Absolute continuity under flows generated by SDE with measurable drift coefficients

We consider the Itô SDE with a non-degenerate diffusion coefficient and a measurable drift coefficient. Under the condition that the gradient of the diffusion coefficient and the divergences of the diffusion and drift coefficients are exponentially integrable with respect to the Gaussian measure, we show that the stochastic flow leaves the reference measure absolutely continuous.     

Large deviations for diffusion processes in duals of nuclear spaces

Motivated by applications to neurophysiological problems, various authors have studied diffusion processes in duals of countably Hilbertian nuclear spaces governed by stochastic differential equations. In these models the diffusion coefficients describe the random stimuli received by spatially extended neurons. In this paper we present a large deviation principle for such processes when the diffusion terms tend to zero in terms of a small parameter. The lower bounds are established by making use of the Girsanov formula in abstract Wiener space. The upper bounds are obtained by Gaussian approximation of the diffusion processes and by taking advantage of the nuclear structure of the state space to pass from compact sets to closed sets.This research was partially supported by the National Science Foundation and the Air Force Office of Scientific Research Grant No. F49620-92-J-0154 and the Army Research Office Grant No. DAAL03-92-G-0008.     

Fast-slow stochastic dynamical system with singular coefficients

This paper aims to study the asymptotic behavior of a fast-slow stochastic dynamical system with singular coefficients, where the fast motion is given by a continuous diffusion process while the slow component is driven by an α-stable noise with α ∈ [1, 2). Using Zvonkin’s transformation and the technique of the Poisson equation, we have that both the strong and weak convergences in the averaging principle are established, which can be viewed as a functional law of large numbers. Then we study t...     

Optimal Control Approach to Nonlinear Diffusion Equations Driven by Wiener Noise

The stochastic nonlinear infinite-dimensional equations of gradient type and with additive Wiener noise can be reduced to an optimal convex control problem via Brezis–Ekeland duality device. This approach is illustrated here on a few classes of nonlinear stochastic parabolic equations which are relevant as diffusion models under stochastic Gaussian perturbations, and image restoring technique.     

Simulation accuracy of stationary Gaussian stochastic processes inL 2(0,T)

Models of stationary Gaussian stochastic processes with discrete and continuous spectra are constructed. Simulation of stationary Gaussian processes with a continuous spectrum is considered for the following cases: when the covariance function of the stochastic process is expandable in a Fourier series with positive coefficients; when the spectrum of the stationary Gaussian stochastic process is concentrated on the interval [0, ]; and in the general case. The stationary Gaussian process is simulated with prescribed reliability and accuracy in L²(0, T).Kiev University. Translated from Vychislitel'naya i Prikladnaya Matematika, No. 75, pp. 108–115, 1991.     

Advanced MCMC methods for sampling on diffusion pathspace

The need to calibrate increasingly complex statistical models requires a persistent effort for further advances on available, computationally intensive Monte-Carlo methods. We study here an advanced version of familiar Markov-chain Monte-Carlo (MCMC) algorithms that sample from target distributions defined as change of measures from Gaussian laws on general Hilbert spaces. Such a model structure arises in several contexts: we focus here at the important class of statistical models driven by diffusion paths whence the Wiener process constitutes the reference Gaussian law. Particular emphasis is given on advanced Hybrid Monte-Carlo (HMC) which makes large, derivative-driven steps in the state space (in contrast with local-move Random-walk-type algorithms) with analytical and experimental results. We illustrate its computational advantages in various diffusion processes and observation regimes; examples include stochastic volatility and latent survival models. In contrast with their standard MCMC counterparts, the advanced versions have mesh-free mixing times, as these will not deteriorate upon refinement of the approximation of the inherently infinite-dimensional diffusion paths by finite-dimensional ones used in practice when applying the algorithms on a computer.     

Convergence of diffusion processes

For stochastic diffusion equations with coefficients depending on a parameter, necessary and sufficient conditions of the weak convergence of solutions to the solution of a stochastic diffusion equation are obtained.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 44, No. 2, pp. 284–289, February, 1992.     

Mixing “In the sense of Ibragimov.” Estimate for the rate of approach of a family of integral functionals of a solution of a differential equation with periodic coefficients to a family of wiener processes. Some applications. I

We prove that a bounded 1-periodic function of a solution of a time-homogeneous diffusion equation with 1-periodic coefficients forms a process that satisfies the condition of uniform strong mixing. We obtain an estimate for the rate of approach of a certain normalized integral functional of a solution of an ordinary time-homogeneous stochastic differential equation with 1-periodic coefficients to a family of Wiener processes in probability in the metric of space C [0, T]. As an example, we consider an ordinary differential equation perturbed by a rapidly oscillating centered process that is a 1-periodic function of a solution of a time-homogeneous stochastic differential equation with 1-periodic coefficients. We obtain an estimate for the rate of approach of a solution of this equation to a solution of the corresponding It? stochastic equation.     

Multiple stochastic integrals constructed by special expansions of products of the integrating stochastic processes

The paper deals with problems of constructing multiple stochastic integrals in the case when the product of increments of the integrating stochastic process admits an expansion as a finite sum of series with random coefficients. This expansion was obtained for a sufficiently wide class including centered Gaussian processes. In the paper, some necessary and sufficient conditions are obtained for the existence of multiple stochastic integrals defined by an expansion of the product of Wiener processes. It was obtained a recurrent representation for the Wiener stochastic integral as an analog of the Hu–Meyer formula.     

Modeling of non-stationary ground motion using the mean reverting stochastic process

This paper presents a new method for modeling amplitude and frequency non-stationary earthquake ground motions using a scalar first order dynamic mean reverting stochastic differential equation driven by Brownian motion with parametric time varying coefficients. It determines the proper relationship between these time varying parametric coefficients and presents the statistical and probability distribution characteristics of the response solution. It demonstrates the applicability of the method by presenting some simulations of amplitude and frequency non-stationary earthquake ground motions. The verification of the amplitude and frequency non-stationary contents of the mean reverting stochastic ground motions is demonstrated using the Hilbert–Huang transform method. Also a corresponding interpretation between the coefficients of the proposed model and the coefficients of the usual oscillatory second order differential equation driven by white Gaussian noise is presented along with some comments how it can be applied to simulate ground motions consistent with acceleration target records such as boxcar, trapezoidal, other exponential functions, or compound and target records at source, near field, and far field distances.     

Modelling of stochastic heat transfer in a solid

The unsteady partial differential equations for expectation and correlation distributions of the stochastic temperature distribution in a solid are obtained, when the coefficients and the source term in the stochastic heat transfer equations are white Gaussian processes. Some solutions of the unsteady partial differential equations for expectation and correlation distributions of stochastic heat transfer are presented.     

An infinite horizon stochastic maximum principle for discounted control problem with Lipschitz coefficients

In the present work, a stochastic maximum principle for discounted control of a certain class of degenerate diffusion processes with global Lipschitz coefficient is investigated. The value function is given by a discounted performance functional, leading to a stochastic maximum principle of semi-couple forward–backward stochastic differential equation with non-smooth coefficients. The proof is based on the approximation of the Lipschitz coefficients by smooth ones and the approximation of the infinite horizon adjoint process.     

Averaging of randomly perturbed evolutionary equations

Evolutionary equations with coefficients perturbed by diffusion processes are considered. It is proved that the solutions of these equations converge weakly in distribution, as a small parameter tends to zero, to a unique solution of a martingale problem that corresponds to an evolutionary stochastic equation in the case where the powers of a small parameter are inconsistent.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 45, No. 7, pp. 963–971, July, 1993.     

Almost sure exponential behaviour for a parabolic SPDE on a manifold

We derive an upper bound on the large-time exponential behavior of the solution to a stochastic partial differential equation on a compact manifold with multiplicative noise potential. The potential is a random field that is white-noise in time, and Hölder-continuous in space. The stochastic PDE is interpreted in its evolution (semigroup) sense. A Feynman–Kac formula is derived for the solution, which is an expectation of an exponential functional of Brownian paths on the manifold. The main analytic technique is to discretize the Brownian paths, replacing them by piecewise-constant paths. The error committed by this replacement is controlled using Gaussian regularity estimates; these are also invoked to calculate the exponential rate of increase for the discretized Feynman–Kac formula. The error is proved to be negligible if the diffusion coefficient in the stochastic PDE is small enough. The main result extends a bound of Carmona and Viens (Stochast. Stochast. Rep. 62 (3–4) (1998) 251) beyond flat space to the case of a manifold.     

Approximation for non-smooth functionals of stochastic differential equations with irregular drift

This paper aims at developing a systematic study for the weak rate of convergence of the Euler–Maruyama scheme for stochastic differential equations with very irregular drift and constant diffusion coefficients. We apply our method to obtain the rates of approximation for the expectation of various non-smooth functionals of both stochastic differential equations and killed diffusion. We also apply our method to the study of the weak approximation of reflected stochastic differential equations whose drift is Hölder continuous.     

Gaussian skewness approximation for dynamic rate multi-server queues with abandonment

The multi-server queue with non-homogeneous Poisson arrivals and customer abandonment is a fundamental dynamic rate queueing model for large scale service systems such as call centers and hospitals. Scaling the arrival rates and number of servers arises naturally when a manager updates a staffing schedule in response to a forecast of increased customer demand. Mathematically, this type of scaling ultimately gives us the fluid and diffusion limits as found in Mandelbaum et al., Queueing Syst 30:149–201 (1998) for Markovian service networks. The asymptotics used here reduce to the Halfin and Whitt, Oper Res 29:567–588 (1981) scaling for multi-server queues. The diffusion limit suggests a Gaussian approximation to the stochastic behavior of this queueing process. The mean and variance are easily computed from a two-dimensional dynamical system for the fluid and diffusion limiting processes. Recent work by Ko and Gautam, INFORMS J Comput, to appear (2012) found that a modified version of these differential equations yield better Gaussian estimates of the original queueing system distribution. In this paper, we introduce a new three-dimensional dynamical system that is based on estimating the mean, variance, and third cumulant moment. This improves on the previous approaches by fitting the distribution from a quadratic function of a Gaussian random variable.     

On the Stochastic Maximum Principle in Optimal Control of Degenerate Diffusions with Lipschitz Coefficients

We establish a stochastic maximum principle in optimal control of a general class of degenerate diffusion processes with global Lipschitz coefficients, generalizing the existing results on stochastic control of diffusion processes. We use distributional derivatives of the coefficients and the Bouleau Hirsh flow property, in order to define the adjoint process on an extension of the initial probability space. This work is partially supported by MENA Swedish Algerian Research Partnership Program (348-2002-6874) and by French Algerian Cooperation, Accord Programme Tassili, 07 MDU 0705.     

Theorems of comparison and stability with probability 1 for one-dimensional stochastic differential equations

We prove the comparison theorems for scalar stochastic differential equations in the case of different diffusion coefficients. Conditions are given of stability with probability 1 with respect to the trivial solution to stochastic differential equations with random coefficients. The results remain valid for deterministic analogs of stochastic differential equations with symmetric integrals.     

Wong–Zakai approximation and support theorem for SPDEs with locally monotone coefficients

In this paper we present the Wong–Zakai approximation results for a class of nonlinear SPDEs with locally monotone coefficients and driven by multiplicative Wiener noise. This model extends the classical monotone one and includes examples like stochastic 2d Navier–Stokes equations, stochastic porous medium equations, stochastic p-Laplace equations and stochastic reaction–diffusion equations. As a corollary, our approximation results also describe the support of the distribution of solutions.     

A property of stable systems of linear stochastic equations

The relationship between exponential mean square stability of systems of linear ordinary differential equations with Gaussian coefficients and the same stability of the corresponding linear stochastic Ito differential equations is obtained.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 42, No. 2, pp. 147–152, February, 1990.