Showing papers in "Journal of Applied Probability in 2007"
TL;DR: This work considers basic ergodicity properties of adaptive Markov chain Monte Carlo algorithms under minimal assumptions, using coupling constructions and proves convergence in distribution and a weak law of large numbers.
Abstract: We consider basic ergodicity properties of adaptive Markov chain Monte Carlo algorithms under minimal assumptions, using coupling constructions. We prove convergence in distribution and a weak law of large numbers. We also give counterexamples to demonstrate that the assumptions we make are not redundant.
436 citations
TL;DR: In this article, the authors provide a distributional analysis of the solution to the classical control problem due to De Finetti (1957), Gerber (1969), Azcue and Muller (2005), and Avram et al. (2006) which concerns the optimal payment of dividends from an insurance risk process prior to ruin.
Abstract: We provide a distributional study of the solution to the classical control problem due to De Finetti (1957), Gerber (1969), Azcue and Muller (2005) and Avram et al. (2006) which concerns the optimal payment of dividends from an insurance risk process prior to ruin. Specifically we build on recent work in the actuarial literature concerning calculations for the n-th moment of the net present value of dividends paid out in the optimal strategy as well as the moments of the deficit at ruin and the Laplace transform of the red period. The calculations we present go much further than existing literature in that our calculations are
125 citations
TL;DR: In this paper, a stochastic failure model in a random environment and the effect of the environmental factors on the failure process of the system is investigated. But the model is restricted to the case where the external environmental conditions under which the system was being operated.
Abstract: Most devices (systems) are operated under different environmental conditions. The failure process of a system not only depends on the intrinsic characteristics of the system itself but also on the external environmental conditions under which the system is being operated. In this paper we study a stochastic failure model in a random environment and investigate the effect of the environmental factors on the failure process of the system.
77 citations
TL;DR: This result is based on a local-time argument that enables us to give an alternative proof of the smooth-fit principle and presents a class of optimal stopping problems for which the propagation of convexity fails.
Abstract: In this paper we investigate sufficient conditions that ensure the optimality of threshold strategies for optimal stopping problems with finite or perpetual maturities. Our result is based on a local-time argument that enables us to give an alternative proof of the smooth-fit principle. Moreover, we present a class of optimal stopping problems for which the propagation of convexity fails.
74 citations
TL;DR: In this paper, the non-negativity of the increments of the driving Levy process was used to study the properties of a highly efficient estimation procedure for the parameters when observations are available of the CAR(1) process at uniformly spaced times 0, h,...,Nh.
Abstract: Continuous time autoregressive moving average (CARMA) processes with a non-negative kernel and driven by a non-decreasing Levy process constitute a very general class of stationary non-negative continuous-time processes. In financial econometrics a stationary Ornstein-Uhlenbeck (or CAR(1)) process, driven by a non-decreasing Levy process, was introduced by Barndorff-Nielsen and Shephard (2001)
as a model for stochastic volatility to allow for a wide variety of possible marginal distributions and the possibility of jumps. For such processes we take advantage of the non-negativity of the increments of the driving Levy process to study the properties of a highly efficient estimation procedure for the parameters when observations are available of the CAR(1) process at uniformly spaced times 0, h, ...,Nh. We also show how to reconstruct the background driving Levy process from a continuously observed realization of the process and use this result to estimate the increments of the Levy process itself when h is small. Asymptotic properties of the
coefficient estimator are derived and the results illustrated using a simulated gamma-driven Ornstein-Uhlenbeck process.
73 citations
TL;DR: In this article, the authors show how fluctuation identities for Levy processes with no positive jumps yield the distribution of the present value of dividends paid until ruin in a Levy insurance risk model with a dividend barrier.
Abstract: In this short paper, we show how fluctuation identities for Levy processes with no positive jumps yield the distribution of the present value of dividends paid until ruin in a Levy insurance risk model with a dividend barrier.
67 citations
TL;DR: In this article, the reliability properties of consecutive k-out-of-n systems with exchangeable components are studied and the reliability functions of these systems can be written as negative mixtures of two series (or parallel) systems.
Abstract: In this paper we study reliability properties of consecutive-k-out-of-n systems with exchangeable components. For 2k ≥ n, we show that the reliability functions of these systems can be written as negative mixtures (i.e. mixtures with some negative weights) of two series (or parallel) systems. Some monotonicity and asymptotic properties for the mean residual lifetime function are obtained and some ordering properties between these systems are established. We prove that, under some assumptions, the mean residual lifetime function of the consecutive-k-out-of-n: G system (i.e. a system that functions if and only if at least k consecutive components function) is asymptotically equivalent to that of a series system with k components. When the components are independent and identically distributed, we show that consecutive-k-out-of-n systems are ordered in the likelihood ratio order and, hence, in the mean residual lifetime order, for 2k ≥ n. However, we show that this is not necessarily true when the components are dependent.
62 citations
TL;DR: In this article, the authors studied the tail behavior of discounted aggregate claims in a continuous-time renewal model and established a tail asymptotic formula, which holds uniformly in time.
Abstract: We study the tail behavior of discounted aggregate claims in a continuous-time renewal model. For the case of Pareto-type claims, we establish a tail asymptotic formula, which holds uniformly in time.
59 citations
TL;DR: In this article, the tail behavior of the distribution of the supremum of a process with regenerative increments is investigated in four qualitatively different regimes involving both light and heavy tails, and illustrated with examples arising in queueing theory and insurance risk.
Abstract: We give precise asymptotic estimates of the tail behavior of the distribution of the supremum of a process with regenerative increments. Our results cover four qualitatively different regimes involving both light tails and heavy tails, and are illustrated with examples arising in queueing theory and insurance risk.
48 citations
TL;DR: In this paper, the authors give a simple and direct treatment of insensitivity in stochastic networks which is quite general and provides probabilistic insight into the phenomenon, and generalise those of Bonald and Proutiere (2002, 2003).
Abstract: We give a simple and direct treatment of insensitivity in stochastic networks which is quite general and provides probabilistic insight into the phenomenon. In the case of multi-class networks, our results generalise those of Bonald and Proutiere (2002), (2003).
47 citations
TL;DR: In this article, the authors considered the tail behavior of the product of two independent nonnegative random variables X and Y and investigated when the condition on Y can be weakened and applied their findings to analyze a class of random difference equations.
Abstract: We consider the tail behavior of the product of two independent nonnegative random variables X and Y. Breiman (1965) has considered this problem, assuming that X is regularly varying with index a and that E{Y α+e } 0. We investigate when the condition on Y can be weakened and apply our findings to analyze a class of random difference equations.
TL;DR: In this paper, corrections are made to formulations and proofs of some theorems about convolution equivalence closure for random sum distributions, which arise because of the falsity of a much used asymptotic equivalence lemma.
Abstract: Corrections are made to formulations and proofs of some theorems about convolution equivalence closure for random sum distributions. These arise because of the falsity of a much used asymptotic equivalence lemma, and they impinge on the convolution equivalence closure theorem for general infinitely divisible laws.
TL;DR: In this paper, the authors present closed form solutions of some discounted optimal stopping problems for the maximum process in a model driven by a Brownian motion and a compound Poisson process with exponential jumps.
Abstract: In this paper we present closed form solutions of some discounted optimal stopping problems for the maximum process in a model driven by a Brownian motion and a compound Poisson process with exponential jumps. The method of proof is based on reducing the initial problems to integro-differential free-boundary problems, where the normal-reflection and smooth-fit conditions may break down and the latter then replaced by the continuous-fit condition. We show that, under certain relationships on the parameters of the model, the optimal stopping boundary can be uniquely determined as a component of the solution of a two-dimensional system of nonlinear ordinary differential equations. The obtained results can be interpreted as pricing perpetual American lookback options with fixed and floating strikes in a jump-diffusion model.
TL;DR: In this paper, the authors investigated large deviations for both partial sums S(k; n 1,..., n k ) = Σ k i=1 Σ ni j=1 X ij and random sums S (k; t) = ξ k i = 1 Σ Ni(t) j = 1 X iJ, where Nj(t), i = 0, k, are counting processes for the claim number.
Abstract: Assume that there are k types of insurance contracts in an insurance company. The ith related claims are denoted by {X ij , j ≥ 1), i = 1,..., k. In this paper we investigate large deviations for both partial sums S(k; n 1 ,..., n k ) = Σ k i=1 Σ ni j=1 X ij and random sums S(k; t) = Σ k i=1 Σ Ni(t) j=1 X ij , where Nj(t), i = 1,..., k, are counting processes for the claim number. The obtained results extend some related classical results.
TL;DR: In this article, the authors considered an M/G/1 retrial queue, where the service time distribution has a finite exponential moment, and they showed that the tail of the queue size distribution is asymptotically given by a geometric function multiplied by a power function.
Abstract: We consider an M/G/1 retrial queue, where the service time distribution has a finite exponential moment. We show that the tail of the queue size distribution is asymptotically given by a geometric function multiplied by a power function. The result is obtained by investigating analytic properties of probability generating functions for the queue size and the server state.
TL;DR: An instantaneous leverage effect can be shown for the exponential continuous-time GARCH(p, p) model and stationarity, mixing, and moment properties of the new model are investigated.
Abstract: In this paper we introduce an exponential continuous time GARCH(p, q) process. It is defined in such a way that it is a continuous time extension of the discrete time EGARCH(p, q) process. We investigate stationarity and moment properties of the new model. An instantaneous leverage effect can be shown for the exponential continuous time GARCH(p, p) model.
TL;DR: In this article, the authors generalize some recent results on spectrally negative Levy processes on the real line to study the excursions of reflected processes, and derive the distributions of (r(a), S τ(a), τ_(a), Y τ (a) ) and of (κ(a, I κ(a ), κ_(a)).
Abstract: For a spectrally negative Levy process X on the real line, let S denote its supremum process and let I denote its infimum process. For a > 0, let r(a) and K(a) denote the times when the reflected processes Y:= S - X and Y:= X - I first exit level a, respectively; let τ_(a) and κ_(a) denote the times when X first reaches S τ(a) and I κ(a) , respectively. The main results of this paper concern the distributions of (r(a), S τ(a) , τ_(a), Y τ(a) ) and of (κ(a), I κ(a) , κ_(a)). They generalize some recent results on spectrally negative Levy processes. Our approach relies on results concerning the solution to the two-sided exit problem for X. Such an approach is also adapted to study the excursions for the reflected processes. More explicit expressions are obtained when X is either a Brownian motion with drift or a completely asymmetric stable process.
TL;DR: In this paper, the authors considered the problem of choosing the parameter of the output stream to minimize the expected long run damage in a large Poisson dam model, where the input stream is described by a Poisson process and the output is state-dependent.
Abstract: A large dam model is an object of study of this paper. The parameters L lower and L upper are its lower and upper levels, L = L upper L lower is large, and if a current level of water is between these bounds, then the dam is assumed to be in normal state. Passage one or other bound leads to damage. It is assumed that input stream of water is described by a Poisson process, while the output stream is state- dependent (the exact formulation of the problem is given in the paper). Let Lt denote the dam level at time t, and let p1 = limt!1 P{Lt = L lower }, p2 = limt!1 P{Lt > L upper } exist. Then the expected long- run damage J = p1J1+p2J2 for the long time interval T proportional to L (J1 and J2 are the corresponding damage costs per time T associated with passage the bounds) is a performance measure, and the aim of the paper is to choose the parameter of output stream (exactly specified in the paper) minimizing J.
TL;DR: In this paper, a stochastic model for the spread of an SEIR (susceptible → exposed → infective → removed) epidemic among a population of individuals partitioned into households is presented.
Abstract: We consider a stochastic model for the spread of an SEIR (susceptible → exposed → infective → removed) epidemic among a population of individuals partitioned into households. The model incorporates both vaccination and isolation in response to the detection of cases. When the infectious period is exponential, we derive an explicit formula for a threshold parameter, and analytic results that enable computation of the probability of the epidemic taking off. These quantities are found to be independent of the exposure period distribution. An approximation for the expected final size of an epidemic that takes off is obtained, evaluated numerically, and found to be reasonably accurate in large populations. When the infectious period is not exponential, but has an increasing hazard rate, we obtain stochastic comparison results in the case where the exposure period is fixed. Our main result shows that as the exposure period increases, both the severity of the epidemic in a single household and the threshold parameter decrease, under certain assumptions concerning isolation. Corresponding results for infectious periods with decreasing hazard rates are also derived.
TL;DR: In this article, it was shown that for large enough k, the D 2 statistic is approximately normal as n gets large, when k = 1, and when n = 2, the variance of D 2 was shown to be approximately normal.
Abstract: Given two sequences of length n over a finite alphabet A of size |A| = d, the D 2 statistic is the number of k-letter word matches between the two sequences. This statistic is used in bioinformatics for EST sequence database searches. Under the assumption of independent and identically distributed letters in the sequences, Lippert, Huang and Waterman (2002) raised questions about the asymptotic behavior of D 2 when the alphabet is uniformly distributed. They expressed a concern that the commonly assumed normality may create errors in estimating significance. In this paper we answer those questions. Using Stein's method, we show that, for large enough k, the D 2 statistic is approximately normal as n gets large. When k = 1, we prove that, for large enough d, the D 2 statistic is approximately normal as n gets large. We also give a formula for the variance of D 2 in the uniform case.
TL;DR: In this paper, the optimal portfolio problem for an insider is studied in terms of the logarithm of the terminal wealth minus a term measuring the roughness and the growth of the portfolio.
Abstract: We study the optimal portfolio problem for an insider, in the case where the performance is measured in terms of the logarithm of the terminal wealth minus a term measuring the roughness and the growth of the portfolio. We give explicit solutions in some cases. Our method uses stochastic calculus of forward integrals.
TL;DR: In a sequence of independent Bernoulli trials, the probability for success in the kth trial is Pk k = 1, 2 as discussed by the authors, where k is the number of strings with a given number of failures between two subsequent successes.
Abstract: In a sequence of independent Bernoulli trials the probability for success in the kth trial is Pk k = 1, 2..... The number of strings with a given number of failures between two subsequent successes ...
TL;DR: In this article, the authors consider the ageing properties of a general repair process and show that the expected age at the beginning of the next cycle in this process is smaller than the initial age of the previous cycle.
Abstract: We consider ageing properties of a general repair process. Under certain assumptions we prove that the expectation of an age at the beginning of the next cycle in this process is smaller than the initial age of the previous cycle. Using this reasoning, we show that the sequence of random ages at the start (end) of consecutive cycles is stochastically increasing and is converging to a limiting distribution. We discuss possible applications and interpretations of our results.
TL;DR: In this paper, the authors consider the performance of the delta hedging strategy obtained from a local volatility model when using as input the physical prices instead of the model price process, and show that robustness holds for a standard Black-Scholes model whenever we hedge a path-dependent derivative with a convex payoff function.
Abstract: We consider the performance of the delta hedging strategy obtained from a local volatility model when using as input the physical prices instead of the model price process. This hedging strategy is called robust if it yields a superhedge as soon as the local volatility model overestimates the market volatility. We show that robustness holds for a standard Black-Scholes model whenever we hedge a path-dependent derivative with a convex payoff function. In a genuine local volatility model the situation is shown to be less stable: robustness can break down for many relevant convex payoffs including average-strike Asian options, lookback puts, floating-strike forward starts, and their aggregated cliquets. Furthermore, we prove that a sufficient condition for the robustness in every local volatility model is the directional convexity of the payoff function.
TL;DR: In this paper, a class of reversible infinite-dimensional diffusion processes with GEM distribution as the reversible measure is constructed, and log-Sobolev inequalities are established for these diffusion processes, leading to the exponential convergence of corresponding reversible measures in the entropy.
Abstract: Starting from a sequence of independent Wright-Fisher diffusion processes on [0, 1], we construct a class of reversible infinite-dimensional diffusion processes on Δ ∞ := {x ∈ [0, 1] N : Σ i≥1 x i = 1} with GEM distribution as the reversible measure. Log-Sobolev inequalities are established for these diffusions, which lead to the exponential convergence of the corresponding reversible measures in the entropy. Extensions are made to a class of measure-valued processes over an abstract space S. This provides a reasonable alternative to the Fleming-Viot process, which does not satisfy the log-Sobolev inequality when S is infinite as observed by Stannat (2000).
TL;DR: In this paper, the authors employ the theory of large deviations to study the probability of large errors in risk measures, and solve the minimization problem explicitly for shortfall risk and average value at risk.
Abstract: The simulation of distributions of financial assets is an important issue for financial institutions. If risk measures are evaluated for a simulated distribution instead of the model-implied distribution, the errors in the risk measurements need to be analyzed. For distribution-invariant risk measures which are continuous on compacts, we employ the theory of large deviations to study the probability of large errors. If the approximate risk measurements are based on the empirical distribution of independent samples, then the rate function equals the minimal relative entropy under a risk measure constraint. We solve this minimization problem explicitly for shortfall risk and average value at risk.
TL;DR: Bounds on the decay parameter for absorbing birth–death processes adapted from results of Chen (2000), (2001) are presented and numerical issues associated with computing these bounds are addressed.
Abstract: We present bounds on the decay parameter for absorbing birth-death processes adapted from results of Chen (2000), (2001). We address numerical issues associated with computing these bounds, and assess their accuracy for several models, including the stochastic logistic model, for which estimates of the decay parameter have been obtained previously by Nasell (2001).
TL;DR: In this article, a stopping rule adapted to a sequence of n independent and identically distributed observations is defined, where the loss is defined to be E[ q ( R τ )], where R j is the rank of the j th observation and q is a nondecreasing function of the rank.
Abstract: For τ, a stopping rule adapted to a sequence of n independent and identically distributed observations, we define the loss to be E[ q ( R τ )], where R j is the rank of the j th observation and q is a nondecreasing function of the rank. This setting covers both the best-choice problem, with q ( r ) = 1 ( r > 1), and Robbins' problem, with q ( r ) = r . As n tends to ∞, the stopping problem acquires a limiting form which is associated with the planar Poisson process. Inspecting the limit we establish bounds on the stopping value and reveal qualitative features of the optimal rule. In particular, we show that the complete history dependence persists in the limit; thus answering a question asked by Bruss (2005) in the context of Robbins' problem.
TL;DR: This paper studies Markov chain models derived from protocols in the TCP paradigm and proves weak convergence results, after appropriate rescaling of time and space, as p → 0.
Abstract: The transmission control protocol (TCP) is a transport protocol used in the Internet. In Ott (2005), a more general class of candidate transport protocols called ‘protocols in the TCP paradigm’ was introduced. The long-term objective of studying this class is to find protocols with promising performance characteristics. In this paper we study Markov chain models derived from protocols in the TCP paradigm. Protocols in the TCP paradigm, as TCP, protect the network from congestion by decreasing the ‘congestion window’ (i.e. the amount of data allowed to be sent but not yet acknowledged) when there is packet loss or packet marking, and increasing it when there is no loss. When loss of different packets are assumed to be independent events and the probability p of loss is assumed to be constant, the protocol gives rise to a Markov chain {Wn}, where Wn is the size of the congestion window after the transmission of the nth packet. For a wide class of such Markov chains, we prove weak convergence results, after appropriate rescaling of time and space, as p → 0. The limiting processes are defined by stochastic differential equations. Depending on certain parameter values, the stochastic differential equation can define an Ornstein–Uhlenbeck process or can be driven by a Poisson process.
TL;DR: It is proved that there exists a constant ĉ > 0 such that the iterated tour partitioning heuristic given by Haimovich and Rinnooy Kan (1985) is a (2 - ĉ)-approximation algorithm with probability arbitrarily close to 1 as the number of customers goes to ∞.
Abstract: We analyze the unit-demand Euclidean vehicle routeing problem, where n customers are modeled as independent, identically distributed uniform points and have unit demand. We show new lower bounds on the optimal cost for the metric vehicle routeing problem and analyze them in this setting. We prove that there exists a constant ĉ > 0 such that the iterated tour partitioning heuristic given by Haimovich and Rinnooy Kan (1985) is a (2 - ĉ )-approximation algorithm with probability arbitrarily close to 1 as the number of customers goes to ∞. It has been a longstanding open problem as to whether one can improve upon the factor of 2 given by Haimovich and Rinnooy Kan. We also generalize this, and previous results, to the multidepot case.