### Abstract

We construct an efficient integrator for stochastic differential systems driven by Lévy 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 Lévy processes.

Original language | English |
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Article number | 20180567 |

Journal | Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences |

Volume | 475 |

Issue number | 2221 |

DOIs | |

Publication status | Published - 23 Jan 2019 |

### Fingerprint

### Keywords

- Efficient integrators
- Lévy processes
- Quasi-shuffle algebra

### ASJC Scopus subject areas

- Mathematics(all)
- Engineering(all)
- Physics and Astronomy(all)

### Cite this

*Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences*,

*475*(2221), [20180567]. https://doi.org/10.1098/rspa.2018.0567

}

*Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences*, vol. 475, no. 2221, 20180567. https://doi.org/10.1098/rspa.2018.0567

**Algebraic structures and stochastic differential equations driven by Lévy processes.** / Curry, Charles; Ebrahimi-Fard, Kurusch; Malham, Simon John A.; Wiese, Anke.

Research output: Contribution to journal › Article

TY - JOUR

T1 - Algebraic structures and stochastic differential equations driven by Lévy processes

AU - Curry, Charles

AU - Ebrahimi-Fard, Kurusch

AU - Malham, Simon John A.

AU - Wiese, Anke

PY - 2019/1/23

Y1 - 2019/1/23

N2 - We construct an efficient integrator for stochastic differential systems driven by Lévy 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 Lévy processes.

AB - We construct an efficient integrator for stochastic differential systems driven by Lévy 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 Lévy processes.

KW - Efficient integrators

KW - Lévy processes

KW - Quasi-shuffle algebra

UR - http://www.scopus.com/inward/record.url?scp=85061331212&partnerID=8YFLogxK

U2 - 10.1098/rspa.2018.0567

DO - 10.1098/rspa.2018.0567

M3 - Article

VL - 475

JO - Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences

JF - Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences

SN - 1364-5021

IS - 2221

M1 - 20180567

ER -