Algebraic Structures and Stochastic Differential Equations driven by Levy processes

Charles Curry, Kurusch Ebrahimi-Fard, Simon John A. Malham, Anke Wiese

Research output: Contribution to journalArticle

Abstract

We construct an efficient integrator for stochastic differential systems driven by Levy processes. An efficient integrator is a strong approximation that is more accurate than the corresponding stochastic Taylor approximation, to all orders and independent of the governing vector fields. This holds provided the driving processes possess moments of all orders and the vector fields are sufficiently smooth. Moreover the efficient integrator in question is optimal within a broad class of perturbations for half-integer global root mean-square orders of convergence. We obtain these results using the quasi-shuffle algebra of multiple iterated integrals of independent Levy processes.
LanguageEnglish
Article number20180567
JournalProceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences
Volume475
Issue number2221
DOIs
StatePublished - 23 Jan 2019

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Lévy Process
Algebraic Structure
Stochastic Equations
Differential equation
Vector Field
Iterated integral
Strong Approximation
Shuffle
Order of Convergence
Stochastic Systems
Differential System
Mean Square
Roots
Moment
Perturbation
Algebra
Integer
Approximation

Cite this

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abstract = "We construct an efficient integrator for stochastic differential systems driven by Levy processes. An efficient integrator is a strong approximation that is more accurate than the corresponding stochastic Taylor approximation, to all orders and independent of the governing vector fields. This holds provided the driving processes possess moments of all orders and the vector fields are sufficiently smooth. Moreover the efficient integrator in question is optimal within a broad class of perturbations for half-integer global root mean-square orders of convergence. We obtain these results using the quasi-shuffle algebra of multiple iterated integrals of independent Levy processes.",
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Algebraic Structures and Stochastic Differential Equations driven by Levy processes. / Curry, Charles; Ebrahimi-Fard, Kurusch; Malham, Simon John A.; Wiese, Anke.

In: Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, Vol. 475, No. 2221, 20180567, 23.01.2019.

Research output: Contribution to journalArticle

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