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

CNRS U.M.R. 7599
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``Probabilités et Modèles Aléatoires''
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**Auteur(s): **

**Code(s) de Classification MSC:**

- 60H10 Stochastic ordinary differential equations [See also 34F05]
- 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*