Université Paris 6
Pierre et Marie Curie
Université Paris 7
Denis Diderot

CNRS U.M.R. 7599
``Probabilités et Modèles Aléatoires''

The approximate Euler method for Lévy driven stochastic differential equations


Code(s) de Classification MSC:

Résumé: 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 L\'evy process $Y$. More precisely we consider an Euler scheme or an ``approximate'' Euler scheme with stepsize $1/n$, giving rise to a simulable variable $X^n_t$, and we study the error $\de_n(g)=\E(g(X^n_t))-\E(g(X_t))$. For a genuine Euler scheme we typically get that $\de_n(g)$ 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 $\de_n(g)$ 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.

Mots Clés: Euler scheme ; stochastic differential equations ; simulations ; approximate simulations

Date: 2003-06-13

Prépublication numéro: PMA-830