Random Ordinary Differential Equations and Their Numerical Solution by Xiaoying Han & Peter E. Kloeden

Author:Xiaoying Han & Peter E. Kloeden
Language: eng
Format: epub
Publisher: Springer Singapore, Singapore

with the support function

The corresponding RODE scheme is given componentwise by

which is the same as the Euler–Maruyama and Milstein schemes.

The componentwise order 1.5 explicit strong scheme is

where

Since 0 for the -component gives the RODE scheme

where

10.2 Implicit Strong Schemes

There are many differential equations that arise often in practice for which explicit schemes may perform poorly, while implicit schemes offer better numerical stability properties.

The order 1.0 implicit strong Taylor scheme given in Kloeden and Platen [91] is a drift-implicit version of the Milstein scheme. For the vector SODE (10.11) with single Wiener process, the componentwise order 1.0 implicit strong Taylor scheme reads

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