Showing papers in "International Journal of Stochastic Analysis in 2012"
TL;DR: In this paper, the authors studied the allelic partition of the population at time t and proved pathwise convergence to an explicit limit, as t goes to infinity, of the relative numbers of types younger than some given age and carried by a given number of individuals (small families).
Abstract: In this paper, 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 birth of individuals or at a constant rate during their lives. In both models, we study the allelic partition of the population at time t. We give closed formulae for the expected frequency spectrum at t and prove pathwise convergence to an explicit limit, as t goes to infinity, of the relative numbers of types younger than some given age and carried by a given number of individuals (small families). We also provide 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.
27 citations
TL;DR: In this paper, the authors provide existence and uniqueness of 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 conditions.
Abstract: 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.
25 citations
TL;DR: This paper analyzes an 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, and its mean waiting time in the stationary regime computed.
Abstract: This paper analyzes an 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.
24 citations
TL;DR: In this article, the general optimal control problem for fully coupled forward-backward stochastic differential equations with random jumps (FBSDEJs) was studied, where the control domain is not assumed to be convex, and the control variable appears in both diffusion and jump coefficients of the forward equation.
Abstract: This paper deals with the general optimal control problem for fully coupled forward-backward
stochastic differential equations with random jumps (FBSDEJs). The control domain is not assumed to be convex, and the control variable appears in both diffusion and jump coefficients of the forward equation. Necessary conditions of Pontraygin's type for the optimal controls are derived by means of spike variation technique and Ekeland variational principle. A linear quadratic stochastic optimal control problem is discussed as an illustrating example.
21 citations
TL;DR: In this article, a power-law asymptotics of the probability that a self-similar process does not exceed a fixed level during a long time is defined, and the exponent in such aymptotic is estimated for some Gaussian processes, including the fractional Brownian motion (FBM) and the integrated FBM in,.
Abstract: 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 , .
20 citations
TL;DR: In this paper, the M-Wright function is used to model anomalous diffusion in complex media, which can be described as the result of a superposition mechanism reflecting the homogeneity and nonstationarity properties of the medium.
Abstract: 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 Erdelyi-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.
19 citations
TL;DR: In this article, the authors studied the survival probability in critical and subcritical cases in a population with two types of individuals, with each occupying an island, where individuals of type 1 live in a variable environment and type 2 dwell on the other island, in a constant environment.
Abstract: 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.
16 citations
TL;DR: In this paper, the authors proposed a new model to obtain asymptotic distributions near zero and compute the limiting distribution for the Ornstein-Uhlenbeck process driven by a fractional Brownian motion.
Abstract: 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.
14 citations
TL;DR: In this article, optimal control problems for one-dimensional diffusion processes in the interval (i.e., the interval) of the diffusion process are considered, where the aim is either to maximize or to minimize the time spent by the controlled processes.
Abstract: 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.
10 citations
TL;DR: In this article, the authors 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.
Abstract: 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.
9 citations
TL;DR: In this article, a general stochastic epidemic process with memory was proposed to model the spread of any rare transmissible disease with a random incubation period and a Reed-Frost type infection.
Abstract: 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.
TL;DR: In this paper, the optimal geometric mean return of a stock and its option is proved by using binomial option pricing model and continuous stochastic models with self-financing assumption.
Abstract: 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.
TL;DR: In this paper, the authors considered the general one-dimensional time-homogeneous regular diffusion process between two reflecting barriers and derived the differential equations with boundary conditions for the Laplace transform of the first passage time and the value function.
Abstract: We consider the general one-dimensional time-homogeneous regular diffusion process between two reflecting barriers. An approach based on the Ito 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.
TL;DR: In this article, a dependent hidden Markov model of credit quality is proposed, where the "true" credit quality was not observed directly but only through noisy observations given by posted credit ratings and the model is formulated in discrete time with a Markov chain observed in martingale noise.
Abstract: 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.
TL;DR: In this article, the conditional full support (CFS) property of a continuous process in an open connected domain is shown to imply the existence of a martingale M under an equivalent probability measure such that M lies in the ǫg0 neighborhood of the process under the supremum norm.
Abstract: 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 𝜖g0 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.
TL;DR: In this article, the authors consider the class of semi-Markov modulated jump diffusions (sMMJDs) whose operator turns out to be an integro-partial differential operator.
Abstract: 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.
TL;DR: In this article, the probabilistic sensitivity analysis of Integrated Force Method was further investigated for the assessment of uncertainty effect in system optimization and identification, and a set of stochastic sensitivity analysis formulation of integrated force method was developed using the perturbation method.
Abstract: 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.
TL;DR: In this article, the stability of the solutions of stochastic differential equations driven by fractional Brownian motions with Hurst parameter greater than half was studied and it was shown 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 solutions of the corresponding equations converge in H¨ older norm to the solution of a Stochastic differential equation.
Abstract: 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 H¨ older 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.
TL;DR: A rigorous study of the MAVAR log-regression estimator is presented, under the assumption that the signal process is a fractional Brownian motion, which proves that it is consistent and asymptotically normally distributed.
Abstract: 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.
TL;DR: In this paper, it was shown that solving a one-dimensional stochastic equation is equivalent to solving a 2-dimensional (1 − ε)-stochastic equation with measurable coefficients.
Abstract: We consider a one-dimensional stochastic equation 𝑑 𝑋 𝑡 = 𝑏 ( 𝑡 , 𝑋 𝑡 − ) 𝑑 𝑍 𝑡 + 𝑎 ( 𝑡 , 𝑋 𝑡 ) 𝑑 𝑡 , 𝑡 ≥ 0 , with respect to a symmetric stable process 𝑍 of index 0 𝛼 ≤ 2 . It is shown that solving this equation is equivalent to solving of a 2-dimensional stochastic equation 𝑑 𝐿 𝑡 = 𝐵 ( 𝐿 𝑡 − ) 𝑑 𝑊 𝑡 with respect to the semimartingale 𝑊 = ( 𝑍 , 𝑡 ) and corresponding matrix 𝐵 . In the case of 1 ≤ 𝛼 2 we provide new sufficient conditions for the existence of solutions of both equations with measurable coefficients. The
existence proofs are established using the method of Krylov's estimates
for processes satisfying the 2-dimensional equation. On another hand,
the Krylov's estimates are based on some analytical facts of independent
interest that are also proved in the paper.
TL;DR: In this paper, the authors introduced a generalization of the Feynman-Kac theorem in Hilbert spaces and proved connection between solutions to the abstract stochastic differential equation and solutions to deterministic partial differential equation for the probability characteristic.
Abstract: 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.
TL;DR: In this paper, the authors considered a retrial queueing system with two types of batch arrivals, called type I and type II customers, and the joint distribution of the number of customers in the priority queue and in the retrial group was obtained in a closed form.
Abstract: 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.
TL;DR: In this paper, the authors considered the best-choice problem with disorder and imperfect observation, and derived an optimal rule in the class of==================Bayes' strategies, where the decision-maker can choose at most one of the observations.
Abstract: We consider the best-choice problem with disorder
and imperfect observation. The decision-maker observes sequentially
a known number of i.i.d random variables from a known distribution
with the object of choosing the largest. At the random time the distribution
law of observations is changed. The random variables cannot
be perfectly observed. Each time a random variable is sampled the
decision-maker is informed only whether it is greater than or less than
some level specified by him. The decision-maker can choose at most
one of the observation. The optimal rule is derived in the class of
Bayes' strategies.
IBM1
TL;DR: In this article, the performance of a production system with correlated demand through diffusion approximation is investigated, and the key performance metric under consideration is the extreme points that this system can reach, which is mapped to a problem of characterizing the joint probability density of a two-dimensional Brownian motion and its coordinate running maximum.
Abstract: 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.