Numerical Methods in the Weak Sense for Stochastic Differential Equations with Small Noise

We propose a new approach to constructing weak numerical methods for finding solutions to stochastic systems with small noise. For these methods we prove an error estimate in terms of products hiεj(h is a time increment, ε is a small parameter). We derive various efficient weak schemes for systems w...

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Bibliographic Details
Published in:SIAM journal on numerical analysis 1997-12, Vol.34 (6), p.2142-2167
Main Authors: Milstein, G. N., M. V. Tret'Yakov
Format: Article
Language:English
Subjects:
Approximation
Coefficients
Computer simulation
Differential equations
Differentials
Error rates
Estimation methods
Human error
Inequality
Methods
Noise
Numerical analysis
Numerical methods
Random variables
Runge Kutta method
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Summary:We propose a new approach to constructing weak numerical methods for finding solutions to stochastic systems with small noise. For these methods we prove an error estimate in terms of products hiεj(h is a time increment, ε is a small parameter). We derive various efficient weak schemes for systems with small noise and study the Talay-Tubaro expansion of their global error. An efficient approach to reducing the Monte-Carlo error is presented. Some of the proposed methods are tested by calculating the Lyapunov exponent of a linear system with small noise.
ISSN:0036-1429
1095-7170
DOI:10.1137/S0036142996278967