The approximate Euler method for Lévy driven stochastic differential equations

This paper is concerned with the numerical approximation of the expected value E(g(X_t), where g is a suitable test function and X is the solution of a stochastic differential equation driven by a Lévy process Y. More precisely we consider an Euler scheme or an "approximate" Euler scheme with stepsize 1/n, giving rise to a simulable variable X_t ⁿ, and we study the error δ_n(g) = E(g(X_t ⁿ)) - E(g(X_t)). For a genuine Euler scheme we typically get that δ_n(g) is of order 1/n, and we even have an expansion of this error in successive powers of 1/n, and the assumptions are some integrability condition on the driving process and appropriate smoothness of the coefficient of the equation and of the test function g. For an approximate Euler scheme, that is we replace the non-simulable increments of X by a simulable variable close enough to the desired increment, the order of magnitude of δ_n(g) is the supremum of 1/N and a kind of "distance" between the increments of Y and the actually simulated variable. In this situation, a second order expansion is also available.