18,532 research outputs found
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
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