Showing papers in "Stochastic Analysis and Applications in 1998"
TL;DR: In this paper, the authors combine two ways for representing uncertainty through stochastic differential inclusions: a "stochastic uncertainty" driven by a Wiener process, and a "contingent uncertainty"driven by a set-valued map, as well as to consider stochastically control problems with continuous dynamir and state dependent controls.
Abstract: The aim of this paper is to combine two ways for representing uncertainty through stochastic differential inclusions:a “stochastic uncertainty”, driven by a Wiener process, and a “contingent uncertainty”, driven by a set-valued map, as well as to consider stochastic control problems with continuous dynamir and st.ate dependent controls. This paper is also devoted to viability of a dosed su hset under stochastic differential inclusions, characterized in terms of stochastic tangent sets to closed subsets
129 citations
TL;DR: In this article, a new local linearization method was proposed which approximates a nonlinear stochastic differential equation by a linear Stochastic equation using the maximum likelihood technique.
Abstract: This paper proposes a new local linearization method which approximates a nonlinear stochastic differential equation by a linear stochastic differential equation. Using this method, we can estimate parameters of the nonlinear stochastic differential equation from discrete observations by the maximum likelihood technique. We conduct the numerical experiments to evaluate the finite sample performance of identification of the new method, and compare it with the two known methods: the original local linearization method and the Euler methods. From the results of experiments, the new method shows much better performance than the other two methods particularly when the sampling interval is large
119 citations
TL;DR: In this paper, the authors considered the almost sure Lyapunov exponent of a semilinear stochastic heat equation with finite delays. And they presented the existence theorem, and the estimates of moment and almost sure exponent of the above equation.
Abstract: In this paper we consider the almost sure a symptotic behavior of mild solutions of the semilinear stochastic evolution equation with finite delays: where f, g have the Lipschitz condition and the linear growth condition. That is, we present the existence theorem, and the estimates of moment and almost sure Lyapunov exponent of the above equation. For illustrating the theorem we discuss a semilinear stochastic heat equation with finite delays
54 citations
TL;DR: In this paper, an interacting branching particle system is considered in which particles move in a random medium on at time t. In between branching times, the motion is governed by a singular, degenerate diffusion coefficient; each particle has mass 1/θn and branches at rate, where γ ≥ 0 and θ ≥ 2 are fixed constants.
Abstract: For any positive integer n, an interacting branching particle system is considered in which particles move in a random medium on at time t. In between branching times, the motion is governed by a singular, degenerate diffusion coefficient; each particle has mass 1/θnand branches at rate , where γ≥ 0 and θ ≥ 2 are fixed constants. As n →∞, the existence, uniqueness, Markovian property, and continuity of the limiting measure-valued processes are investigated
50 citations
TL;DR: In this article, a Kalman type system of integral equations was obtained for the linear filtering problem in which the noise generating the signal is a fractional Brownian motion with long-range dependence, and the error in applying the usual Kalman filter to this problem was determined explicitly for a simple example.
Abstract: A Kalman type system of integral equations is obtained for the linear filtering problem in which the noise generating the signal is a fractional Brownian motion with long-range dependence The error in applying the usual Kalman filter to this problem is determined explicitly for a simple example
33 citations
TL;DR: In this article, basic results on differentiability of probability functions are reviewed and systematically presented, and some basic theorems are informally proved and illustrated with examples, and the results can be used for sensitivity and uncertainty analysis of stochastic models.
Abstract: Basic results on differentiability of probability functions are reviewed and systematically presented. Strict mathematical statements are formulated, some basic theorems are informally proved and illustrated with examples. Probability function, from formal point of view, is an expectation of an indicator function or an integral over the domain depending upon the parameter. Gradient of probability function is represented in different forms (integral over the surface, volume, or sum of surface and volume integrals). These results can be used for sensitivity and uncertainty analysis of stochastic models, reliability analysis, Probabilistic Risk Analysis, optimization of the stochastic systems, and other applications involving uncertainties in parameters.
32 citations
TL;DR: In this paper, Girsanov's theorem for continuous orthogonal martingale measures is used to establish a one-to-one correspondence between solutions of two space-time SDEs differing only by a drift coefficient.
Abstract: We prove Girsanov's theorem for continuous orthogonal martingale measures. We then define space-time SDEs, and use Girsanov's theorem to establish a oneto- one correspondence between solutions of two space-time SDEs differing only by a drift coefficient. For such stochastic equations, we give necessary conditions under which the laws of their solutions are absolutely continuous with respect to each other. Using Girsanov's theorem again, we prove additional existence and uniqueness results for space-time SDEs. The same one-to-one correspondence and absolute continuity theorems are also proved for the stochastic heat and wave equations
31 citations
TL;DR: In this paper, exact solutions to an unsolved class of jump-diffusion stochastic differential equations and derive efficient numerical schemes for the general non-linear cases are presented, and the generator and Taylor expansion in the expectation semi-group are verified under weaker conditions.
Abstract: This paper presents exact solutions to an unsolved class of jump-diffusion stochastic differential equations and derives efficient numerical schemes for the general non-linear cases. It is proved that even the second order mean square efficient schemes may not be second order efficient in the weak sense. The generator and Taylor expansion in the expectationssemi-group are verified under weaker conditions and applied to derive new doubly efficient schemes which are proved to converge with the best possible order rate in both senses. A class of direct jump-adapted schemes are als o presented. Comparative simulations are consistent with the findings.
28 citations
TL;DR: In this article, the authors prove hypercontractivity of nonsymmetric Ornstein-Uhlenbeck semigroups in Hilbert spaces, using direct probabilistic arguments, and imply exponential convergence at infinity for the semigroup.
Abstract: We prove hypercontractivity of nonsymmetric Ornstein-Uhlenbeck semigroups in Hilbert spaces, using direct probabilistic arguments. Our results imply exponential convergence at infinity for the semigroup. We show by examples that in our setting logarithmic Sobolev inequalities do not hold in general
25 citations
TL;DR: In this paper, the authors define a set-valued stochastic process and an integral integral of a such process with respect to a semimartingale and prove some results on the existence of its solutions.
Abstract: We define a set-valued stochastic process and an integral of a such process with respect to a semimartingale. Next we consider a stochastic integral inclusion. Using fixed point methods we prove some results on the existence of its solutions
21 citations
TL;DR: In this paper, an input-output linear time-varying differential system with homogeneous jump Markov parameters and mean square exponential stable evolution is considered, and it is proved that a parametrized by γ differential Riccati type system has a unique global bounded and stablizing solution.
Abstract: An input-output linear time-varying differential system with homogeneous jump Markov parameters and mean square exponential stable evolution is considered. We define a family T(t), t ≥ 0 of linear bounded input-output operators. It is proved that if sup ‖T(t)‖<γ then a parametrized by γ differential Riccati type system has a unique global bounded and stablizing solution. An application to the estimate of a stability radius is given
TL;DR: A class of explicit Runge-Kutta methods for numerical solution of stochastic differential equations is described, analyzed, and numerically tested and it is shown that this class is of second-order accuracy in the weak sense.
Abstract: A class of explicit Runge-Kutta methods for numerical solution of stochastic differential equations is described, analyzed, and numerically tested. It is shown that this class is of second-order accuracy in the weak sense. Also, a varaince reduction technique is presented that reduces the stochastic error involved when computing expectations. Numerical examples are presented to support the theoretical results.
TL;DR: In this article, shape-preserving wavelet approximators of cumulative distribution functions and densities with a priori prescribed smoothness properties are introduced and evaluated for a variety of loss functions and analyzed their asymptotic behavior.
Abstract: In a previous paper we introduced a general class of shapepreserving wavelet approximating operators (approximators) which transform cumulative distribution functions (cdf) and densities into functions of the same type. Empirical versions of these operators are used in this paper to introduce, in an unified way, shape- preserving wavelet estimators of cdf and densities, with a priori prescribed smoothness properties. We evaluate their risk for a variety of loss functions and analyze their asymptotic behavior. We study the convergence rates depending on minimal additional assumptions about the cdf/ density. These assumptions are in terms of the function belonging to certain homogeneous Besov or Triebel- Lizorkin spaces and others. As a main evaluation tool the integral p-modulus of smoothness is used
TL;DR: In this article, the Girsanow and stochastic Feynman-Kac formulae are treated in a unified way, using a suitable generalization of a bar.
Abstract: By using a suitable generalization of a bar.kwa.rd Ito formula, we treat in a. unified way Girsanow and stochastic Feynman-Kac formulae.
Journal Article•
TL;DR: In this article, an input-output linear time-varying discrete system with statedependent noise and mean square exponential stable evolution is considered, and it is proved that if the norm of the input output operator is less than γ then a corresponding parametrized by γ Riccati equation has a unique global bounded and stabilizing solution.
Abstract: An input-output linear time-varying discrete system with statedependent noise and mean square exponential stable evolution is considered. It is proved that if the norm of the input-output operator is less than γ then a corresponding parametrized by γ Riccati equation has a unique global bounded and stabilizing solution. An application to the estimate of a stability radius is given
TL;DR: In this article, the authors studied two-player stochastic differential games with multiple modes and established the existence of optimal strategies for both players for zero-sum and non-zero-sum games.
Abstract: We have studied two person stochastic differential games with multiple modes. For the zero-sum game we have established the existence of optimal strategies for both players. For the non zerosum case we have proved the existence of a Nash equilibrium
TL;DR: In this article, a (C o)-group generated by the Levy Laplacian acting on a domain in the space of Hida distributions is discussed and a relationship between the analytic extension of this group and the adjoint operator of an infinite dimensional Fourier-Mehler transform introduced by H.-H. Kuo is given.
Abstract: In this paper, we discuss a (C o)-group generated by the Levy Laplacian acting on a domain in the space of Hida distributions. We also give a relationship between the analytic extension of this (C o)-group and the adjoint operator of an infinite dimensional Fourier-Mehler transform introduced by H.- H. Kuo
TL;DR: In this paper, it was shown that the uniform mean-square ergodic theorem holds for the family of wide sense stationary sequences, as soon as the random process with orthogonal increments, which corresponds to the orthogon stochastic measure generated by means of the spectral representation theorem, is of bounded variation and uniformly continuous at zero in a mean square sense.
Abstract: It is shown that the uniform mean-square ergodic theorem holds for the family of wide sense stationary sequences, as soon as the random process with orthogonal increments, which corresponds to the orthogonal stochastic measure generated by means of the spectral representation theorem, is of bounded variation and uniformly continuous at zero in a mean-square sense. The converse statement is also shown to be valid, whenever the process is sufficiently rich. The method of proof relies upon the spectral representation theorem, integration by parts formula, and estimation of the asymptotic behaviour of total variation of the underlying trigonometric functions. The result extends and generalizes to provide the uniform mean-square ergodic theorem for families of wide sense stationary processes
TL;DR: In this article, the rate of convergence for an almost surely convergent series of random variables (Xn, n ≥ 1) is studied and conditions for each of these conditions to hold for given numerical sequences are provided.
Abstract: The rate of convergence for an almost surely convergent series of random variables {Xn, n ≥ 1} is studied in this paper. More specifically, if Sn converges almost surely to a random variable S then the tail series is a well-defined sequence of random variable with Tn → 0 almost surely. The main result provides conditions for each of to hold for given numerical sequences . As special cases, new results are obtained when {Xn, n ≥ 1} is a martingale difference sequence and when the Xn 's are of the form , where {an, n ≥ 1} is a sequence of constants and {Yn, n ≥ 1} is an orthogonal system of exchangeable random variables. An application to the Polya urn scheme is considered. The current work generalizes and simplifies some of the recent results of Nam and Rosalsky [15]
TL;DR: In this article, a stabilizing feedback law for nonlinear control systems corrupted by noise is provided, which extends Jurdjevic-Quinn's theorem to control stochastic differential equations.
Abstract: In this paper, we provide a formula for a stabilizing feedback law for control stochastic differential equations. This result extends Jurdjevic-Quinn's theorem to nonlinear control systems corrupted by noise
TL;DR: In this article, the authors considered the discrete-time LQ-optimal control problem for the class of linear systems with Markovian jump parameters and additive l 2-stochastic input.
Abstract: In this paper we consider the discrete-time LQ-optimal control problem for the class of linear systems with Markovian jump parameters and additive l 2-stochastic input. The state-space of the Markov chain is assumed to be a countably infinite set. The controller has access to both the state-variable and jump-variable. It is shown that the optimal control law is characterized by a feedback term plus a term defined by the:l 2-stochastic input and Markov chain. An application to the optimal control of a failure prone manufacturing system subject to a random demand for a single type of item is presented.
TL;DR: In this article, the authors gave an Lp stability result for the random matrix solution Zn of where ⊘n is an unbounded regressor having a Markovian representation.
Abstract: We give an Lp stability result for the random matrix solution Zn of where ⊘n is an unbounded regressor having a Markovian representation
TL;DR: In this paper, the authors considered an infinite server queue in continuous time in which arrivals are in batches of variable size X and service is provided in groups of fixed size R and obtained analytical results for the number of busy servers and waiting customers at arbitrary time points.
Abstract: This paper considers an infinite server queue in continuous time in which arrivals are in batches of variable size X and service is provided in groups of fixed size R. We obtain analytical results for the number of busy servers and waiting customers at arbitrary time points. For the number of busy servers, we obtain a recursive relation for the partial binomial moments both in transient and steady states. Special cases are also discussed
TL;DR: In this article, exact constants are given for the rate of convergence of the solution x n of the nonlinear difference equation towards the root θ of f under very general conditions on the steplengths an, on the weighting factors bn and on the random noise vn.
Abstract: In this paper exact constants are given for the rate of convergence of the solution x n of the nonlinear difference equation towards the root θ of f under very general conditions on the steplengths an , on the weighting factors bn and on the random noise vn Moreover, an almost sure representation is developed for the deviation xn – θ by weighted sums of (independent) random variables
TL;DR: In this article, the explicit structure of an optimal sequential allocation policy is obtained under pertinent reward/loss functions, when the experiments are characterized by random variables with distributions from the one parameter exponential family.
Abstract: The unknown performance of a new experiment is to be evaluated and compared with that of an existing one over a finite horizon. The explicit structure of an optimal sequential allocation policy is obtained under pertinent reward/loss functions, when the experiments are characterized by random variables with distributions from the one parameter exponential family.
TL;DR: In this article, a state process is described by either a discrete time Hilbert space valued process, or a stochastic differential equation in Hilbert space, and the state is observed through a finite dimensional process.
Abstract: A state process is described by either a discrete time Hilbert space valued process, or a stochastic differential equation in Hilbert space. The state is observed through a finite dimensional process. Using a change of measure and a Fusive theorem the Zakai equation is obtained in discrete or continuous time. A risk sensitive state estimate is also defined
TL;DR: In this paper, a sufficient condition for the stability property of stochastic inclusion of Ito type is given, and a necessary condition for ito type stability property is given.
Abstract: Let processes X and Y be two arbitrary solutions of stochastic inclusion of Ito type A sufficient condition for the stability property is given
TL;DR: In this paper, the essential self-adjointness of Dirichlet operators with identity diffusion part on M-type 2 Banach spaces via the classical notion of semi inner product was established.
Abstract: In this paper, we study Diriehlet operators on certain smooth Banach spaces. We establish the well-known Kato's inequality in our general infinite dimensional setting. By applying this,we show the essential self-adjointness of Dirichlet operators with non-constant diffusion part on certain smooth Banach spaces. We also provide an approximation criterion for essential self-adjointness of Dirichlet operators with identity diffusion part on M-type 2 Banach spaces via the classical notion of semi inner-product.