91 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
Pathwise Accuracy and Ergodicity of Metropolized Integrators for SDEs
Metropolized integrators for ergodic stochastic differential equations (SDE)
are proposed which (i) are ergodic with respect to the (known) equilibrium
distribution of the SDE and (ii) approximate pathwise the solutions of the SDE
on finite time intervals. Both these properties are demonstrated in the paper
and precise strong error estimates are obtained. It is also shown that the
Metropolized integrator retains these properties even in situations where the
drift in the SDE is nonglobally Lipschitz, and vanilla explicit integrators for
SDEs typically become unstable and fail to be ergodic.Comment: 46 pages, 5 figure
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
Loss of regularity for Kolmogorov equations
The celebrated H\"{o}rmander condition is a sufficient (and nearly necessary)
condition for a second-order linear Kolmogorov partial differential equation
(PDE) with smooth coefficients to be hypoelliptic. As a consequence, the
solutions of Kolmogorov PDEs are smooth at all positive times if the
coefficients of the PDE are smooth and satisfy H\"{o}rmander's condition even
if the initial function is only continuous but not differentiable. First-order
linear Kolmogorov PDEs with smooth coefficients do not have this smoothing
effect but at least preserve regularity in the sense that solutions are smooth
if their initial functions are smooth. In this article, we consider the
intermediate regime of nonhypoelliptic second-order Kolmogorov PDEs with smooth
coefficients. The main observation of this article is that there exist
counterexamples to regularity preservation in that case. More precisely, we
give an example of a second-order linear Kolmogorov PDE with globally bounded
and smooth coefficients and a smooth initial function with compact support such
that the unique globally bounded viscosity solution of the PDE is not even
locally H\"{o}lder continuous. From the perspective of probability theory, the
existence of this example PDE has the consequence that there exists a
stochastic differential equation (SDE) with globally bounded and smooth
coefficients and a smooth function with compact support which is mapped by the
corresponding transition semigroup to a function which is not locally
H\"{o}lder continuous. In other words, degenerate noise can have a roughening
effect. A further implication of this loss of regularity phenomenon is that
numerical approximations may converge without any arbitrarily small polynomial
rate of convergence to the true solution of the SDE. More precisely, we prove
for an example SDE with globally bounded and smooth coefficients that the
standard Euler approximations converge to the exact solution of the SDE in the
strong and numerically weak sense, but at a rate that is slower then any power
law.Comment: Published in at http://dx.doi.org/10.1214/13-AOP838 the Annals of
Probability (http://www.imstat.org/aop/) by the Institute of Mathematical
Statistics (http://www.imstat.org
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