scholarly journals On the Existence and the Applications of Modified Equations for Stochastic Differential Equations

2011 ◽  
Vol 33 (1) ◽  
pp. 102-130 ◽  
Author(s):  
K. C. Zygalakis
Keyword(s):  
Differential Equations ◽  
Stochastic Differential Equations ◽  
Modified Equations

scholarly journals High Weak Order Methods for Stochastic Differential Equations Based on Modified Equations

2012 ◽  
Vol 34 (3) ◽  
pp. A1800-A1823 ◽  
Author(s):  
Assyr Abdulle ◽  
David Cohen ◽  
Gilles Vilmart ◽  
Konstantinos C. Zygalakis
Keyword(s):  
Differential Equations ◽  
Stochastic Differential Equations ◽  
Weak Order ◽  
Modified Equations

scholarly journals Improving multilevel Monte Carlo for stochastic differential equations with application to the Langevin equation

2015 ◽  
Vol 471 (2176) ◽  
pp. 20140679 ◽  
Author(s):  
Eike H. Müller ◽  
Rob Scheichl ◽  
Tony Shardlow
Keyword(s):  
Monte Carlo ◽  
Differential Equations ◽  
Stochastic Differential Equations ◽  
Langevin Equation ◽  
Operator Splitting ◽  
Random Variables ◽  
Weak Approximation ◽  
Multilevel Monte Carlo ◽  
Discrete Random Variables ◽  
Modified Equations

This paper applies several well-known tricks from the numerical treatment of deterministic differential equations to improve the efficiency of the multilevel Monte Carlo (MLMC) method for stochastic differential equations (SDEs) and especially the Langevin equation. We use modified equations analysis as an alternative to strong-approximation theory for the integrator, and we apply this to introduce MLMC for Langevin-type equations with integrators based on operator splitting. We combine this with extrapolation and investigate the use of discrete random variables in place of the Gaussian increments, which is a well-known technique for the weak approximation of SDEs. We show that, for small-noise problems, discrete random variables can lead to an increase in efficiency of almost two orders of magnitude for practical levels of accuracy.

Modified Equations for Stochastic Differential Equations

2006 ◽  
Vol 46 (1) ◽  
pp. 111-125 ◽  
Author(s):  
Tony Shardlow
Keyword(s):  
Differential Equations ◽  
Stochastic Differential Equations ◽  
Modified Equations

Stochastic Differential Equations for Power Law Behaviors

2012 ◽  
Author(s):  
Bo Jiang ◽  
Roger Brockett ◽  
Weibo Gong ◽  
Don Towsley
Keyword(s):  
Differential Equations ◽  
Stochastic Differential Equations ◽  
Power Law

scholarly journals Strong Solution Existence for a Class of Degenerate Stochastic Differential Equations

2020 ◽  
Vol 53 (2) ◽  
pp. 2220-2224
Author(s):  
William M. McEneaney ◽  
Hidehiro Kaise ◽  
Peter M. Dower ◽  
Ruobing Zhao
Keyword(s):  
Differential Equations ◽  
Stochastic Differential Equations ◽  
Strong Solution ◽  
Solution Existence
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