10,320 research outputs found
Accurate stationary densities with partitioned numerical methods for stochastic partial differential equations
We consider the numerical solution, by finite differences, of second-order-in-time stochastic partial differential equations (SPDEs) in one space dimension. New timestepping methods are introduced by generalising recently-introduced methods for second-order-in-time stochastic differential equations to multidimensional systems. These stochastic methods, based on leapfrog and Runge–Kutta methods, are designed to give good approximations to the stationary variances and the correlations in the position and velocity variables. In particular, we introduce the reverse leapfrog method and stochastic Runge–Kutta Leapfrog methods, analyse their performance applied to linear SPDEs and perform numerical experiments to examine their accuracy applied to a type of nonlinear SPDE
A Fourier interpolation method for numerical solution of FBSDEs: Global convergence, stability, and higher order discretizations
The implementation of the convolution method for the numerical solution of
backward stochastic differential equations (BSDEs) introduced in [19] uses a
uniform space grid. Locally, this approach produces a truncation error, a space
discretization error, and an additional extrapolation error. Even if the
extrapolation error is convergent in time, the resulting absolute error may be
high at the boundaries of the uniform space grid. In order to solve this
problem, we propose a tree-like grid for the space discretization which
suppresses the extrapolation error leading to a globally convergent numerical
solution for the (F)BSDE. On this alternative grid the conditional expectations
involved in the BSDE time discretization are computed using Fourier analysis
and the fast Fourier transform (FFT) algorithm as in the initial
implementation. The method is then extended to higher-order time
discretizations of FBSDEs. Numerical results demonstrating convergence are also
presented.Comment: 28 pages, 8 figures; Previously titled 'Global convergence and
stability of a convolution method for numerical solution of BSDEs'
(1410.8595v1
Numerical Methods for Stochastic Differential Equations
Stochastic differential equations (sdes) play an important role in physics
but existing numerical methods for solving such equations are of low accuracy
and poor stability. A general strategy for developing accurate and efficient
schemes for solving stochastic equations in outlined here. High order numerical
methods are developed for integration of stochastic differential equations with
strong solutions. We demonstrate the accuracy of the resulting integration
schemes by computing the errors in approximate solutions for sdes which have
known exact solutions
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