International Journal of Stochastic Analysis The latest articles from Hindawi Publishing Corporation © 2014 , Hindawi Publishing Corporation . All rights reserved. Efficient Variable Step Size Approximations for Strong Solutions of Stochastic Differential Equations with Additive Noise and Time Singularity Wed, 02 Jul 2014 10:48:33 +0000 We consider stochastic differential equations with additive noise and conditions on the coefficients in those equations that allow a time singularity in the drift coefficient. Given a maximum step size, , we specify variable (adaptive) step sizes relative to which decrease as the time node points approach the singularity. We use an Euler-type numerical scheme to produce an approximate solution and estimate the error in the approximation. When the solution is restricted to a fixed closed time interval excluding the singularity, we obtain a global pointwise error of order . An order of error for any is obtained when the approximation is run up to a time within of the singularity for an appropriate choice of exponent . We apply this scheme to Brownian bridge, which is defined as the nonanticipating solution of a stochastic differential equation of the type under consideration. In this special case, we show that the global pointwise error is of order , independent of how close to the singularity the approximation is considered. Harry Randolph Hughes and Pathiranage Lochana Siriwardena Copyright © 2014 Harry Randolph Hughes and Pathiranage Lochana Siriwardena. All rights reserved. A Two-Mode Mean-Field Optimal Switching Problem for the Full Balance Sheet Sun, 25 May 2014 09:28:32 +0000 We consider the problem of switching a large number of production lines between two modes, high production and low production. The switching is based on the optimal expected profit and cost yields of the respective production lines and considers both sides of the balance sheet. Furthermore, the production lines are all assumed to be interconnected through a coupling term, which is the average of all optimal expected yields. Intuitively, this means that each individual production line is compared to the average of all its peers which acts as a benchmark. Due to the complexity of the problem, we consider the aggregated optimal expected yields, where the coupling term is approximated with the mean of the optimal expected yields. This turns the problem into a two-mode optimal switching problem of mean-field type, which can be described by a system of Snell envelopes where the obstacles are interconnected and nonlinear. The main result of the paper is a proof of a continuous minimal solution to the system of Snell envelopes, as well as the full characterization of the optimal switching strategy. Boualem Djehiche and Ali Hamdi Copyright © 2014 Boualem Djehiche and Ali Hamdi. All rights reserved. Influence of Gestation Delay and Predator’s Interference in Predator-Prey Interaction under Stochastic Environment Tue, 08 Apr 2014 00:00:00 +0000 Previous experimental and theoretical studies suggest that predator’s interference in predator-prey relationship provides better descriptions of predator’s feeding over a range of predator-prey abundances. Also biological delays and environmental stochasticity play an important role to describe the system and its values. In this present study, I consider a Gaussian white-noise induced stochastic predator-prey model with the Beddington-DeAngelis functional response and gestation delay. Stochastic stability is measured by second order moment terms by calculating the nonequilibrium fluctuation of the nondelayed system and Fourier transform technique depicts the fluctuation of stochastic stability by introducing time lag. Different dynamical behaviors for both situations have been illustrated numerically also. The biological implications of my analytical and numerical findings are discussed critically. Debaldev Jana Copyright © 2014 Debaldev Jana. All rights reserved. with Setup Time, Bernoulli Vacation, Break Down, and Delayed Repair Mon, 31 Mar 2014 07:21:43 +0000 We present a single server in which customers arrive in batches and the server provides service one by one. The server provides two heterogeneous service stages such that service time of both stages is different and mandatory to all arriving customers in such a way that, after the completion of first stage, the second stage should also be provided to the customers. The server may subject to random breakdowns with brake down rate and, after break down, it should be repaired but it has to wait for being repaired and such waiting time is called delay time. Both the delay time and repair time follow exponential distribution. Upon the completion of the second stage service, the server will go for vacation with probability or stay back in the system probability , if any. The vacation time follows general (arbitrary) distribution. Before providing service to a new customer or a batch of customers that joins the system in the renewed busy period, the server enters into a random setup time process such that setup time follows exponential distribution. We discuss the transient behavior and the corresponding steady state results with the performance measures of the model. G. Ayyappan and S. Shyamala Copyright © 2014 G. Ayyappan and S. Shyamala. All rights reserved. From Pseudorandom Walk to Pseudo-Brownian Motion: First Exit Time from a One-Sided or a Two-Sided Interval Wed, 26 Mar 2014 07:28:39 +0000 Let be a positive integer, a positive constant and be a sequence of independent identically distributed pseudorandom variables. We assume that the ’s take their values in the discrete set and that their common pseudodistribution is characterized by the (positive or negative) real numbers for any . Let us finally introduce the associated pseudorandom walk defined on by and for . In this paper, we exhibit some properties of . In particular, we explicitly determine the pseudodistribution of the first overshooting time of a given threshold for as well as that of the first exit time from a bounded interval. Next, with an appropriate normalization, we pass from the pseudorandom walk to the pseudo-Brownian motion driven by the high-order heat-type equation . We retrieve the corresponding pseudodistribution of the first overshooting time of a threshold for the pseudo-Brownian motion (Lachal, 2007). In the same way, we get the pseudodistribution of the first exit time from a bounded interval for the pseudo-Brownian motion which is a new result for this pseudoprocess. Aimé Lachal Copyright © 2014 Aimé Lachal. All rights reserved. Diffusion Processes Satisfying a Conservation Law Constraint Tue, 04 Mar 2014 12:08:07 +0000 We investigate coupled stochastic differential equations governing N nonnegative continuous random variables that satisfy a conservation principle. In various fields a conservation law requires a set of fluctuating variables to be nonnegative and (if appropriately normalized) sum to one. As a result, any stochastic differential equation model to be realizable must not produce events outside of the allowed sample space. We develop a set of constraints on the drift and diffusion terms of such stochastic models to ensure that both the nonnegativity and the unit-sum conservation law constraints are satisfied as the variables evolve in time. We investigate the consequences of the developed constraints on the Fokker-Planck equation, the associated system of stochastic differential equations, and the evolution equations of the first four moments of the probability density function. We show that random variables, satisfying a conservation law constraint, represented by stochastic diffusion processes, must have diffusion terms that are coupled and nonlinear. The set of constraints developed enables the development of statistical representations of fluctuating variables satisfying a conservation law. We exemplify the results with the bivariate beta process and the multivariate Wright-Fisher, Dirichlet, and Lochner’s generalized Dirichlet processes. J. Bakosi and J. R. Ristorcelli Copyright © 2014 J. Bakosi and J. R. Ristorcelli. All rights reserved. SPDEs with -Stable Lévy Noise: A Random Field Approach Tue, 04 Feb 2014 12:18:58 +0000 This paper is dedicated to the study of a nonlinear SPDE on a bounded domain in , with zero initial conditions and Dirichlet boundary, driven by an -stable Lévy noise with , , and possibly nonsymmetric tails. To give a meaning to the concept of solution, we develop a theory of stochastic integration with respect to this noise. The idea is to first solve the equation with “truncated” noise (obtained by removing from the jumps which exceed a fixed value ), yielding a solution , and then show that the solutions coincide on the event , for some stopping times converging to infinity. A similar idea was used in the setting of Hilbert-space valued processes. A major step is to show that the stochastic integral with respect to satisfies a th moment inequality. This inequality plays the same role as the Burkholder-Davis-Gundy inequality in the theory of integration with respect to continuous martingales. Raluca M. Balan Copyright © 2014 Raluca M. Balan. All rights reserved. The Relationship between the Stochastic Maximum Principle and the Dynamic Programming in Singular Control of Jump Diffusions Thu, 09 Jan 2014 13:11:32 +0000 The main objective of this paper is to explore the relationship between the stochastic maximum principle (SMP in short) and dynamic programming principle (DPP in short), for singular control problems of jump diffusions. First, we establish necessary as well as sufficient conditions for optimality by using the stochastic calculus of jump diffusions and some properties of singular controls. Then, we give, under smoothness conditions, a useful verification theorem and we show that the solution of the adjoint equation coincides with the spatial gradient of the value function, evaluated along the optimal trajectory of the state equation. Finally, using these theoretical results, we solve explicitly an example, on optimal harvesting strategy, for a geometric Brownian motion with jumps. Farid Chighoub and Brahim Mezerdi Copyright © 2014 Farid Chighoub and Brahim Mezerdi. All rights reserved. Measure-Dependent Stochastic Nonlinear Beam Equations Driven by Fractional Brownian Motion Tue, 31 Dec 2013 17:44:11 +0000 We study a class of nonlinear stochastic partial differential equations arising in the mathematical modeling of the transverse motion of an extensible beam in the plane. Nonlinear forcing terms of functional-type and those dependent upon a family of probability measures are incorporated into the initial-boundary value problem (IBVP), and noise is incorporated into the mathematical description of the phenomenon via a fractional Brownian motion process. The IBVP is subsequently reformulated as an abstract second-order stochastic evolution equation driven by a fractional Brownian motion (fBm) dependent upon a family of probability measures in a real separable Hilbert space and is studied using the tools of cosine function theory, stochastic analysis, and fixed-point theory. Global existence and uniqueness results for mild solutions, continuous dependence estimates, and various approximation results are established and applied in the context of the model. Mark A. McKibben Copyright © 2013 Mark A. McKibben. All rights reserved. Sharp Large Deviation for the Energy of -Brownian Bridge Sun, 08 Dec 2013 13:12:22 +0000 We study the sharp large deviation for the energy of -Brownian bridge. The full expansion of the tail probability for energy is obtained by the change of measure. Shoujiang Zhao, Qiaojing Liu, Fuxiang Liu, and Hong Yin Copyright © 2013 Shoujiang Zhao et al. All rights reserved. Some Limit Properties of the Harmonic Mean of Transition Probabilities for Markov Chains in Markovian Environments Indexed by Cayley's Trees Thu, 05 Dec 2013 18:20:23 +0000 We prove some limit properties of the harmonic mean of a random transition probability for finite Markov chains indexed by a homogeneous tree in a nonhomogeneous Markovian environment with finite state space. In particular, we extend the method to study the tree-indexed processes in deterministic environments to the case of random enviroments. Huilin Huang Copyright © 2013 Huilin Huang. All rights reserved. Foundations of the Theory of Semilinear Stochastic Partial Differential Equations Wed, 27 Nov 2013 08:38:41 +0000 The goal of this review article is to provide a survey about the foundations of semilinear stochastic partial differential equations. In particular, we provide a detailed study of the concepts of strong, weak, and mild solutions, establish their connections, and review a standard existence and uniqueness result. The proof of the existence result is based on a slightly extended version of the Banach fixed point theorem. Stefan Tappe Copyright © 2013 Stefan Tappe. All rights reserved. Online Stochastic Convergence Analysis of the Kalman Filter Thu, 21 Nov 2013 16:09:40 +0000 This paper presents modifications to the stochastic stability lemma which is then used to estimate the convergence rate and persistent error of the linear Kalman filter online without using knowledge of the true state. Unlike previous uses of the stochastic stability lemma for stability proof, this new convergence analysis technique considers time-varying parameters, which can be calculated online in real-time to monitor the performance of the filter. Through simulation of an example problem, the new method was shown to be effective in determining a bound on the estimation error that closely follows the actual estimation error. Different cases of assumed process and measurement noise covariance matrices were considered in order to study their effects on the convergence and persistent error of the Kalman filter. Matthew B. Rhudy and Yu Gu Copyright © 2013 Matthew B. Rhudy and Yu Gu. All rights reserved. Analysis of Queue-Length Dependent Vacations and P-Limited Service in BMAP/G/1/N Systems: Stationary Distributions and Optimal Control Sun, 10 Nov 2013 09:08:15 +0000 We consider a finite-buffer single server queueing system with queue-length dependent vacations where arrivals occur according to a batch Markovian arrival process (BMAP). The service discipline is P-limited service, also called E-limited with limit variation (ELV) where the server serves until either the system is emptied or a randomly chosen limit of customers has been served. Depending on the number of customers present in the system, the server will monitor his vacation times. Queue-length distributions at various epochs such as before, arrival, arbitrary and after, departure have been obtained. Several other service disciplines like Bernoulli scheduling, nonexhaustive service, and E-limited service can be treated as special cases of the P-limited service. Finally, the total expected cost function per unit time is considered to determine locally optimal values of or a maximum limit of as the number of customers served during a service period at a minimum cost. A. D. Banik Copyright © 2013 A. D. Banik. All rights reserved. The BALM Copula Thu, 26 Sep 2013 16:03:18 +0000 The class of probability distributions possessing the almost-lack-of-memory property appeared about 20 years ago. It reasonably took place in research and modeling, due to its suitability to represent uncertainty in periodic random environment. Multivariate version of the almost-lack-of-memory property is less known, but it is not less interesting. In this paper we give the copula of the bivariate almost-lack-of-memory (BALM) distributions and discuss some of its properties and applications. An example shows how the Marshal-Olkin distribution can be turned into BALM and what is its copula. Boyan Dimitrov and Nikolai Kolev Copyright © 2013 Boyan Dimitrov and Nikolai Kolev. All rights reserved. Stability Analysis of a Stochastic SIR Epidemic Model with Specific Nonlinear Incidence Rate Sun, 22 Sep 2013 12:10:25 +0000 We investigate a stochastic SIR epidemic model with specific nonlinear incidence rate. The stochastic model is derived from the deterministic epidemic model by introducing random perturbations around the endemic equilibrium state. The effect of random perturbations on the stability behavior of endemic equilibrium is discussed. Finally, numerical simulations are presented to illustrate our theoretical results. Jihad Adnani, Khalid Hattaf, and Noura Yousfi Copyright © 2013 Jihad Adnani et al. All rights reserved. The LMI Approach for Stabilizing of Linear Stochastic Systems Thu, 29 Aug 2013 08:56:12 +0000 Stochastic linear systems subjected both to Markov jumps and to multiplicative white noise are considered. In order to stabilize such type of stochastic systems, the so-called set of generalized discrete-time algebraic Riccati equations has to be solved. The LMI approach for computing the stabilizing symmetric solution (which is in fact the equilibrium point) of this system is studied. We construct a new modification of the standard LMI approach, and we show how to apply the new modification. Computer realizations of all modifications are compared. Numerical experiments are given where the LMI modifications are numerically compared. Based on the experiments the main conclusion is that the new LMI modification is faster than the standard LMI approach. Ivan Ivanov Copyright © 2013 Ivan Ivanov. All rights reserved. A Stochastic Diffusion Process for the Dirichlet Distribution Wed, 10 Apr 2013 17:23:06 +0000 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. The Itô Integral with respect to an Infinite Dimensional Lévy Process: A Series Approach Thu, 04 Apr 2013 11:15:21 +0000 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. Filtering for Discrete-Time Stochastic Systems with Nonlinear Sensor and Time-Varying Delay Tue, 19 Mar 2013 15:10:47 +0000 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. Applications of Stochastic Processes in Biology and Medicine Tue, 19 Mar 2013 14:19:20 +0000 Charles J. Mode, Rick Durrett, Fima Klebaner, and Peter Olofsson Copyright © 2013 Charles J. Mode et al. All rights reserved. Risk of Infectious Disease Outbreaks by Imported Cases with Application to the European Football Championship 2012 Tue, 19 Mar 2013 11:21:39 +0000 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. Modeling Neutral Evolution Using an Infinite-Allele Markov Branching Process Sun, 17 Mar 2013 08:10:38 +0000 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. Simulating the Emergence of Mutations and Their Subsequent Evolution in an Age-Structured Stochastic Self-Regulating Process with Two Sexes Mon, 11 Mar 2013 15:17:58 +0000 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. Asymptotic Behavior of Densities for Stochastic Functional Differential Equations Thu, 28 Feb 2013 10:59:58 +0000 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. Time Reversal of Volterra Processes Driven Stochastic Differential Equations Wed, 27 Feb 2013 16:02:29 +0000 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. A Decomposable Branching Process in a Markovian Environment Mon, 31 Dec 2012 17:54:21 +0000 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. Birth and Death Processes with Neutral Mutations Mon, 31 Dec 2012 14:51:17 +0000 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. Performance Analysis of Production Systems with Correlated Demand via Diffusion Approximations Mon, 31 Dec 2012 09:26:03 +0000 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. Probabilistic Solution of the General Robin Boundary Value Problem on Arbitrary Domains Sun, 30 Dec 2012 11:06:23 +0000 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.