http://www.hindawi.com The latest articles from Hindawi Publishing Corporation © 2013 , Hindawi Publishing Corporation . All rights reserved. Wed, 10 Apr 2013 17:23:06 +0000 http://www.hindawi.com/journals/ijsa/2013/842981/ The method of potential solutions of Fokker-Planck equations is used to develop a transport equation for the joint probability of N coupled stochastic variables with the Dirichlet distribution as its asymptotic solution. To ensure a bounded sample space, a coupled nonlinear diffusion process is required: the Wiener processes in the equivalent system of stochastic differential equations are multiplicative with coefficients dependent on all the stochastic variables. Individual samples of a discrete ensemble, obtained from the stochastic process, satisfy a unit-sum constraint at all times. The process may be used to represent realizations of a fluctuating ensemble of N variables subject to a conservation principle. Similar to the multivariate Wright-Fisher process, whose invariant is also Dirichlet, the univariate case yields a process whose invariant is the beta distribution. As a test of the results, Monte Carlo simulations are used to evolve numerical ensembles toward the invariant Dirichlet distribution. J. Bakosi and J. R. Ristorcelli Copyright © 2013 J. Bakosi and J. R. Ristorcelli. All rights reserved. Thu, 04 Apr 2013 11:15:21 +0000 http://www.hindawi.com/journals/ijsa/2013/703769/ We present an alternative construction of the infinite dimensional Itô integral with respect to a Hilbert space valued Lévy process. This approach is based on the well-known theory of real-valued stochastic integration, and the respective Itô integral is given by a series of Itô integrals with respect to standard Lévy processes. We also prove that this stochastic integral coincides with the Itô integral that has been developed in the literature. Stefan Tappe Copyright © 2013 Stefan Tappe. All rights reserved. Tue, 19 Mar 2013 15:10:47 +0000 http://www.hindawi.com/journals/ijsa/2013/306707/ The filtering problem for a class of discrete-time stochastic systems with nonlinear sensor and time-varying delay is investigated. By using the Lyapunov stability theory, sufficient conditions are proposed to guarantee the asymptotical stablity with an prescribe performance level of the filtering error systems. These conditions are dependent on the lower and upper bounds of the discrete time-varying delays and are obtained in terms of a linear matrix inequality (LMI). Finally, two numerical examples are provided to illustrate the effectiveness of the proposed methods. Mingang Hua, Pei Cheng, Juntao Fei, Jianyong Zhang, and Junfeng Chen Copyright © 2013 Mingang Hua et al. All rights reserved. Tue, 19 Mar 2013 14:19:20 +0000 http://www.hindawi.com/journals/ijsa/2013/790625/ Charles J. Mode, Rick Durrett, Fima Klebaner, and Peter Olofsson Copyright © 2013 Charles J. Mode et al. All rights reserved. Tue, 19 Mar 2013 11:21:39 +0000 http://www.hindawi.com/journals/ijsa/2013/576381/ The European Centre for Disease Prevention and Control called the attention in March 2012 to the risk of measles in Ukraine among visitors to the 2012 UEFA European Football Championship. Large populations of supporters travelled to various locations in Poland and Ukraine, depending on the schedule of Euro 2012 and the outcome of the games, possibly carrying the disease from one location to another. In the present study, we propose a novel two-phase multitype branching process model with immigration to describe the risk of a major epidemic in connection with large-scale sports-related mass gathering events. By analytic means, we calculate the expected number and the variance of imported cases and the probability of a major epidemic caused by the imported cases in their home country. Applying our model to the case study of Euro 2012 we demonstrate that the results of the football games can be highly influential to the risk of measles outbreaks in the home countries of supporters. To prevent imported epidemics, it should be emphasized that vaccinating travellers would most efficiently reduce the risk of epidemic, while requiring the minimum doses of vaccines as compared to other vaccination strategies. Our theoretical framework can be applied to other future sport tournaments too. Attila Dénes, Péter Kevei, Hiroshi Nishiura, and Gergely Röst Copyright © 2013 Attila Dénes et al. All rights reserved. Sun, 17 Mar 2013 08:10:38 +0000 http://www.hindawi.com/journals/ijsa/2013/963831/ We consider an infinite-allele Markov branching process (IAMBP). Our main focus is the frequency spectrum of this process, that is, the proportion of alleles having a given number of copies at a specified time point. We derive the variance of the frequency spectrum, which is useful for interval estimation and hypothesis testing for process parameters. In addition, for a class of special IAMBP with birth and death offspring distribution, we show that the mean of its limiting frequency spectrum has an explicit form in terms of the hypergeometric function. We also derive an asymptotic expression for convergence rate to the limit. Simulations are used to illustrate the results for the birth and death process. Xiaowei Wu and Marek Kimmel Copyright © 2013 Xiaowei Wu and Marek Kimmel. All rights reserved. Mon, 11 Mar 2013 15:17:58 +0000 http://www.hindawi.com/journals/ijsa/2013/826321/ The stochastic process under consideration is intended to be not only part of the working paradigm of evolutionary and population genetics but also that of applied probability and stochastic processes with an emphasis on computer intensive methods. In particular, the process is an age-structured self-regulating multitype branching process with a genetic component consisting of an autosomal locus with two alleles for females and males. It is within this simple context that mutation will be quantified in terms of probabilities that a given allele mutates to the other per meiosis. But, unlike many models that are currently being used in mathematical population genetics, in which natural selection is often characterized in terms of parameters called fitness by genotype or phenotype, in this paper the parameterization of submodules of the model provides a framework for characterizing natural selection in terms of some of its components. One of these modules consists of reproductive success that is quantified in terms of the total expected number of offspring a female contributes to the population throughout her fertile years. Another component consists of survival probabilities that characterize an individual’s ability to compete for limited environmental resources. A third module consists of a parametric function that expresses the probabilities of survival in a birth cohort of individuals by age for both females and males. A forth module of the model as an acceptance matrix of conditional probabilities such female may show a preference for the genotype or phenotype as her male sexual partner. It is assumed that any force of natural selection acts at the level of the three genotypes under consideration for each sex. By assigning values of the parameters in each of the modules under consideration, it is possible to conduct Monte Carlo simulation experiments designed to study the effects of each component of selection separately or in any combination on a population evolving from a given initial population over some specified period of time. Charles J. Mode, Candace K. Sleeman, and Towfique Raj Copyright © 2013 Charles J. Mode et al. All rights reserved. Thu, 28 Feb 2013 10:59:58 +0000 http://www.hindawi.com/journals/ijsa/2013/537023/ Consider stochastic functional differential equations depending on whole past histories in a finite time interval, which determine non-Markovian processes. Under the uniformly elliptic condition on the coefficients of the diffusion terms, the solution admits a smooth density with respect to the Lebesgue measure. In the present paper, we will study the large deviations for the family of the solution process and the asymptotic behaviors of the density. The Malliavin calculus plays a crucial role in our argument. Akihiro Kitagawa and Atsushi Takeuchi Copyright © 2013 Akihiro Kitagawa and Atsushi Takeuchi. All rights reserved. Wed, 27 Feb 2013 16:02:29 +0000 http://www.hindawi.com/journals/ijsa/2013/790709/ We consider stochastic differential equations driven by some Volterra processes. Under time reversal, these equations are transformed into past-dependent stochastic differential equations driven by a standard Brownian motion. We are then in position to derive existence and uniqueness of solutions of the Volterra driven SDE considered at the beginning. L. Decreusefond Copyright © 2013 L. Decreusefond. All rights reserved. Mon, 31 Dec 2012 17:54:21 +0000 http://www.hindawi.com/journals/ijsa/2012/694285/ A population has two types of individuals, with each occupying an island. One of those, where individuals of type 1 live, offers a variable environment. Type 2 individuals dwell on the other island, in a constant environment. Only one-way migration () is possible. We study then asymptotics of the survival probability in critical and subcritical cases. Vladimir Vatutin, Elena Dyakonova, Peter Jagers, and Serik Sagitov Copyright © 2012 Vladimir Vatutin et al. All rights reserved. Mon, 31 Dec 2012 14:51:17 +0000 http://www.hindawi.com/journals/ijsa/2012/569081/ We review recent results of ours concerning branching processes with general lifetimes and neutral mutations, under the infinitely many alleles model, where mutations can occur either at the birth of particles or at a constant rate during their lives. In both models, we study the allelic partition of the population at time . We give closed-form formulae for the expected frequency spectrum at and prove a pathwise convergence to an explicit limit, as , of the relative numbers of types younger than some given age and carried by a given number of particles (small families). We also provide the convergences in distribution of the sizes or ages of the largest families and of the oldest families. In the case of exponential lifetimes, population dynamics are given by linear birth and death processes, and we can most of the time provide general formulations of our results unifying both models. Nicolas Champagnat, Amaury Lambert, and Mathieu Richard Copyright © 2012 Nicolas Champagnat et al. All rights reserved. Mon, 31 Dec 2012 09:26:03 +0000 http://www.hindawi.com/journals/ijsa/2012/109417/ We investigate the performance of a production system with correlated demand through diffusion approximation. The key performance metric under consideration is the extreme points that this system can reach. This problem is mapped to a problem of characterizing the joint probability density of a two-dimensional Brownian motion and its coordinate running maximum. To achieve this goal, we obtain the stationary distribution of a reflected Brownian motion within the positive quarter-plane, which is of independent interest, through investigating a solution of an extended Helmhotz equation. Yingdong Lu Copyright © 2012 Yingdong Lu. All rights reserved. Sun, 30 Dec 2012 11:06:23 +0000 http://www.hindawi.com/journals/ijsa/2012/163096/ Using a capacity approach and the theory of the measure’s perturbation of the Dirichlet forms, we give the probabilistic representation of the general Robin boundary value problems on an arbitrary domain Ω, involving smooth measures, which give rise to a new process obtained by killing the general reflecting Brownian motion at a random time. We obtain some properties of the semigroup directly from its probabilistic representation, some convergence theorems, and also a probabilistic interpretation of the phenomena occurring on the boundary. Khalid Akhlil Copyright © 2012 Khalid Akhlil. All rights reserved. Tue, 25 Dec 2012 07:38:16 +0000 http://www.hindawi.com/journals/ijsa/2012/236327/ We provide existence and uniqueness of global (and local) mild solutions for a general class of semilinear stochastic partial differential equations driven by Wiener processes and Poisson random measures under local Lipschitz and linear growth (or local boundedness, resp.) conditions. The so-called “method of the moving frame” allows us to reduce the SPDE problems to SDE problems. Stefan Tappe Copyright © 2012 Stefan Tappe. All rights reserved. Mon, 24 Dec 2012 15:14:11 +0000 http://www.hindawi.com/journals/ijsa/2012/498050/ The optimal geometric mean return is an important property of an asset. As a derivative of the underlying asset, the option also has this property. In this paper, we show that the optimal geometric mean returns of a stock and its option are the same from Kelly criterion. It is proved by using binomial option pricing model and continuous stochastic models with self-financing assumption. A simulation study reveals the same result for the continuous option pricing model. Guoyi Zhang Copyright © 2012 Guoyi Zhang. All rights reserved. Mon, 10 Dec 2012 09:37:54 +0000 http://www.hindawi.com/journals/ijsa/2012/673642/ A retrial queueing system with two types of batch arrivals, called type I and type II customers, is considered. Type I customers and type II customers arrive in batches of variable sizes according to two different Poisson processes. Service time distributions are identical and independent and are different for both types of customers. If the arriving customers are blocked due to the server being busy, type I customers are queued in a priority queue of infinite capacity, whereas type II customers enter into a retrial group in order to seek service again after a random amount of time. A type I customer who has received service departs the system with a preassigned probability or returns to the priority queue for reservice with the complement probability. A type II call who has received service leaves the system with a preassigned probability or rejoins the retrial group with complement probability. For this model, the joint distribution of the number of customers in the priority queue and in the retrial group is obtained in a closed form. Some particular models and operating characteristics are obtained. A numerical study is also carried out. R. Kalyanaraman Copyright © 2012 R. Kalyanaraman. All rights reserved. Wed, 05 Dec 2012 14:57:50 +0000 http://www.hindawi.com/journals/ijsa/2012/281474/ We study the stability of the solutions of stochastic differential equations driven by fractional Brownian motions with Hurst parameter greater than half. We prove that when the initial conditions, the drift, and the diffusion coefficients as well as the fractional Brownian motions converge in a suitable sense, then the sequence of the solutions of the corresponding equations converge in Hölder norm to the solution of a stochastic differential equation. The limit equation is driven by the limit fractional Brownian motion and its coefficients are the limits of the sequence of the coefficients. Bruno Saussereau Copyright © 2012 Bruno Saussereau. All rights reserved. Tue, 04 Dec 2012 15:42:09 +0000 http://www.hindawi.com/journals/ijsa/2012/598701/ We present a stochastic methodology to study the decay phase of an epidemic. It is based on a general stochastic epidemic process with memory, suitable to model the spread in a large open population with births of any rare transmissible disease with a random incubation period and a Reed-Frost type infection. This model, which belongs to the class of multitype branching processes in discrete time, enables us to predict the incidences of cases and to derive the probability distributions of the extinction time and of the future epidemic size. We also study the epidemic evolution in the worst-case scenario of a very late extinction time, making use of the Q-process. We provide in addition an estimator of the key parameter of the epidemic model quantifying the infection and finally illustrate this methodology with the study of the Bovine Spongiform Encephalopathy epidemic in Great Britain after the 1988 feed ban law. Sophie Pénisson and Christine Jacob Copyright © 2012 Sophie Pénisson and Christine Jacob. All rights reserved. Thu, 29 Nov 2012 16:14:40 +0000 http://www.hindawi.com/journals/ijsa/2012/137271/ The problem is a power-law asymptotics of the probability that a self-similar process does not exceed a fixed level during long time. The exponent in such asymptotics is estimated for some Gaussian processes, including the fractional Brownian motion (FBM) in , and the integrated FBM in , . G. Molchan Copyright © 2012 G. Molchan. All rights reserved. Tue, 27 Nov 2012 15:39:48 +0000 http://www.hindawi.com/journals/ijsa/2012/905082/ In order to estimate the memory parameter of Internet traffic data, it has been recently proposed a log-regression estimator based on the so-called modified Allan variance (MAVAR). Simulations have shown that this estimator achieves higher accuracy and better confidence when compared with other methods. In this paper we present a rigorous study of the MAVAR log-regression estimator. In particular, under the assumption that the signal process is a fractional Brownian motion, we prove that it is consistent and asymptotically normally distributed. Finally, we discuss its connection with the wavelets estimators. Alessandra Bianchi, Massimo Campanino, and Irene Crimaldi Copyright © 2012 Alessandra Bianchi et al. All rights reserved. Sat, 10 Nov 2012 12:50:56 +0000 http://www.hindawi.com/journals/ijsa/2012/268568/ Consider an Ornstein-Uhlenbeck process driven by a fractional Brownian motion. It is an interesting problem to find criteria for whether the process is stable or has a unit root, given a finite sample of observations. Recently, various asymptotic distributions for estimators of the drift parameter have been developed. We illustrate through computer simulations and through a Stein's bound that these asymptotic distributions are inadequate approximations of the finite-sample distribution for moderate values of the drift and the sample size. We propose a new model to obtain asymptotic distributions near zero and compute the limiting distribution. We show applications to regression analysis and obtain hypothesis tests and their asymptotic power. Michael Moers Copyright © 2012 Michael Moers. All rights reserved. Tue, 23 Oct 2012 11:25:02 +0000 http://www.hindawi.com/journals/ijsa/2012/971212/ We consider the general one-dimensional time-homogeneous regular diffusion process between two reflecting barriers. An approach based on the Itô formula with corresponding boundary conditions allows us to derive the differential equations with boundary conditions for the Laplace transform of the first passage time and the value function. As examples, the explicit solutions of them for several popular diffusions are obtained. In addition, some applications to risk theory are considered. Chuancun Yin and Huiqing Wang Copyright © 2012 Chuancun Yin and Huiqing Wang. All rights reserved. Tue, 16 Oct 2012 16:31:59 +0000 http://www.hindawi.com/journals/ijsa/2012/427383/ The Master Equation approach to model anomalous diffusion is considered. Anomalous diffusion in complex media can be described as the result of a superposition mechanism reflecting inhomogeneity and nonstationarity properties of the medium. For instance, when this superposition is applied to the time-fractional diffusion process, the resulting Master Equation emerges to be the governing equation of the Erdélyi-Kober fractional diffusion, that describes the evolution of the marginal distribution of the so-called generalized grey Brownian motion. This motion is a parametric class of stochastic processes that provides models for both fast and slow anomalous diffusion: it is made up of self-similar processes with stationary increments and depends on two real parameters. The class includes the fractional Brownian motion, the time-fractional diffusion stochastic processes, and the standard Brownian motion. In this framework, the M-Wright function (known also as Mainardi function) emerges as a natural generalization of the Gaussian distribution, recovering the same key role of the Gaussian density for the standard and the fractional Brownian motion. Gianni Pagnini, Antonio Mura, and Francesco Mainardi Copyright © 2012 Gianni Pagnini et al. All rights reserved. Tue, 02 Oct 2012 08:49:15 +0000 http://www.hindawi.com/journals/ijsa/2012/803724/ Optimal control problems for one-dimensional diffusion processes in the interval () are considered. The aim is either to maximize or to minimize the time spent by the controlled processes in (). Exact solutions are obtained when the processes are symmetrical with respect to . Approximate solutions are derived in the asymmetrical case. The one-barrier cases are also treated. Examples are presented. Mario Lefebvre and Foued Zitouni Copyright © 2012 Mario Lefebvre and Foued Zitouni. All rights reserved. Wed, 26 Sep 2012 10:14:08 +0000 http://www.hindawi.com/journals/ijsa/2012/858736/ The aim of the paper is to introduce a generalization of the Feynman-Kac theorem in Hilbert spaces. Connection between solutions to the abstract stochastic differential equation and solutions to the deterministic partial differential (with derivatives in Hilbert spaces) equation for the probability characteristic is proved. Interpretation of objects in the equations is given. I. V. Melnikova and V. S. Parfenenkova Copyright © 2012 I. V. Melnikova and V. S. Parfenenkova. All rights reserved. Mon, 27 Aug 2012 13:47:57 +0000 http://www.hindawi.com/journals/ijsa/2012/687376/ Let 𝑋𝑡 be any d-dimensional continuous process that takes values in an open connected domain 𝒪 in ℝ𝑑. In this paper, we give equivalent formulations of the conditional full support (CFS) property of 𝑋𝑡 in 𝒪. We use them to show that the CFS property of X in 𝒪 implies the existence of a martingale M under an equivalent probability measure such that M lies in the 𝜖>0 neighborhood of 𝑋𝑡 for any given 𝜖 under the supremum norm. The existence of such martingales, which are called consistent price systems (CPSs), has relevance with absence of arbitrage and hedging problems in markets with proportional transaction costs as discussed in the recent paper by Guasoni et al. (2008), where the CFS property is introduced and shown sufficient for CPSs for processes with certain state space. The current paper extends the results in the work of Guasoni et al. (2008), to processes with more general state space. Florian Maris and Hasanjan Sayit Copyright © 2012 Florian Maris and Hasanjan Sayit. All rights reserved. Thu, 23 Aug 2012 09:03:14 +0000 http://www.hindawi.com/journals/ijsa/2012/185474/ We consider the class of semi-Markov modulated jump diffusions (sMMJDs) whose operator turns out to be an integro-partial differential operator. We find conditions under which the solutions of this class of switching jump-diffusion processes are almost surely exponentially stable and moment exponentially stable. We also provide conditions that imply almost sure convergence of the trivial solution when the moment exponential stability of the trivial solution is guaranteed. We further investigate and determine the conditions under which the trivial solution of the sMMJD-perturbed nonlinear system of differential equations 𝑑𝑋𝑡/𝑑𝑡=𝑓(𝑋𝑡) is almost surely exponentially stable. It is observed that for a one-dimensional state space, a linear unstable system of differential equations when stabilized just by the addition of the jump part of an sMMJD process does not get destabilized by any addition of a Brownian motion. However, in a state space of dimension at least two, we show that a corresponding nonlinear system of differential equations stabilized by jumps gets destabilized by addition of Brownian motion. Amogh Deshpande Copyright © 2012 Amogh Deshpande. All rights reserved. Mon, 13 Aug 2012 16:03:24 +0000 http://www.hindawi.com/journals/ijsa/2012/719237/ We propose a dependent hidden Markov model of credit quality. We suppose that the "true" credit quality is not observed directly but only through noisy observations given by posted credit ratings. The model is formulated in discrete time with a Markov chain observed in martingale noise, where "noise" terms of the state and observation processes are possibly dependent. The model provides estimates for the state of the Markov chain governing the evolution of the credit rating process and the parameters of the model, where the latter are estimated using the EM algorithm. The dependent dynamics allow for the so-called "rating momentum" discussed in the credit literature and also provide a convenient test of independence between the state and observation dynamics. Małgorzata Wiktoria Korolkiewicz Copyright © 2012 Małgorzata Wiktoria Korolkiewicz. All rights reserved. Wed, 01 Aug 2012 10:07:01 +0000 http://www.hindawi.com/journals/ijsa/2012/145867/ This paper analyzes an 𝑀/𝑀/2 queueing system with two heterogeneous servers, one of which is always available but the other goes on vacation in the absence of customers waiting for service. The vacationing server, however, returns to serve at a low rate as an arrival finds the other server busy. The system is analyzed in the steady state using matrix geometric method. Busy period of the system is analyzed and mean waiting time in the stationary regime computed. Conditional stochastic decomposition of stationary queue length is obtained. An illustrative example is also provided. A. Krishnamoorthy and C. Sreenivasan Copyright © 2012 A. Krishnamoorthy and C. Sreenivasan. All rights reserved. Thu, 07 Jun 2012 11:47:05 +0000 http://www.hindawi.com/journals/ijsa/2012/249201/ As a new formulation in structural analysis, Integrated Force Method has been successfully applied to many structures for civil, mechanical, and aerospace engineering due to the accurate estimate of forces in computation. Right now, it is being further extended to the probabilistic domain. For the assessment of uncertainty effect in system optimization and identification, the probabilistic sensitivity analysis of IFM was further investigated in this study. A set of stochastic sensitivity analysis formulation of Integrated Force Method was developed using the perturbation method. Numerical examples are presented to illustrate its application. Its efficiency and accuracy were also substantiated with direct Monte Carlo simulations and the reliability-based sensitivity method. The numerical algorithm was shown to be readily adaptable to the existing program since the models of stochastic finite element and stochastic design sensitivity are almost identical. X. F. Wei and S. N. Patnaik Copyright © 2012 X. F. Wei and S. N. Patnaik. All rights reserved.