1,713 research outputs found
Metropolis Integration Schemes for Self-Adjoint Diffusions
We present explicit methods for simulating diffusions whose generator is
self-adjoint with respect to a known (but possibly not normalizable) density.
These methods exploit this property and combine an optimized Runge-Kutta
algorithm with a Metropolis-Hastings Monte-Carlo scheme. The resulting
numerical integration scheme is shown to be weakly accurate at finite noise and
to gain higher order accuracy in the small noise limit. It also permits to
avoid computing explicitly certain terms in the equation, such as the
divergence of the mobility tensor, which can be tedious to calculate. Finally,
the scheme is shown to be ergodic with respect to the exact equilibrium
probability distribution of the diffusion when it exists. These results are
illustrated on several examples including a Brownian dynamics simulation of DNA
in a solvent. In this example, the proposed scheme is able to accurately
compute dynamics at time step sizes that are an order of magnitude (or more)
larger than those permitted with commonly used explicit predictor-corrector
schemes.Comment: 54 pages, 8 figures, To appear in MM
The Euler-Maruyama approximation for the absorption time of the CEV diffusion
A standard convergence analysis of the simulation schemes for the hitting
times of diffusions typically requires non-degeneracy of their coefficients on
the boundary, which excludes the possibility of absorption. In this paper we
consider the CEV diffusion from the mathematical finance and show how a weakly
consistent approximation for the absorption time can be constructed, using the
Euler-Maruyama scheme
Efficient estimation of one-dimensional diffusion first passage time densities via Monte Carlo simulation
We propose a method for estimating first passage time densities of
one-dimensional diffusions via Monte Carlo simulation. Our approach involves a
representation of the first passage time density as expectation of a functional
of the three-dimensional Brownian bridge. As the latter process can be
simulated exactly, our method leads to almost unbiased estimators. Furthermore,
since the density is estimated directly, a convergence of order ,
where is the sample size, is achieved, the last being in sharp contrast to
the slower non-parametric rates achieved by kernel smoothing of cumulative
distribution functions.Comment: 14 pages, 2 figure
First Passage Time Distribution for Anomalous Diffusion
We study the first passage time (FPT) problem in Levy type of anomalous
diffusion. Using the recently formulated fractional Fokker-Planck equation, we
obtain an analytic expression for the FPT distribution which, in the large
passage time limit, is characterized by a universal power law. Contrasting this
power law with the asymptotic FPT distribution from another type of anomalous
diffusion exemplified by the fractional Brownian motion, we show that the two
types of anomalous diffusions give rise to two distinct scaling behavior.Comment: 11 pages, 2 figure
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