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:**

- 65U05 Numerical methods in probability and statistics
- 60Hxx Stochastic analysis, see also {58G32}
- 65C05 Monte Carlo methods
- 35K20 Boundary value problems for second-order, parabolic equations

**Résumé:** We study the weak approximation of a
multidimensional diffusion $(X_t)_{0\leq t \leq T}$ killed as it leaves an open set
$D$, when the diffusion is approximated by its continuous Euler scheme
$(\tX_{t})_{0\leq t\leq T}$, with discretization step $T/N$. If we set
$\tau:=\inf\{t>0:X_t\notin D\}$ and $\tt:=\inf\{t>0:\tX_{t}\notin D\}$, we prove
that the discretization error
$\E_x\left[\1_{T<\tt}\;f(\tX_T)\right]-\E_x\left[\1_{T<\tau}\;f(X_T)\right]$
can be expanded to the first order in $N^{-1}$, provided that $f$ is a bounded
measurable function with support strictly included in $D$. The support condition on
$f$ can be weakened if $f$ is smooth enough. In the second part
of this work, we will study the weak approximation using a discrete Euler scheme:
under some conditions, the approximation error is of order $N^{-1/2}$.

**Mots Clés:** *weak approximation ; killed diffusion ; Euler scheme ; Malliavin calculus ; error's expansion*

**Date:** 1999-05-04

**Prépublication numéro:** *PMA-501*