Complexity and effective dimension of discrete Lévy areas

Discretisation methods to simulate stochastic differential equations belong to the main tools in mathematical finance. For Itô processes, there exist several Euler- or Runge–Kutta-like methods which are analogues of well-known approximation schemes in the nonstochastic case. In the multidimensional case, there appear several difficulties, caused by the mixed second order derivatives. These mixed terms (or more precisely their differences) correspond to special random variables called Lévy stochastic area terms. In the present paper, we compare three approximation methods for such random variables with respect to computational complexity and the so-called effective dimension.