Measures For Stochastic Differential Equations Research Articles

Stationary measures for stochastic differential equations with degenerate damping

Stationary measures for stochastic differential equations with degenerate damping

Existence of geometric ergodic periodic measures of stochastic differential equations

Periodic measures are the time-periodic counterpart to invariant measures for dynamical systems and can be used to characterise the long-term periodic behaviour of stochastic systems. This paper gives sufficient conditions for the existence, uniqueness and geometric convergence of a periodic measure for time-periodic Markovian processes on a locally compact metric space in great generality. In particular, we apply these results in the context of time-periodic weakly dissipative stochastic differential equations, gradient stochastic differential equations as well as Langevin equations. We will establish the Fokker-Planck equation that the density of the periodic measure sufficiently and necessarily satisfies. Applications to physical problems shall be discussed with specific examples.

The backward Euler-Maruyama method for invariant measures of stochastic differential equations with super-linear coefficients

The backward Euler-Maruyama (BEM) method is employed to approximate the invariant measure of stochastic differential equations, where both the drift and the diffusion coefficient are allowed to grow super-linearly. The existence and uniqueness of the invariant measure of the numerical solution generated by the BEM method are proved and the convergence of the numerical invariant measure to the underlying one is shown. Simulations are provided to illustrate the theoretical results and demonstrate the application of our results in the area of system control.

Ergodic numerical approximation to periodic measures of stochastic differential equations

In this paper, we consider numerical approximation to periodic measure of a time periodic stochastic differential equations (SDEs) under weakly dissipative condition. For this we first study the existence of the periodic measure ρt and the large time behaviour of U(t+s,s,x)≔Eϕ(Xts,x)−∫ϕdρt, where Xts,x is the solution of the SDEs and ϕ is a test function being smooth and of polynomial growth at infinity. We prove U and all its spatial derivatives decay to 0 with exponential rate on time t in the sense of average on initial time s. We also prove the existence and the geometric ergodicity of the periodic measure of the discretised semi-flow from the Euler–Maruyama scheme and moment estimate of any order when the time step is sufficiently small (uniform for all orders). We thereafter obtain that the weak error for the numerical scheme of infinite horizon is of the order 1 in terms of the time step. We prove that the choice of step size can be uniform for all test functions ϕ. Subsequently we are able to estimate the average periodic measure with ergodic numerical schemes.

Random quasi-periodic paths and quasi-periodic measures of stochastic differential equations

In this paper, we define random quasi-periodic paths for random dynamical systems and quasi-periodic measures for Markovian semigroups. We give a sufficient condition for the existence and uniqueness of random quasi-periodic paths and quasi-periodic measures for stochastic differential equations and a sufficient condition for the density of the quasi-periodic measure to exist and to satisfy the Fokker-Planck equation. We obtain an invariant measure by considering lifted flow and semigroup on cylinder and the tightness of the average of lifted quasi-periodic measures. We further prove that the invariant measure is unique, and thus ergodic.

A criterion on a repeller being a null set of any limit measure for stochastic differential equations

This paper studies limit behaviors of stationary measures for stochastic ordinary differential equations with nondegenerate noise and presents a criterion to guarantee that a repeller with zero Lebesgue measure is a null set of any limit measure. Using this criterion, we first provide a series of nontrivial concrete examples to show that their repelling limit cycles or quasi-periodic orbits are null sets for all limit measures, which deduces that all their limit measures are concentrated on stable equilibria and stable limit cycles or quasi-periodic orbits, and saddles. Interesting open questions on exact supports of limit measures are proposed.

Multi-level Monte Carlo methods for the approximation of invariant measures of stochastic differential equations

We develop a framework that allows the use of the multi-level Monte Carlo (MLMC) methodology (Giles in Acta Numer. 24:259–328, 2015. https://doi.org/10.1017/S096249291500001X) to calculate expectations with respect to the invariant measure of an ergodic SDE. In that context, we study the (over-damped) Langevin equations with a strongly concave potential. We show that when appropriate contracting couplings for the numerical integrators are available, one can obtain a uniform-in-time estimate of the MLMC variance in contrast to the majority of the results in the MLMC literature. As a consequence, a root mean square error of mathcal {O}(varepsilon ) is achieved with mathcal {O}(varepsilon ^{-2}) complexity on par with Markov Chain Monte Carlo (MCMC) methods, which, however, can be computationally intensive when applied to large datasets. Finally, we present a multi-level version of the recently introduced stochastic gradient Langevin dynamics method (Welling and Teh, in: Proceedings of the 28th ICML, 2011) built for large datasets applications. We show that this is the first stochastic gradient MCMC method with complexity mathcal {O}(varepsilon ^{-2}|log {varepsilon }|^{3}), in contrast to the complexity mathcal {O}(varepsilon ^{-3}) of currently available methods. Numerical experiments confirm our theoretical findings.

Invariant measures of the Milstein method for stochastic differential equations with commutative noise

In this paper, the Milstein method is used to approximate invariant measures of stochastic differential equations with commutative noise. The decay rate of the transition probability kernel generated by the Milstein method to the unique invariant measure of the method is observed to be exponential with respect to the time variable. The convergence rate of the numerical invariant measure to the underlying counterpart is shown to be one. Numerical simulations are presented to demonstrate the theoretical results.

Large deviations for invariant measures of stochastic differential equations with jumps

ABSTRACTLarge deviation principle (LDP) for the invariant measures of stochastic differential equations with jumps is obtained. First, we prove given as the solutions of the stochastic differential equations have invariant measures . In order to ensure the uniqueness of the invariant measures, the strong Feller property and irreducibility for the Markov semigroup are proved. Moreover, the LDP for the invariant measures is established. The proof of the LDP for is based on the LDP for , which have been studied by Budhiraja et al. [5].

A Note on Evolution Systems of Measures of Stochastic Differential Equations in Infinite Dimensional Hilbert Spaces

A Note on Evolution Systems of Measures of Stochastic Differential Equations in Infinite Dimensional Hilbert Spaces

An estimator for the relative entropy rate of path measures for stochastic differential equations

We address the problem of estimating the relative entropy rate (RER) for two stochastic processes described by stochastic differential equations. For the case where the drift of one process is known analytically, but one has only observations from the second process, we use a variational bound on the RER to construct an estimator.

A Class of Lévy Driven SDEs and their Explicit Invariant Measures

We describe a class of explicit invariant measures for stochastic differential equations driven by Levy noise. We relate them to the corresponding Fokker Planck equation. In the symmetric case, we point out the relation with the theory of Dirichlet forms and generalized Schrodinger type operators.

Stationary measures for stochastic differential equations with jumps

In the paper, stationary measures of stochastic differential equations with jumps are considered. Under some general conditions, existence of stationary measures is proved through Markov measures and Lyapunov functions. Moreover, for two special cases, stationary measures are given by solutions of Fokker–Planck equations and long time limits for the distributions of system states.

The Mean Stability Criteria in terms of Two Measures for Stochastic Differential Equations with Coefficient’s Uncertainty

This paper extends the stochastic stability criteria of two measures to the mean stability and proves the stability criteria for a kind of stochastic Itô’s systems. Moreover, by applying optimal control approaches, the mean stability criteria in terms of two measures are also obtained for the stochastic systems with coefficient’s uncertainty.

Uniqueness of Invariant Measures of Infinite Dimensional Stochastic Differential Equations Driven by Lévy Noises

In this paper, we are attempting to study the uniqueness of invariant measures of a stochastic differential equation driven by a Levy type noise in a real separable Hilbert space. To investigate this problem, we study the strong Feller property and irreducibility of the corresponding Markov transition semigroup respectively. To show the strong Feller property, we generalize a Bismut–Elworthy–Li type formula to our Markov transition semigroup under a non-degeneracy condition of the coefficient of the Wiener process.

Existence of global solutions and invariant measures for stochastic differential equations driven by Poisson type noise with non-Lipschitz coefficients

The purpose of this paper is twofold. Firstly, we investigate the problem of existence and uniqueness of solutions to stochastic differential equations with one sided dissipative drift driven by semi-martingales. Secondly, we investigate the problem of existence of an invariant measure for such equations when the coefficients are time independent.

Stability in Terms of Two Measures for Stochastic Differential Equations with Markovian Switching

In this paper, we investigate the stability in terms of two measures for stochastic differential equations with Markovian switching by using the method of Lyapunov functions. Our new theory can not only be used to show a given system to be stochastically stable in the classical sense, but can also be used to deal with some situations where the classical stability theory is not applicable.

Uniqueness and stability of invariant measures for stochastic differential equations in hilbert spaces

The asymptotic behaviour of solutions for stochastic differential equations in Hilbert spaces is investigated. Liapunov type stability and attractivity of the solutions in the space of probability distributions with weak and strong topologies are studied. Also, several statements on uniqueness of invariant measures (stationary distributions) are included

Setwise convergence of solution measures of stochastic differential equations

We study the setwise convergence of solution measures corresponding to stochastic differential equations of the form dx t = f( t, x t ) dt + σ( t, x t ) dW t , x 0 = c, t ϵ [0, 1]. (1) Here W t is a Brownian motion defined on some probability space (Ω, A , μ). We show under quite general conditions, that if the drift term coefficients converge, or the initial conditions x n converge to x 0 then the corresponding solution measures μ x n converge setwise to μ x0 .

Stability of invariant measure of a stochastic differential equation describing molecular rotation

Stability of an invariant measure of stochastic differential equation with respect to bounded pertubations of its coefficients is investigated. The results as well as some earlier author's results on Liapunov type stability of the invariant measure are applied to a system describing molecular rotation.