Showing papers in "Journal of Applied Probability in 2012"
TL;DR: This work forms the problem of twenty questions with noisy answers as a dynamic program and shows that any policy optimizing the one-step expected reduction in entropy is also optimal over the full horizon.
Abstract: We consider the problem of 20 questions with noisy answers, in which we seek to nd a target by repeatedly choosing a set, asking an oracle whether the target lies in this set, and obtaining an answer corrupted by noise. Starting with a prior distribution on the target’s location, we seek to minimize the expected entropy of the posterior distribution. We formulate this problem as a dynamic program and show that any policy optimizing the one-step expected reduction in entropy is also optimal over the full horizon. Two such Bayes-optimal policies are presented: one generalizes the probabilistic bisection policy due to Horstein and the other asks a deterministic set of questions. We study the structural properties of the latter, and illustrate its use in a computer vision application.
94 citations
TL;DR: In this paper, the authors considered de Finetti's control problem with the condition that the underlying Lebesgue measure has a completely monotone density and established an explicit optimal strategy for this case that envelopes the existing results.
Abstract: In the last few years there has been renewed interest in the classical control problem of de Finetti (1957) for the case where the underlying source of randomness is a spectrally negative Levy process. In particular, a significant step forward was made by Loeffen (2008), who showed that a natural and very general condition on the underlying Levy process which allows one to proceed with the analysis of the associated Hamilton-Jacobi-Bellman equation is that its Levy measure is absolutely continuous, having completely monotone density. In this paper we consider de Finetti's control problem, but with the restriction that control strategies are absolutely continuous with respect to the Lebesgue measure. This problem has been considered by Asmussen and Taksar (1997), Jeanblanc-Picque and Shiryaev (1995), and Boguslavskaya (2006) in the diffusive case, and Gerber and Shiu (2006) for the case of a Cramer-Lundberg process with exponentially distributed jumps. We show the robustness of the condition that the underlying Levy measure has a completely monotone density and establish an explicit optimal strategy for this case that envelopes the aforementioned existing results. The explicit optimal strategy in question is the so-called refraction strategy.
56 citations
TL;DR: The concept of complete mixability is relevant to some problems of optimal couplings with important applications in quantitative risk management as discussed by the authors, where the authors prove new properties of the set of completely mixable distributions, including a completeness and a decomposition theorem.
Abstract: The concept of complete mixability is relevant to some problems of optimal couplings with important applications in quantitative risk management. In this paper we prove new properties of the set of completely mixable distributions, including a completeness and a decomposition theorem. We also show that distributions with a concave density and radially symmetric distributions are completely mixable.
46 citations
TL;DR: In this paper, the authors study the asymptotic properties of the canonical plugin estimates for coherent risk measures and prove a central limit theorem for independent and identically distributed data, and then extend it to weakly dependent data.
Abstract: In this paper we study the asymptotic properties of the canonical plugin estimates for law-invariant coherent risk measures. Under rather mild conditions not relying on the explicit representation of the risk measure under consideration, we first prove a central limit theorem for independent and identically distributed data, and then extend it to the case of weakly dependent data. Finally, a number of illustrating examples is presented.
33 citations
TL;DR: In this paper, the authors derive asymptotics for the ruin probability under multivariate regularly variation and derive them from Breiman's Theorem extensions, and then derive the value-at-risk in terms of the value at risk.
Abstract: Modeling insurance risks is a task that received an increasing attention because of Solvency Capital Requirements. The ruin probability has become a standard risk measure to assess regulatory capital. In this paper we focus on discrete time models for nite time horizon. Several results are available in the literature allowing to calibrate the ruin probability by means of the sum of the tail probabilities of individual claim amounts. The aim of this work is to obtain asymptotics for such probabilities under multivariate regularly variation and, more precisely, to derive them from Breiman's Theorem extensions. We thus exhibit new situations where the ruin probability admits computable equivalents. Consequences are also derived in terms of the Value-at-Risk.
32 citations
TL;DR: In this paper, the authors extend the class of tempered stable distributions, which were first introduced in Rosinski (2007), to allow for more structure and more variety of the tail behaviors, and characterize the possible tails, giving detailed results about finiteness of various moments.
Abstract: We extend the class of tempered stable distributions, which were first introduced in Rosinski (2007). Our new class allows for more structure and more variety of the tail behaviors. We discuss various subclasses and the relations between them. To characterize the possible tails, we give detailed results about finiteness of various moments. We also give necessary and sufficient conditions for the tails to be regularly varying. This last part allows us to characterize the domain of attraction to which a particular tempered stable distribution belongs.
31 citations
TL;DR: In this article, the existence of a unique quasistationary distribution and of the Yaglom limit on a drifted Brownian motion killed at 0 was proved. But this was only for the case where + ∞ is an entrance boundary and 0 is an exit boundary.
Abstract: We study quasistationary distributions on a drifted Brownian motion killed at 0, when +∞ is an entrance boundary and 0 is an exit boundary. We prove the existence of a unique quasistationary distribution and of the Yaglom limit.
30 citations
TL;DR: The first passage time distribution of skew Brownian motion was derived in this article, where the authors generalize results of Pitman and Yor (2011) and Csaki and Hu (2004) to derive formulae for the distribution of ranked excursion heights.
Abstract: Nearly fifty years after the introduction of skew Brownian motion by Ito and McKean (1963), the first passage time distribution remains unknown. In this paper we first generalize results of Pitman and Yor (2011) and Csaki and Hu (2004) to derive formulae for the distribution of ranked excursion heights of skew Brownian motion, and then use these results to derive the first passage time distribution.
27 citations
TL;DR: In this article, a class of financial market models which are based on telegraph processes with alternating tendencies and jumps is proposed. But these models are typically incomplete, but the set of equivalent martingale measures can be described in detail.
Abstract: In this paper we propose a class of financial market models which are based on telegraph processes with alternating tendencies and jumps. It is assumed that the jumps have random sizes and that they occur when the tendencies are switching. These models are typically incomplete, but the set of equivalent martingale measures can be described in detail. We provide additional suggestions which permit arbitrage-free option prices as well as hedging strategies to be obtained.
27 citations
TL;DR: In this paper, the distributions of X n and Y n, and their limit behaviors as n → ∞ are discussed for both the critical and subcritical cases for the Galton-Watson branching process.
Abstract: In a Galton-Watson branching process that is not extinct by the nth generation and has at least two individuals, pick two individuals at random by simple random sampling without replacement. Trace their lines of descent back in time till they meet. Call that generation X n a pairwise coalescence time. Similarly, let Y n denote the coalescence time for the whole population of the nth generation conditioned on the event that it is not extinct. In this paper the distributions of X n and Y n , and their limit behaviors as n → ∞ are discussed for both the critical and subcritical cases.
27 citations
TL;DR: In this paper, finite Markov chain imbedding is extended to compute the distribution of the conditional discrete scan statistic, defined from a sequence of Bernoulli trials, and it is shown that the continuous scan statistic induced by a Poisson process defined on (0, 1) is a limiting distribution of weighted distributions of Conditional Discrete Scan Statistics.
Abstract: The distributions for continuous, discrete, and conditional discrete scan statistics are studied. The approach of finite Markov chain imbedding, which has been applied to random permutations as well as to runs and patterns, is extended to compute the distribution of the conditional discrete scan statistic, defined from a sequence of Bernoulli trials. It is shown that the distribution of the continuous scan statistic induced by a Poisson process defined on (0, 1] is a limiting distribution of weighted distributions of conditional discrete scan statistics. Comparisons of rates of convergence as well as numerical comparisons of various bounds and approximations are provided to illustrate the theoretical results.
TL;DR: The Hamilton method is a natural and common method to distribute seats proportionally between states (or parties) in a parliament as discussed by the authors, but it has been abandoned due to some drawbacks, in particula...
Abstract: Hamilton's method is a natural and common method to distribute seats proportionally between states (or parties) in a parliament. In the USA it has been abandoned due to some drawbacks, in particula ...
TL;DR: In this article, a general definition of the signature for systems with ν = 3 has been proposed, and several properties and properties of such a general signature definition have been analyzed.
Abstract: The notion of the signature is a basic concept and a powerful tool in the analysis of networks and reliability systems of binary type. An appropriate definition of this concept has recently been introduced for systems that have ν possible states (with ν ≥ 3). In this paper we analyze in detail several properties and the most relevant aspects of such a general definition. For simplicity's sake, we focus our attention on the case ν = 3. Our analysis will however provide a number of hints for understanding the basic aspects of the general case.
TL;DR: In this article, the authors considered a coherent system consisting of n components with independent and identically distributed components and proposed two time-dependent criteria to investigate the inactivity times of the failed components of the coherent system still functioning though some of its components have failed.
Abstract: In the study of the reliability of technical systems in reliability engineering, coherent systems play a key role. In this paper we consider a coherent system consisting of n components with independent and identically distributed components and propose two time-dependent criteria. The first criterion is a measure of the residual lifetime of live components of a coherent system having some of the components alive when the system fails at time t. The second criterion is a time-dependent measure which enables us to investigate the inactivity times of the failed components of a coherent system still functioning though some of its components have failed. Several ageing and stochastic properties of the proposed measures are then established.
TL;DR: In this paper, a general bivariate Levy-driven risk model is considered and the finite-time and infinite-time ruin probabilities for the case in which the loss process P has a Levy measure of extended regular variation and the stochastic exponential of R fulfills a moment condition.
Abstract: Consider a general bivariate Levy-driven risk model. The surplus process Y, starting with Y 0=x > 0, evolves according to dY t = Y t- dR t -dP t for t > 0, where P and R are two independent Levy processes respectively representing a loss process in a world without economic factors and a process describing the return on investments in real terms. Motivated by a conjecture of Paulsen, we study the finite-time and infinite-time ruin probabilities for the case in which the loss process P has a Levy measure of extended regular variation and the stochastic exponential of R fulfills a moment condition. We obtain a simple and unified asymptotic formula as x→∞, which confirms Paulsen's conjecture.
TL;DR: In this paper, the authors investigate a family of distributions having a property of stability underaddition, provided that the number of added-up random variables in the random sum is also a random variable.
Abstract: We investigate a family of distributions having a property of stability-underaddition, provided that the number ” of added-up random variables in the random sum is also a random variable. We call the corresponding property a ”-stability and investigate the situation with the semigroup generated by the generating function of ” is commutative. Using results from the theory of iterations of analytic functions, we show that the characteristic function of such a ”-stable distribution can be represented in terms of Chebyshev polynomials, and for the case of ”-normal distribution, the resulting characteristic function corresponds to the hyperbolic secant distribution. We discuss some speciflc properties of the class and present particular examples.
TL;DR: In this paper, a continuous time stochastic process with moments that satisfy an exact scaling relation, including odd order moments, is proposed for the price of a financial asset, based on a natural extension of the MRW construction described in [3].
Abstract: We present the construction of a continuous time stochastic process which has moments that satisfy an exact scaling relation, including odd order moments. It is based on a natural extension of the MRW construction described in [3]. This allows us to propose a continuous time model for the price of a financial asset that reflects most major stylized facts observed on real data, including asymmetry and multifractal scaling.
TL;DR: In this article, the authors define backward coalescence times for these kind of processes, which allow them to construct perfect simulation algorithms under weaker conditions than in Comets, Fernandez and Ferrari (2002).
Abstract: This paper is devoted to the perfect simulation of a stationary process with an at most countable state space. The process is specified through a kernel, prescribing the probability of the next state conditional to the whole past history. We follow the seminal work of Comets, Fernandez and Ferrari (2002), who gave sufficient conditions for the construction of a perfect simulation algorithm. We define backward coalescence times for these kind of processes, which allow us to construct perfect simulation algorithms under weaker conditions than in Comets, Fernandez and Ferrari (2002). We discuss how to construct backward coalescence times (i) by means of information depths, taking into account some a priori knowledge about the histories that occur; and (ii) by identifying suitable coalescing events.
TL;DR: In this paper, a general model for inhomogeneous random digraphs with labeled vertices is presented, where the arcs are generated independently, and the probability of inserting an arc depends on the labels of its endpoints and its orientation.
Abstract: We present and investigate a general model for inhomogeneous random digraphs with labeled vertices, where the arcs are generated independently, and the probability of inserting an arc depends on the labels of its endpoints and its orientation. For this model the critical point for the emergence of a giant component is determined via a branching process approach. key words: inhomogeneous digraph, phase transition, giant component. 2010 Mathematics Subject Classification 05C82, 90B15, 60C05
TL;DR: In this article, the ergodicity and exponential egodicity of Levy-driven Ornstein-Uhlenbeck processes were established based on the explicit coupling property, and they were shown to have the same eigenvectors.
Abstract: Based on the explicit coupling property, the ergodicity and the exponential ergodicity of Levy-driven Ornstein--Uhlenbeck processes are established.
TL;DR: In this article, the distributions of occupation times of a Markov-modulated Brownian motion (MMBM) in separate intervals before a first passage time or an exit from an interval are derived in terms of their Laplace transforms.
Abstract: In this paper we determine the distributions of occupation times of a Markov-modulated Brownian motion (MMBM) in separate intervals before a first passage time or an exit from an interval. We derive the distributions in terms of their Laplace transforms, and we also distinguish between occupation times in different phases. For MMBMs with strictly positive variation parameters, we further propose scale functions.
TL;DR: In this paper, the correlation function of {X n (t), t∈[0,∞]], n∈ℕ, which satisfies the local and long-range strong dependence conditions, was established.
Abstract: Let {X n (t), t∈[0,∞)}, n∈ℕ, be standard stationary Gaussian processes. The limit distribution of t∈[0,T(n)]|X n (t)| is established as r n (t), the correlation function of {X n (t), t∈[0,∞)}, n∈ℕ, which satisfies the local and long-range strong dependence conditions, extending the results obtained in Seleznjev (1991).
TL;DR: In this article, it was shown that fractional Brownian motion with H < 1/2 can arise as a limit of a simple class of traffic processes called scheduled traffic models.
Abstract: This paper shows that fractional Brownian motion with H < 1=2 can arise as a limit of a simple class of traffic processes that we call “scheduled traffic models”. To our knowledge, this paper provides the first simple traffic model leading to fractional Brownnian motion with H < 1=2: We also discuss some immediate implications of this result for queues fed by scheduled traffic, including a heavytraffic limit theorem.
TL;DR: In this paper, it was shown that a number of the identities in [7] are still valid for a much more general class of rate functions : [0,∞) → R.
Abstract: In this article we show that a number of the identities in [7] are still valid for a much more general class of rate functions : [0,∞) → R. Moreover, we show that, with appropriately chosen , the perturbed process can pass continuously (ie. creep) into (−∞,0) in two different ways.
TL;DR: In this article, a change of variable formula in the framework of functions of bounded variation is used to derive an explicit formula for the Fourier transform of the crossing function of shot noise processes with jumps, which is then able to study the behavior of the mean number of crossings as the intensity of the Poisson point process of the shot noise goes to infinity.
Abstract: We use here a change of variable formula in the framework of functions of bounded variation to derive an explicit formula for the Fourier transform of the crossing function of shot noise processes with jumps. We illustrate the result on some examples and give some applications. In particular we are then able to study the behavior of the mean number of crossings as the intensity of the Poisson point process of the shot noise goes to infinity.
TL;DR: In this article, the authors study the asymptotic behavior of a general class of product-form closed queueing networks as the population size grows large and derive new, computationally simple approximations for performance metrics.
Abstract: In this paper we study the asymptotic behavior of a general class of product-form closed queueing networks as the population size grows large. We first characterize the asymptotic behavior of the normalization constant for the stationary distribution of the network in exact order. This result then enables us to establish the asymptotic behavior of the system performance metrics, which extends a number of well-known asymptotic results to exact order. We further derive new, computationally simple approximations for performance metrics that significantly improve upon existing approximations for large-scale networks. In addition to their direct use for the analysis of large networks, these new approximations are particularly useful for reformulating large-scale queueing network optimization problems into more easily solvable forms, which we demonstrate with an optimal capacity planning example.
TL;DR: In this paper, the smooth-fit property of the American put price with finite maturity in an exponential Levy model was studied, where the underlying stock pays dividends at a continuous rate.
Abstract: We study the smooth-fit property of the American put price with finite maturity in an exponential Levy model when the underlying stock pays dividends at a continuous rate. As in the perpetual case, a regularity property is sufficient for smooth fit to occur. We also derive conditions on the Levy measure under which smooth fit fails.
TL;DR: In this article, the authors present the Lie algebraic method, and apply it to three biologically well-motivated examples, and show that the result of this is a solution form that is often highly computationally advantageous.
Abstract: Many natural populations are well modelled through time-inhomogeneous stochastic processes. Such processes have been analysed in the physical sciences using a method based on Lie algebras, but this methodology is not widely used for models with ecological, medical, and social applications. In this paper we present the Lie algebraic method, and apply it to three biologically well-motivated examples. The result of this is a solution form that is often highly computationally advantageous.