Algebraic Structures and Stochastic Differential Equations driven by Levy processes

We define a new numerical integration scheme for stochastic differential equations driven by Levy processes that has a leading order error coefficient less than that of the scheme of the same strong order of convergence obtained by truncating the stochastic Taylor series. This holds for all such equations where the driving processes possess moments of all orders and the coefficients are sufficiently smooth. The results are obtained using the quasi-shuffle algebra of multiple iterated integrals of independent Levy processes. Our findings generalize recent results concerning equations driven by Wiener processes.