Strong convergence of an explicit numerical method for SDEs with nonglobally Lipschitz continuous coefficients

On the one hand, the explicit Euler scheme fails to converge strongly to the exact solution of a stochastic differential equation (SDE) with a superlinearly growing and globally one-sided Lipschitz continuous drift coefficient. On the other hand, the implicit Euler scheme is known to converge strongly to the exact solution of such an SDE. Implementations of the implicit Euler scheme, however, require additional computational effort. In this article we therefore propose an explicit and easily implementable numerical method for such an SDE and show that this method converges strongly with the standard order one-half to the exact solution of the SDE. Simulations reveal that this explicit strongly convergent numerical scheme is considerably faster than the implicit Euler scheme.Comment: Published in at http://dx.doi.org/10.1214/11-AAP803 the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org

Divergence of the multilevel Monte Carlo Euler method for nonlinear stochastic differential equations

The Euler-Maruyama scheme is known to diverge strongly and numerically weakly when applied to nonlinear stochastic differential equations (SDEs) with superlinearly growing and globally one-sided Lipschitz continuous drift coefficients. Classical Monte Carlo simulations do, however, not suffer from this divergence behavior of Euler's method because this divergence behavior happens on rare events. Indeed, for such nonlinear SDEs the classical Monte Carlo Euler method has been shown to converge by exploiting that the Euler approximations diverge only on events whose probabilities decay to zero very rapidly. Significantly more efficient than the classical Monte Carlo Euler method is the recently introduced multilevel Monte Carlo Euler method. The main observation of this article is that this multilevel Monte Carlo Euler method does - in contrast to classical Monte Carlo methods - not converge in general in the case of such nonlinear SDEs. More precisely, we establish divergence of the multilevel Monte Carlo Euler method for a family of SDEs with superlinearly growing and globally one-sided Lipschitz continuous drift coefficients. In particular, the multilevel Monte Carlo Euler method diverges for these nonlinear SDEs on an event that is not at all rare but has probability one. As a consequence for applications, we recommend not to use the multilevel Monte Carlo Euler method for SDEs with superlinearly growing nonlinearities. Instead we propose to combine the multilevel Monte Carlo method with a slightly modified Euler method. More precisely, we show that the multilevel Monte Carlo method combined with a tamed Euler method converges for nonlinear SDEs with globally one-sided Lipschitz continuous drift coefficients and preserves its strikingly higher order convergence rate from the Lipschitz case.Comment: Published in at http://dx.doi.org/10.1214/12-AAP890 the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org

The mean-square dichotomy spectrum and a bifurcation to a mean-square attractor

The dichotomy spectrum is introduced for linear mean-square random dynamical systems, and it is shown that for finite-dimensional mean-field stochastic differential equations, the dichotomy spectrum consists of finitely many compact intervals. It is then demonstrated that a change in the sign of the dichotomy spectrum is associated with a bifurcation from a trivial to a non-trivial mean-square random attractor

Anisotropic Energy Distribution in Three-Dimensional Vibrofluidized Granular Systems

We examine the energy distribution in a three-dimensional model granular system contained in an open cylinder under the influence of gravity. Energy is supplied to the system by a vibrating base. We introduce spatially resolved, partial particle-particle ``dissipations'' for directions parallel and perpendicular to the energy input, respectively. Energy balances show that the total (integrated) ``dissipation'' is less than zero in the parallel direction while greater than zero in the perpendicular directions. The energy supplied to the perpendicular directions is dissipated by particle-wall collisions. We further define a fractional energy transfer, which in the steady state represents the fraction of the power supplied by the vibrating base that is dissipated at the wall. We examine the dependence of the fractional energy transfer on the number of particles, the velocity of the vibrating base, the particle-particle restitution coefficient, and the particle-wall restitution coefficient. We also explore the influence of the system parameters on the spatially dependent partial dissipations.Comment: 10 pages, 10 figures, RevTeX forma

Perturbations in Bouncing Cosmological Models

I describe the features and general properties of bouncing models and the evolution of cosmological perturbations on such backgrounds. I will outline possible observational consequences of the existence of a bounce in the primordial Universe and I will make a comparison of these models with standard long inflationary scenarios.Comment: 9 pages, no figure