Algebraic Structures and Stochastic Differential Equations driven by Levy processes

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.

41 pages, 11 figures

References in corpus (5)

• The Euler scheme for Levy driven stochastic differential equations: limit theorems
• The exponential Lie series for continuous semimartingales
• Flows and stochastic Taylor series in Ito calculus
• Levy Processes and Quasi-Shuffle Algebras
• Approximations for solutions of Lévy-type stochastic differential equations

Cited by in corpus (3)

• The exponential Lie series for continuous semimartingales
• Time-warping invariants of multidimensional time series
• Quasi-geometric rough paths and rough change of variable formula