Numerical approximations to the stationary solutions of stochastic differential equations

This thesis investigates the possibility of approximating stationary solutions of stochastic differential equations using numerical methods. We consider a particular class of stochastic differential equations, which are known to generate random dynamical systems. The existence of stochastic stationary solution is proved using global attractor approach. Euler's numerical method, applied to the stochastic differential equation, is proved to generate a discrete random dynamical system. The existence of stationary solution is proved again using global attractor approach. At last we prove that the approximate stationary point converges in mean-square sense to the exact one as the time step of the numerical scheme diminishes.EThOS - Electronic Theses Online ServiceGBUnited Kingdo

Numerical Approximation of Stationary Distribution for SPDEs

In this paper, we show that the exponential integrator scheme both in spatial discretization and time discretization for a class of stochastic partial differential equations has a unique stationary distribution whenever the stepsize is sufficiently small, and reveal that the weak limit of the law for the exponential integrator scheme is in fact the counterpart for the stochastic partial differential equation considered.Comment: P2

The random periodic solution of a stochastic differential equation with a monotone drift and its numerical approximation

In this paper we study the existence and uniqueness of the random periodic solution for a stochastic differential equation with a one-sided Lipschitz condition (also known as monotonicity condition) and the convergence of its numerical approximation via the backward Euler-Maruyama method. The existence of the random periodic solution is shown as the limits of the pull-back flows of the SDE and discretized SDE respectively. We establish a convergence rate of the strong error for the backward Euler-Maruyama method and obtain the weak convergence result for the approximation of the periodic measure

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

Numerical approximation of random periodic solutions of stochastic differential equations

In this paper, we discuss the numerical approximation of random periodic solutions (r.p.s.) of stochastic differential equations (SDEs) with multiplicative noise. We prove the existence of the random periodic solution as the limit of the pull-back flow when the starting time tends to −∞ along the multiple integrals of the period. As the random periodic solution is not explicitly constructible, it is useful to study the numerical approximation. We discretise the SDE using the Euler-Maruyama scheme and moldi ied Milstein scheme. Subsequently we obtain the existence of the random periodic solution as the limit of the pullback of the discretised SDE. We prove that the latter is an approximated random periodic solution with an error to the exact one at the rate of √∆t in the mean-square sense in Euler- Maruyama method and ∆t in the Milstein method. We also obtain the weak convergence result for the approximation of the periodic measure

Numerical analysis of random periodicity of stochastic differential equations

In this thesis, we discuss the numerical approximation of random periodic solutions (r.p.s.) of stochastic differential equations (SDEs) with multiplicative noise. We prove the existence of the random periodic solution as the limit of the pull-back flow when the starting time tends to $-\infty$ along the multiple integrals of the period. As the random periodic solution is not explicitly constructible, it is useful to study the numerical approximation. We discretise the SDE using the Euler-Maruyama scheme and modified Milstein scheme. Subsequently we obtain the existence of the random periodic solution as the limit of the pull-back of the discretised SDE. We prove that the latter is an approximated random periodic solution with an error to the exact one at the rate of $\sqrt {\Delta t}$ in the mean-square sense in Euler-Maruyama method and $\Delta t$ in the modified Milstein method. We obtain the weak convergence result in infinite horizon for the approximation of the average periodic measure

Further properties on functional SDEs.

In this work, we aim to study some fine properties for functional stochastic differential equation. The results consist of five main parts. In the second chapter, by constructing successful couplings, the derivative formula, gradient estimates and Harnack inequalities are established for the semigroup associated with a class of degenerate functional stochastic differential equations. In the third chapter, by using Malliavin calculus, explicit derivative formulae are established for a class of semi-linear functional stochastic partial differential equations with additive or multiplicative noise. As applications, gradient estimates and Harnack inequalities are derived for the semigroup of the associated segment process. In the fourth chapter, we apply the weak convergence approach to establish a large deviation principle for a class of neutral functional stochastic differential equations with jumps. In particular, we discuss the large deviation principle for neutral stochastic differential delay equations which allow the coefficients to be highly nonlinear with respect to the delay argument. In the fifth chapter, we discuss the convergence of Euler-Maruyama scheme for a class of neutral stochastic partial differential equations driven by alpha-stable processes, where the numerical scheme is based on spatial discretization and time discretization. In the last chapter, we discuss (i) the existence and uniqueness of the stationary distribution of explicit Euler-Maruyama scheme both in time and in space for a class of stochastic partial differential equations whenever the stepsize is sufficiently small, and (ii) show that the stationary distribution of the Euler-Maruyama scheme converges weakly to the counterpart of the stochastic partial differential equation

Numerical approximations to the stationary solutions of stochastic differential equations

This article was published in the journal, SIAM Journal on Numerical Analysis [© Society for Industrial and Applied Mathematics] and the definitive version is available at: http://dx.doi.org/10.1137/100797886In this paper, we investigate the possibility of approximating the stationary solution of a stochastic differential equation (SDE). We start with the random dynamical system generated by the SDE with the multiplicative noise. We prove that the pullback flow has a stationary point. However, the stationary point is not constructible explicitly; therefore, we look at the numerical approximation. We prove that the discrete time random dynamical system also has a stationary point. Finally, we prove mean-square convergence of the approximate stationary solution to the exact stationary solution as the time step diminishes, as well as almost surely convergence when the time step is rational