paper, the copula of a bivariate random vector is computed when partial information is available, such as the value of a copula on a given subset of [0, 1]2, or the values of a functional of the copulum, monotone with respect to the concordance order.
Abstract: Improved bounds on the copula of a bivariate random vector are computed when partial information is available, such as the values of the copula on a given subset of [0, 1]2, or the value of a functional of the copula, monotone with respect to the concordance order. These results are then used to compute model-free bounds on the prices of two-asset options which make use of extra information about the dependence structure, such as the price of another two-asset option.
67 citations
TL;DR: In this article, the authors considered extensions of signature-based representations of the reliability functions of coherent systems with independent and identically distributed component lifetimes to systems with heterogeneous components.
Abstract: Signature-based representations of the reliability functions of coherent systems with independent and identically distributed component lifetimes have proven very useful in studying the ageing characteristics of such systems and in comparing the performance of different systems under varied criteria. In this paper we consider extensions of these results to systems with heterogeneous components. New representation theorems are established for both the case of components with independent lifetimes and the case of component lifetimes under specific forms of dependence. These representations may be used to compare the performance of systems with homogeneous and heterogeneous components.
57 citations
TL;DR: In this paper, the authors considered a discrete-time insurance risk model where the insurer makes both risk-free and risky investments, leading to an overall stochastic discount factor Y i from time i to time i - 1.
Abstract: Consider a discrete-time insurance risk model. Within period i, the net insurance loss is denoted by a real-valued random variable X i . The insurer makes both risk-free and risky investments, leading to an overall stochastic discount factor Y i from time i to time i - 1. Assume that (X i , Y i ), i ∈ ℕ, form a sequence of independent and identically distributed random pairs following a common bivariate Farlie-Gumbel-Morgenstern distribution with marginal distribution functions F and G. When F is subexponential and G fulfills some constraints in order for the product convolution of F and G to be subexponential too, we derive a general asymptotic formula for the finite-time ruin probability. Then, for special cases in which F belongs to the Frechet or Weibull maximum domain of attraction, we improve this general formula to be transparent.
54 citations
TL;DR: In this article, the authors studied the leading term in the small-time asymptotics of atthe-money call option prices when the stock price process S follows a general martingale.
Abstract: We study the leading term in the small-time asymptotics of atthe-money call option prices when the stock price process S follows a general martingale. This is equivalent to studying the first centered absolute moment of S. We show that if S has a continuous part, the leading term is of order p T in time T and depends only on the initial value of the volatility. Furthermore, the term is linear in T if and only if S is of finite variation. The leading terms for pure-jump processes with infinite variation are between these two cases; we obtain their exact form for stable-like small jumps. To derive these results, we use a natural approximation of S so that calculations are necessary only for the class of Levy processes.
49 citations
TL;DR: In this article, a stochastic perpetuity model is proposed for generating samples of the stationary distribution of the Markov chain defined recursively by D n+1=A n D n +B n, n≥0, where A n =e Y n ; D ∞ then satisfies the stochastically fixed-point equation D∞DAD ∞+B, where B n and A n are independent copies of the A n and B n (and independent of D ∆ on the right hand side).
Abstract: A stochastic perpetuity takes the form D∞=∑ n=0 ∞ exp(Y 1+⋯+Y n )B n , where Y n :n≥0) and (B n :n≥0) are two independent sequences of independent and identically distributed random variables (RVs). This is an expression for the stationary distribution of the Markov chain defined recursively by D n+1=A n D n +B n , n≥0, where A n =e Y n ; D ∞ then satisfies the stochastic fixed-point equation D ∞DAD ∞+B, where A and B are independent copies of the A n and B n (and independent of D ∞ on the right-hand side). In our framework, the quantity B n , which represents a random reward at time n, is assumed to be positive, unbounded with EB n p 0, and have a suitably regular continuous positive density. The quantity Y n is assumed to be light tailed and represents a discount rate from time n to n-1. The RV D ∞ then represents the net present value, in a stochastic economic environment, of an infinite stream of stochastic rewards. We provide an exact simulation algorithm for generating samples of D ∞. Our method is a variation of dominated coupling from the past and it involves constructing a sequence of dominating processes.
43 citations
TL;DR: In this paper, the authors studied the dependence of individual reproduction upon the size of the whole population in a general branching process context, and the particular feature under scrutiny is that of reproduction changing from supercritical in small populations to subcritical in large ones.
Abstract: Dependence of individual reproduction upon the size of the whole population is studied in a general branching process context. The particular feature under scrutiny is that of reproduction changing from supercritical in small populations to subcritical in large ones. The transition occurs when population size passes a critical threshold, known in ecology as the carrying capacity. We show that populations either die out directly, never coming close to the carrying capacity, or else they grow quickly towards the latter, subsequently lingering around it for a time that is expected to be exponentially long in terms of a carrying capacity tending to infinity.
41 citations
TL;DR: In this article, the authors extend this study to the general k-out-of-n system for the case when there are only two types of component in the system and show that a parallel system with heterogeneous exponential component lifetimes is more skewed (according to the convex transform order) than a system with independent and identically distributed exponential components.
Abstract: Kochar and Xu (2009) proved that a parallel system with heterogeneous exponential component lifetimes is more skewed (according to the convex transform order) than the system with independent and identically distributed exponential components. In this paper we extend this study to the general k-out-of-n systems for the case when there are only two types of component in the system. An open problem proposed in Pǎltǎnea (2008) is partially solved.
39 citations
TL;DR: The signature is an important structural characteristic of a coherent system, however, its computation is often rather involved and complex as mentioned in this paper, and the signature can be reduced by reducing the complexity of the signature.
Abstract: The signature is an important structural characteristic of a coherent system. Its computation, however, is often rather involved and complex. We analyze several cases where this complexity can be considerably reduced. These are the cases when a ‘large’ coherent system is obtained as a series, parallel, or recurrent structure built from ‘small’ modules with known signature. Corresponding formulae can be obtained in terms of cumulative notions of signatures. An algebraic closure property of families of homogeneous polynomials plays a substantial role in our derivations.
TL;DR: In this paper, an extension of the two-dimensional risk model introduced by Avram et al. (2008a) is considered and the Laplace transform of the time until at least one insurer is ruined is derived when the claim sizes follow a general distribution.
Abstract: In this paper, we consider an extension of the two-dimensional risk model introduced by Avram et al. (2008a). To this end, we assume two insurers in which the flrst is subject to claims arising from two independent compound Poisson processes. The second insurer, that can be viewed as a difierent line of business of the same insurer or as a reinsurer, covers a proportion of the claims caused by one of these two compound Poisson processes. The Laplace transform of the time until at least one insurer is ruined is derived when the claim sizes follow a general distribution. The surplus level of the flrst insurer when the second one is ruined flrst is discussed in the end in connection with a few open questions.
TL;DR: In this paper, the authors compared the lifetimes of series (parallel) systems arising out of different allocations of one or two standby redundancies with respect to the increasing concave (convex) order, the hazard rate order, and the stochastic precedence order.
Abstract: To enhance the performance of a system, a common practice employed by reliability engineers is to use redundant components in the system. In this paper we compare lifetimes of series (parallel) systems arising out of different allocations of one or two standby redundancies. These comparisons are made with respect to the increasing concave (convex) order, the hazard rate order, and the stochastic precedence order. The main results extend some related conclusions in the literature.
TL;DR: In this article, it was shown that the empirical profile of the unique invariant measure for the Fleming-Viot process with N particles converges as N -> infinity to the unique quasistationary distribution of the one-particle motion.
Abstract: Consider a continuous-time Markov process with transition rates matrix Q in the state space Lambda boolean OR {0}. In In the associated Fleming-Viot process N particles evolve independently in A with transition rates matrix Q until one of them attempts to jump to state 0. At this moment the particle jumps to one of the positions of the other particles, chosen uniformly at random. When Lambda is finite, we show that the empirical distribution of the particles at a fixed time converges as N -> infinity to the distribution of a single particle at the same time conditioned on not touching {0}. Furthermore, the empirical profile of the unique invariant measure for the Fleming-Viot process with N particles converges as N -> infinity to the unique quasistationary distribution of the one-particle motion. A key element of the approach is to show that the two-particle correlations are of order 1/N.
TL;DR: In this paper, the central limit theorem for urn problems was proved for multicolor randomly reinforced urns, and the latter was investigated by paying special attention to multicolored randomly reinforced IBEs.
Abstract: Let Xn be a sequence of integrable real random variables, adapted to a filtration (Gn). Define Cn = √{(1 / n)∑k=1nXk - E(Xn+1 | Gn)} and Dn = √n{E(Xn+1 | Gn) - Z}, where Z is the almost-sure limit of E(Xn+1 | Gn) (assumed to exist). Conditions for (Cn, Dn) → N(0, U) x N(0, V) stably are given, where U and V are certain random variables. In particular, under such conditions, we obtain √n{(1 / n)∑k=1nX_k - Z} = Cn + Dn → N(0, U + V) stably. This central limit theorem has natural applications to Bayesian statistics and urn problems. The latter are investigated, by paying special attention to multicolor randomly reinforced urns.
TL;DR: It is found that in the cases studied both VM and CE methods prescribe the same importance sampling parameters, suggesting that the criterion of minimizing the CE distance is very close, if not asymptotically identical, to minimizing the variance of the associated importance sampling estimator.
Abstract: The variance minimization (VM) and cross-entropy (CE) methods are two versatile adaptive importance sampling procedures that have been successfully applied to a wide variety of difficult rare-event estimation problems. We compare these two methods via various examples where the optimal VM and CE importance densities can be obtained analytically. We find that in the cases studied both VM and CE methods prescribe the same importance sampling parameters, suggesting that the criterion of minimizing the cross- entropy distance might be asymptotically identical to minimizing the variance of the associated importance sampling estimator.
TL;DR: In this paper, the conditional full support (CFS) property was investigated for Gaussian processes with stationary increments and integrability conditions on the spectral measure of such a process were given.
Abstract: We investigate the conditional full support (CFS) property, introduced in Guasoni et al. (2008a), for Gaussian processes with stationary increments. We give integrability conditions on the spectral measure of such a process which ensure that the process has CFS or not. In particular, the general results imply that, for a process with spectral density f such that $f(\lambda) \sim c_1 \lambda^p \rm{e}^{-c_2\lambda^q}$ for ? ? 8 (with necessarily p <1 if q = 0), the CFS property holds if and only if q <1.
TL;DR: In this article, the authors consider the problem of calculating properties of summary statistics (e.g., mean time spent in a state, mean number of jumps between two states, and the distribution of the total number of jump) for discretely observed continuous-time Markov chains.
Abstract: Continuous-time Markov chains are a widely used modelling tool. Applications include DNA sequence evolution, ion channel gating behaviour, and mathematical finance. We consider the problem of calculating properties of summary statistics (e.g. mean time spent in a state, mean number of jumps between two states, and the distribution of the total number of jumps) for discretely observed continuous-time Markov chains. Three alternative methods for calculating properties of summary statistics are described and the pros and cons of the methods are discussed. The methods are based on (i) an eigenvalue decomposition of the rate matrix, (ii) the uniformization method, and (iii) integrals of matrix exponentials. In particular, we develop a framework that allows for analyses of rather general summary statistics using the uniformization method.
TL;DR: This work presents an exact simulation algorithm for the stationary distribution of the customer delay D for first-in–first-out (FIFO) M/G/c queues in which ρ=λ/μ<1.
Abstract: We present an exact simulation algorithm for the stationary distribution of the customer delay D for first-in–first-out (FIFO) M/G/c queues in which ρ=λ/μ y)dy are such that samples of them can be simulated. We further assume that G has a finite second moment. Our method involves the general method of dominated coupling from the past (DCFTP) and we use the single-server M/G/1 queue operating under the processor sharing discipline as an upper bound. Our algorithm yields the stationary distribution of the entire Kiefer–Wolfowitz workload process, the first coordinate of which is D. Extensions of the method to handle simulating generalized Jackson networks in stationarity are also remarked upon.
TL;DR: In this article, the authors revisited the discussion on minimal repair in heterogeneous populations in Finkelstein (2004) and considered the corresponding stochastic intensities (intensity processes) for items in heterogenous populations given available information on their operational history, i.e., the failure (repair) times and the time since the last failure.
Abstract: In this note we revisit the discussion on minimal repair in heterogeneous populations in Finkelstein (2004). We consider the corresponding stochastic intensities (intensity processes) for items in heterogeneous populations given available information on their operational history, i.e. the failure (repair) times and the time since the last failure (repair). Based on the improved definitions, the setup of Finkelstein (2004) is modified and the main results are corrected in accordance with the updating procedure for the conditional frailty distribution.
TL;DR: In this article, it was shown that there exists a constant μ ≥ 1 such that P 0,x (D(0, x)/μ∥x∥ 2 ∉ (1―e, 1+e) | 0, x ∈ C ∞ ) exponentially decreases when ∥x ∥ 2 tends to ∞.
Abstract: Denote the Palm measure of a homogeneous Poisson process H λ with two points 0 and x by P 0,x . We prove that there exists a constant μ ≥ 1 such that P 0,x (D(0, x)/μ∥x∥ 2 ∉ (1―e, 1+e) | 0, x ∈ C ∞ ) exponentially decreases when ∥x∥ 2 tends to ∞, where D(0, x) is the graph distance between 0 and x in the infinite component C ∞ of the random geometric graph G(H λ ; 1). We derive a large deviation inequality for an asymptotic shape result. Our results have applications in many fields and especially in wireless sensor networks.
TL;DR: In this paper, the EM algorithm is compared to a direct Newton-Raphson optimization of the likelihood function. And the Fisher information matrix is calculated for both continuous and discrete phase-type distributions.
Abstract: This paper is concerned with statistical inference for both continuous and discrete phase-type distributions. We consider maximum likelihood estimation, where traditionally the expectation-maximization (EM) algorithm has been employed. Certain numerical aspects of this method are revised and we provide an alternative method for dealing with the E-step. We also compare the EM algorithm to a direct Newton–Raphson optimization of the likelihood function. As one of the main contributions of the paper, we provide formulae for calculating the Fisher information matrix both for the EM algorithm and Newton–Raphson approach. The inverse of the Fisher information matrix provides the variances and covariances of the estimated parameters.
TL;DR: In this article, the authors consider the Λ-coalescent processes with a positive frequency of singleton clusters and show that some large-sample properties of these processes can be derived by coupling the coalescent with an increasing Levy process (subordinator).
Abstract: We consider the Λ-coalescent processes with a positive frequency of singleton clusters. The class in focus covers, for instance, the beta(a, b)-coalescents with a > 1. We show that some large-sample properties of these processes can be derived by coupling the coalescent with an increasing Levy process (subordinator), and by exploiting parallels with the theory of regenerative composition structures. In particular, we discuss the limit distributions of the absorption time and the number of collisions.
TL;DR: In this article, the excursion time of a Brownian motion with drift outside a corridor was studied using a four-state semi-Markov model, and an explicit expression for the Laplace transform of its price was obtained.
Abstract: In this paper we study the excursion time of a Brownian motion with drift outside a corridor by using a four-state semi-Markov model. In mathematical finance, these results have an important application in the valuation of double-barrier Parisian options. We subsequently obtain an explicit expression for the Laplace transform of its price.
TL;DR: In this paper, the maximal number of offspring among all individuals in a critical Galton-Watson process started with k ancestors was investigated, and it was shown that when the reproduction law has a regularly varying tail with index -α for 1 < α < 2, then k-1Mk converges in distribution to a Frechet law with shape parameter 1 and scale parameter depending only on α.
Abstract: We investigate the maximal number Mk of offspring amongst all individuals in a critical Galton-Watson process started with k ancestors. We show that when the reproduction law has a regularly varying tail with index -α for 1 < α < 2, then k-1Mk converges in distribution to a Frechet law with shape parameter 1 and scale parameter depending only on α.
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TL;DR: In this article, the authors study four discrete-time stochastic systems on N, modeling processes of rumor spreading, where involved individuals can either have an active or a passive role, speaking up or asking for the rumor.
Abstract: We study four discrete-time stochastic systems on N, modeling processes of rumor spreading. The involved individuals can either have an active or a passive role, speaking up or asking for the rumor. The appetite for spreading or hearing the rumor is represented by a set of random variables whose distributions may depend on the individuals. Our goal is to understand - based on the distribution of the random variables - whether the probability of having an infinite set of individuals knowing the rumor is positive or not.
TL;DR: Barrieu, Rouault and Yor (2004) determined asymptotics for the logarithm of the distribution function of the Hartman-Watson distribution as discussed by the authors, which can be applied to the pricing of Asian options in the Black-Scholes model.
Abstract: Barrieu, Rouault and Yor (2004) determined asymptotics for the logarithm of the distribution function of the Hartman–Watson distribution. We determine the asymptotics of the density. This refinement can be applied to the pricing of Asian options in the Black–Scholes model.
TL;DR: In this paper, the authors considered several versions of the job assignment problem for an M/M/m queue with servers of different speeds and developed an intuitive proof that the optimal policy that minimizes the mean waiting time has a threshold structure.
Abstract: We consider several versions of the job assignment problem for an M/M/m queue with servers of different speeds. When there are two classes of customers, primary and secondary, the number of secondary customers is infinite, and idling is not permitted, we develop an intuitive proof that the optimal policy that minimizes the mean waiting time has a threshold structure. That is, for each server, there is a server-dependent threshold such that a primary customer will be assigned to that server if and only if the queue length of primary customers meets or exceeds the threshold. Our key argument can be generalized to extend the structural result to models with impatient customers, discounted waiting time, batch arrivals and services, geometrically distributed service times, and a random environment. We show how to compute the optimal thresholds, and study the impact of heterogeneity in server speeds on mean waiting times. We also apply the same machinery to the classical slow-server problem without secondary customers, and obtain more general results for the two-server case and strengthen existing results for more than two servers.
TL;DR: In this paper, the authors consider a single-server queue with Levy input, and their workload process (Qt)t≥0, focusing on its correlation structure, and prove that r(·) is positive, decreasing, and convex, relying on the machinery of completely monotone functions.
Abstract: In this paper we consider a single-server queue with Levy input, and, in particular, its workload process (Qt)t≥0, focusing on its correlation structure. With the correlation function defined as r(t):= cov(Q0, Qt) / varQ0 (assuming that the workload process is in stationarity at time 0), we first study its transform ∫0∞r(t)e-ϑtdt, both for when the Levy process has positive jumps and when it has negative jumps. These expressions allow us to prove that r(·) is positive, decreasing, and convex, relying on the machinery of completely monotone functions. For the light-tailed case, we estimate the behavior of r(t) for large t. We then focus on techniques to estimate r(t) by simulation. Naive simulation techniques require roughly (r(t))-2 runs to obtain an estimate of a given precision, but we develop a coupling technique that leads to substantial variance reduction (the required number of runs being roughly (r(t))-1). If this is augmented with importance sampling, it even leads to a logarithmically efficient algorithm.
TL;DR: In this paper, the first two moments of Z n, the Zagreb index of a random recursive tree of size n, are obtained through a recurrence equation, and the random process {Z n − E[Z n ], n ≥ 1} is a martingale.
Abstract: We investigate the Zagreb index, one of the topological indices, of random recursive trees in this paper. Through a recurrence equation, the first two moments of Z n , the Zagreb index of a random recursive tree of size n, are obtained. We also show that the random process {Z n − E[Z n ], n ≥ 1} is a martingale. Then the asymptotic normality of the Zagreb index of a random recursive tree is given by an application of the martingale central limit theorem. Finally, two other topological indices are also discussed in passing.