Université Paris 6Pierre et Marie Curie Université Paris 7Denis Diderot CNRS U.M.R. 7599 ``Probabilités et Modèles Aléatoires''

### The approximate Euler method for Lévy driven stochastic differential equations

Auteur(s):

Code(s) de Classification MSC:

• 65C30 Stochastic differential and integral equations
• 60G51 Processes with independent increments

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