Stabilized methods for stiff stochastic systems

Assyr Abdulle; Stéphane Cirilli

doi:10.1016/j.crma.2007.10.009

Numerical Analysis

Stabilized methods for stiff stochastic systems

Presented by: Philippe G. Ciarlet

Assyr Abdulle ¹; Stéphane Cirilli ²

¹ School of Mathematics and Maxwell Institute for Mathematical Sciences, University of Edinburgh, Edinburgh EH9 3JZ, UK
² Collège de Genève, DIP, Genève, Switzerland

Comptes Rendus. Mathématique, Volume 345 (2007) no. 10, pp. 593-598.

Abstract
Résumé

Stiff stochastic systems are usually solved numerically by (semi-)implicit methods, since explicit methods, such as the Euler–Maruyama scheme, face severe stepsize reductions. This comes at the cost of solving linear algebra systems at each step and can be expensive for large systems and complicated to implement for complex problems. In this Note, we introduce a new class of explicit methods for stochastic differential equations with multi-dimensional Wiener processes, with much better stability properties (in the mean square sense) than existing explicit methods. These new methods are as easy to implement as standard explicit schemes but much more efficient for handling stiff stochastic problems.

Les équations différentielles stochastiques (EDS) raides sont usuellements résolues par des schémas (semi-)implicites, car l'utilisation de schémas explicites, comme par exemple celui d'Euler–Maruyama, entraîne une sévère réduction du pas de temps. L'utilisation de schémas implicites implique la résolution de systèmes linéaires à chaque pas de temps. Cette procédure est coûteuse pour des grands systèmes d'équations et peut être difficile à réaliser pour des systèmes complexes. Dans cette Note, nous proposons une nouvelle classe de schémas explicites pour des EDS avec un processus de Wiener multidimensionel, qui ont des propriétés de stabilité (en moyenne quadratique) bien plus favorables que les schémas explicites existants. Ces nouveaus schémas sont aussi aisés à réaliser que les schémas explicites traditionnels, mais plus efficaces pour résoudre numériquement des équations différentielles stochastiques raides.

Received: 2007-07-12
Accepted: 2007-09-28
Published online: 2007-11-07

DOI: 10.1016/j.crma.2007.10.009

Author's affiliations:

Assyr Abdulle ¹; Stéphane Cirilli ²

¹ School of Mathematics and Maxwell Institute for Mathematical Sciences, University of Edinburgh, Edinburgh EH9 3JZ, UK
² Collège de Genève, DIP, Genève, Switzerland

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     author = {Assyr Abdulle and St\'ephane Cirilli},
     title = {Stabilized methods for stiff stochastic systems},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {593--598},
     publisher = {Elsevier},
     volume = {345},
     number = {10},
     year = {2007},
     doi = {10.1016/j.crma.2007.10.009},
     language = {en},
}

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Assyr Abdulle; Stéphane Cirilli. Stabilized methods for stiff stochastic systems. Comptes Rendus. Mathématique, Volume 345 (2007) no. 10, pp. 593-598. doi : 10.1016/j.crma.2007.10.009. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2007.10.009/

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