Showing papers in "Stochastic Analysis and Applications in 1988"
TL;DR: In this article, expressions in vector notation are given for the central moments, non-central moments, and the cumulants of arbitrary order of the multivariate normal distribution of the normal distribution.
Abstract: Expressions in vector notation are given for the central moments, the non–central moments and the cumulants of arbitrary order of the multivariate normal distribution
56 citations
TL;DR: Variance-reducing estimators for functionals of the solution of the general Ito stochastic differential equation are derived in this article, which allow to apply variance reduction techniques known from the Monte Carlo theory.
Abstract: Variance–reducing estimators are derived for functionals of the solution of the general Ito stochastic differential equation. These estimators allow to apply variance reduction techniques known from the Monte Carlo theory. In particular, variance–reducing Euler estimators are constructed as well as variance–reducing unbiased estimators. Numerical examples are given. They show that the variance reduction techniques cause an enormous gain in efficiency, reducing the statistical error up to 50 times. They also demonstrate the effect of the unbiased estimators, which allow to evaluate the functionals without reducing the time step.
43 citations
TL;DR: In this article, the authors characterize m-exchangeability for a large class of martingale problems on the usual space of continuous trajectories into (Rd)m, in terms of the drift and diffusion coefficients.
Abstract: We characterize m-exchangeability for a large class of martingale problems on the usual space of continuous trajectories into (Rd)m, in terms of the drift and diffusion coefficients. Under some Lipschitz conditions on the coefficients, the sequence of empirical measures associated with a triangular array of exchangeable diffusions is shown to be tight. The diffusions considered here may be very strongly correlated, in which case tightness entails the existence of “random McKean–Vlasov limits”.
30 citations
TL;DR: The concept of white noise in space and time arising in the context of stochastic partial differential equations is related to Wiener processes with values in Hilbert spaces of distributions.
Abstract: The concept of white noise in space and time arising in the context of stochastic partial differential equations is related to Wiener processes with values in Hilbert spaces of distributions
23 citations
TL;DR: In this article, a class of second order stochastic PDEs with second order drift terms was studied by the stochastically characteristics method, as developped by Kunita for the first order PDE, which was transformed in a family of deterministic parabolic problems.
Abstract: We study a class of second order (in the drift term) stochastic partial differential equations by the stochastic characteristics method, as developped by Kunita for the first order stochastic partial differential equations. With this method the original problem is transformed in a family of deterministic parabolic problems.
20 citations
TL;DR: In this article, a criterion for an infinite dimensional Gaussian process to satisfy a Langevin equation is extended to the case where the evolution term is time dependent, and the Langevin equations are extended to Gaussian processes with time dependent evolution terms.
Abstract: A criterion for an infinite dimensional Gaussian process to satisfy a Langevin equation i s extended to the case where the evolution term is time dependent
18 citations
TL;DR: In this paper, the authors apply the methods of probabilistic potential theory and Dirichlet forms to the study of a class of infinite dimensional diffusions with independent coordinates, including the Ornstein-Uhlenbeck process considered by a number of authors.
Abstract: We apply the methods of probabilistic potential theory and Dirichlet forms to the study of a class of infinite dimensional diffusions with independent coordinates. This class includes the Ornstein-Uhlenbeck process considered by a number of authors. Using the properties of their Dirichlet forms, we obtain criteria for the 12 sample path continuity of the processes under consideration, in terms of their invariant measures and diffusion coefficients. As well, we consider the behaviour along sample paths of the sequence of coordinates
13 citations
TL;DR: In this article, the authors considered a tandem queue model with a single server who can switch instantaneously from one queue to another, where customers arrive according to a Poisson process with rate λ, and the amount of service required by each customer at the ith queue is an exponentially distributed random variable with rate μi.
Abstract: Consider a tandem queue model with a single server who can switch instantaneously from one queue to another. Customers arrive according to a Poisson process with rate λ . The amount of service required by each customer at the ith queue is an exponentially distributed random variable with rate μi. Whenever two or more customers are in the system, the decision as to which customer should be served first depends on the optimzation criterion. In this system all server allocation policies in the finite set of work conserving deterministic policies have the same expected first passage times (makespan) to empty the system of customers from any initial state. However, a unique policy maximizes the first passage probability of empty-ing the system before the number of customers exceeds K, for any value of K, and it stochastically minimizes (he number of customers in the system at any time t > 0 . This policy always assigns the server to the non empty queue closest to the exit
12 citations
TL;DR: In this paper, an account of orthogonal and general series as well as moving average representations of weakly harmonizable processes is presented, together with an operator representation of the spectral functions of these processes.
Abstract: This paper contains an account of orthogonal and general series as well as moving average representations of weakly harmonizable processes. Also oscillatory and periodically correlated second order processes, their relation to harmonizable classes and an operator representation are presented. Fur-ther, supports of the spectral functions of weakly harmonizable processes are considered
6 citations
TL;DR: In this paper, an integral representation of disjointly additive order preserving nonexpansive operators in L 1, using a family {Pt,t ∊ R} of substochastic kernels, was given.
Abstract: We give an integral representation of disjointly additive order preserving nonexpansive operators in L 1, using a family {Pt ,t ∊ R} of substochastic kernels. We also characterize the case when the kernels correspond to positive linear operators Tt in L 1 of norm ≤ 1. For we then obtain where Xtf is the indicator function of {f > t}
5 citations
TL;DR: The mathematical theory of stochastic Volterra-Hamilton systems is used to model cost of terpene production of colonizing soft corals and their highly toxic effects on reef-building corals of the Great Barrier Reef.
Abstract: The mathematical theory of stochastic Volterra-Hamilton systems is used to model cost of terpene production of colonizing soft corals and their highly toxic effects on reef–building corals of the Great Barrier Reef based on recent biological findings of Sammarco, Coll and others. The central question asked is what strategies would enable them to survive soft coral colonization. The model predicts that any factor which reduces soft coral encroachment and direct toxic effects has survival value. Reasoning is based on exact solution of a forward Cauchy problem using Kunita backward calculus and involves the Riemannian sealer curvature of phase space explicitly
TL;DR: In this paper, the transient state probabilities of number of breakdowns and initial busy period distributions of a given number of operatives have been obtained in closed form using Laplace transform and matrix theory.
Abstract: A multi–operative Markovian machine interference model with arbitrary inital number of breakdowns is studied by taking mean breakdown of each machine as λ-1 and the mean repairtime of each machine as μ-1. The transient state probabilities of number of breakdowns and initial busy period distributions of a given number of operatives have been obtained in closed form using Laplace transform and matrix theory. The efficacy of the method is tested by computing the state probabilities and measures of effectiveness such as machine availability function and operating efficiency function for various values of traffic intensity ρ=λ/μ.
TL;DR: In this paper, exact solutions for steadystate distributions of population levels of species in an n-dimensional Lotka-Volterra food chain with stochastic fluctuations at the lowest level are obtained.
Abstract: In this note, exact solutions for steadystate distributions of population levels of species in an n–dimensional Lotka–Volterra food chain with stochastic fluctuations at the lowest level are obtained. The bounds on the amount of stochasticity that can be introduced are shown and extensions to other non-linear structures are made
TL;DR: In this article, it was shown that the assumptions between basic Theorem 1 and 2 and main Theorem 3 and 4, in Chang [1] are equivalent, and several common fixed point theorems which are proved under considerably weaker conditions.
Abstract: In this paper we first prove that the assumptions between basic Theorem 1 and 2, hence main Theorem 3 and 4, in Chang [ 1 ] are equivalent. We then give several common fixed point theorems which are proved under considerably weaker conditions. We finally point out three mistakes in Chang [ 1 ] and correct them. The method used in this paper defers from the method used by Chang [ 1 ]
TL;DR: In this article, singularly perturbed Markov processes are studied and new results in convergence of resolvents, weak convergence, and convergence of semigroups are obtained.
Abstract: In, the paper we study singularly perturbed Markov processes. Some new results in convergence of resolvents, weak convergence and convergence of semigroups are obtained. The results are applied to optima1 stopping and impulse control of singular1y perturbed Markov processes. Examples justifying the assumptions made in text are also presented
TL;DR: In this paper, the authors give procedures for testing hypotheses and estimating the common mean direction of several von Mises populations with differing dispersions, and an exact distribution is given for the maximum likelihood estimator of the Common Mean Direction conditioned on a weighted vector length statistic.
Abstract: We give procedures for testing hypotheses and estimating the common mean direction of several von Mises populations with differing dispersions. An exact distribution is given for the maximum likelihood estimator of the common mean direction conditioned on a weighted vector length statistic. Based on this distribution confidence limits of the common mean direction and tests about the same are derived.
TL;DR: In this paper, a single item inventory model with probabilistic demand is considered, where the replenishment of items is cyclical and occurs every n-periods, and the order for replenishment can be placed at the end of any period during the cycle.
Abstract: A single-item inventory model with probabilistic demand is considered in this paper. The replenishment of items is cyclical and occurs every n-periods. The order for replenishment can be p laced at the end of any period during the cycle . The uni ts ordered fluctuate based on actual and expected demand during the cycle. The general expression of this inventory system is formulated by developing the stationary distribution of the inventory position, units ordered and the probability of placing an order during any period. The expression is stated in terms of the reorder point (s) and the order level (S).
TL;DR: In this article, an exhaustive study of Markov processes defined by forward and backward passages along finitely many directed weighted circuits is given together with a connection with the Kirchhoff' s principle for electrical networks.
Abstract: An exhaustive study of Markov processes defined by forward and backward passages along finitely many directed weighted circuits is given together with a connection with the Kirchhoff' s principle for electrical networks.
TL;DR: In this article, the construction of the minimal and maximal solutions of 1-dimensional stochastic differential equation of Ito's type under mild conditions for coefficients is studied, and the usual monotone iterative technique is used.
Abstract: This paper is concerned with the construction of the minimal and maximal solutions of 1–dimensional stochastic differential equation of Ito's type under rather mild conditions for coefficients. To this, usual monotone iterative technique is used.
TL;DR: In this paper, the mathematical expectation of the number of real roots of the equation where K is any non zero real constant, permitted to be a function of degree n was studied.
Abstract: Let be a sequence of normally distributed independent random variables with mathematical expectation μ≠0, and variance unity. Here for n→∞ we find the asymptotic relation for the mathematical expectation of the number of real roots of the equation where K is any non zero real constant, permitted to be a function of degree n. It is shown that the results are valid even for K→∞ as long as
TL;DR: In this paper, a least squares polynomial regression curve was derived for simultaneously collected starfish/coral state data to set up the prediction preliminaries and the results showed that a more refined mesh for the Mihlstein approximations used to evaluate the Ito stochastic integrals involved in Zakai theory would improve the accuracy of predictions.
Abstract: The Zakai form of nonlinear prediction theory is used to estimate year-to-year state changes in crown-of-thorns starfish populations of the Great Barrier Reef. Taking the previously defined coral-state diffusion as observation process and the starfish-state diffusion as signal process, a least squares polynomial regression curve recently derived for simultaneously collected starfish/coral state data is used to set up the prediction preliminaries. Numerical results are not inconsistant with starfish outbreak values of the last 5 years and yield an expected high value for 1987. However, because of large error in the data it is doubtful that a more refined mesh for the Mihlstein approximations used to evaluate the Ito stochastic integrals involved in Zakai theory would improve the accuracy of predictions.
TL;DR: In this article, the authors studied the near-optimum regulators for multi-time scale singularly perturbed systems by using a slow and fast mode decomposition which provides three lower-order control problems in different time scales.
Abstract: The objective of this paper is to study the near–optimum regulators for multi–time scale singularly perturbed systems. This is accomplished by using a slow and fast mode decomposition which provides three lower order control problems in different time scales. The near–optimal control for the overall control system is obtained in terms of the optimal control laws for three lower order control problems. Furthermore, the near–optimal performance index of the original control problem is obtained in terms of the lower order control problems in separate time scales
TL;DR: For sequences of independent random variables conditions are presented in this paper in which a generalized law of the iterated logarithm cannot hold and the limiting behavior of the normed sequence of partial sums is almost surely established.
Abstract: For sequences of independent random variables conditions are presented in which a generalized law of the iterated logarithm cannot hold. Moreover, the limiting behavior of the normed sequence of partial sums is almost surely established. In addition, a strong law of large numbers is presented where neither the assumptions of independence nor common distribution are required.
TL;DR: In this paper, a long-division technique for finding the similarity transformation matrix and transforming the estimated left MFD to the right MFD is developed; the derivation is given in detail, and the procedures involved are briefly characterized.
Abstract: The design of self-tuning controllers for multivariable stochastic systems is considered analytically A long-division technique for finding the similarity transformation matrix and transforming the estimated left MFD to the right MFD is developed; the derivation is given in detail, and the procedures involved are briefly characterized
TL;DR: In this paper, the authors deal with the problem of convergence of solutions of parabolic stochastic ito equations, and apply Hilbert space methods to an optimal control problem, which is given an application to an optimization problem.
Abstract: The paper deals with a problem of convergence of solutions of parabolic stochastic ito equations. There are applied Hilbert space methods. There is given an application to an optimal control problem.
TL;DR: In this article, the authors consider three σ-algebras that contain information about what happens to a stochastic process prior to a random time T and show that they coincide in a canonical setting but not in general.
Abstract: This note considers three σ-algebras that contain information about what happens to a stochastic process prior to a random time T. It is shown that the three σ-algebras coincide in a canonical setting but not in general.