Algebraic structures in stochastic differential equations

We define a new numerical integration scheme for stochastic differential equations driven by Levy processes with uniformly lower mean square remainder than that of the scheme of the same strong order of convergence obtained by truncating the stochastic Taylor series. In doing so we generalize recent results concerning stochastic differential equations driven by Wiener processes. The aforementioned works studied integration schemes obtained by applying an invertible mapping to the stochastic Taylor series, truncating the resulting series and applying the inverse of the original mapping. The shuffle Hopf algebra and its associated convolution algebra play important roles in the their analysis, arising from the combinatorial structure of iterated Stratonovich integrals. It was recently shown that the algebra generated by iterated It^o integrals of independent Levy processes is isomorphic to a quasi-shuffle algebra. We utilise this to consider map-truncate-invert schemes for Levy processes. To facilitate this, we derive a new form of stochastic Taylor expansion from those of Wagner & Platen, enabling us to extend existing algebraic encodings of integration schemes. We then derive an alternative method of computing map-truncate-invert schemes using a single step, resolving diffculties encountered at the inversion step in previous methods.

Grantor dc:publisher

Heriot-Watt University

Year dc:date.issued

2014

Author dc:creator

• Curry, Charles

Advisors dc:contributor.advisor

• Malham, Simon
• Wiese, Anke

Statement dc:rights

• All items in ROS are protected by the Creative Commons copyright license (http://creativecommons.org/licenses/by-nc-nd/2.5/scotland/), with some rights reserved.

Language dc:language.iso

en

Handle dc:identifier.uri

http://hdl.handle.net/10399/2791

OAI identifier oai:identifier

oai:ros.hw.ac.uk:10399/2791