Random periodic solutions of SDEs: Existence, uniqueness and numerical issues

The aim of this paper is to discuss existence and uniqueness of random periodic solutions to stochastic differential equations (SDEs) with multiplicative noise under a one-sided Lipschitz condition, as well as on their numerical approximation via two classes of stochastic θ-methods, i.e., θ-Maruyama methods with θ∈[1/2,1] and θ-Milstein ones with θ∈[0,1]. The existence of the random periodic solutions as the limit of the pull-back flows of the discretized SDEs and the strong convergence rate of the aforementioned methods are also investigated. Selected numerical experiments confirming the theoretical analysis are also given

Exponential mean-square stability properties of stochastic linear multistep methods

AbstractThe aim of this paper is the analysis of exponential mean-square stability properties of nonlinear stochastic linear multistep methods. In particular it is known that, under certain hypothesis on the drift and diffusion terms of the equation, exponential mean-square contractivity is visible: the qualitative feature of the exact problem is here analysed under the numerical perspective, to understand whether a stochastic linear multistep method can provide an analogous behaviour and which restrictions on the employed stepsize should be imposed in order to reproduce the contractive behaviour. Numerical experiments confirming the theoretical analysis are also given

Mean-square contractivity of stochastic θ\theta-methods

The paper is focused on the nonlinear stability analysis of stochastic $\theta$-methods. In particular, we consider nonlinear stochastic differential equations such that the mean-square deviation between two solutions exponentially decays, i.e., a mean-square contractive behaviour is visible along the stochastic dynamics. We aim to make the same property visible also along the numerical dynamics generated by stochastic $\theta$-methods: this issue is translated into sharp stepsize restrictions depending on parameters of the problem, here accurately estimated. A selection of numerical tests confirming the effectiveness of the analysis and its sharpness is also provided

Strong backward error analysis of symplectic integrators for stochastic Hamiltonian systems

Backward error analysis is a powerful tool in order to detect the long-term conservative behavior of numerical methods. In this work, we present a long-term analysis of symplectic stochastic numerical integrators, applied to Hamiltonian systems with multiplicative noise. We first compute and analyze the associated stochastic modified differential equations. Then, suitable bounds for the coefficients of such equations are provided towards the computation of long-term estimates for the Hamiltonian deviations occurring along the aforementioned numerical dynamics. This result generalizes Benettin-Giorgilli Theorem to the scenario of stochastic symplectic methods. Finally, specific numerical methods are considered, in order to provide a numerical evidence confirming the effectiveness of the theoretical investigation

Highly Stable Multistage Numerical Methods for Functional Equations: Theory and Implementation Issues

2008-2009Functional equations provide the best and most natural way to describe evolution in time and space, also in presence of memory. In fact, the spread of diseases, the growth of biological populations, the brain dynamics, elasticity and plasticity, heat conduction, fluid dynamics, scattering theory, seismology, biomechanics, game theory, control, queuing theory, design of electronic filters and many other problems from physics, chemistry, pharmacology, medicine, economics can be modelled through systems of functional equations, such as Ordinary Differential Equations (ODEs) and Volterra Integral Equations (VIEs). For instance, ODEs based models can be found in the context of evolution of biological populations, mathematical models in physiology and medicine, such as oncogenesis and spread of infections and diseases, economical sciences, analysis of signals. Concerning VIEs based models, the following books and review papers contain sections devoted to this subject in the physical and biological sciences: Brunner, Agarwal and O’Regan, Corduneanu and Sandberg, Zhao. Most of these also include extensive lists of references. Regarding some specific applications of VIEs, they are for example models of population dynamics and spread of epidemics, wave problems, fluido-dynamics, contact problems,electromagnetic signals.Arizona State UniversityVIII n.s

Two-step Runge-Kutta methods with quadratic stability functions

We describe the construction of implicit two-step Runge-Kutta methods with stability properties determined by quadratic stability functions. We will aim for methods which are A-stable and L-stable and such that the coefficients matrix has a one point spectrum. Examples of methods of order up to eight are provided

Drift-preserving numerical integrators for stochastic Hamiltonian systems

The paper deals with numerical discretizations of separable nonlinear Hamiltonian systems with additive noise. For such problems, the expected value of the total energy, along the exact solution, drifts linearly with time. We present and analyze a time integrator having the same property for all times. Furthermore, strong and weak convergence of the numerical scheme along with efficient multilevel Monte Carlo estimators are studied. Finally, extensive numerical experiments illustrate the performance of the proposed numerical scheme