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 $(X_t)_{t \geq 0}$ killed as it leaves an open set $D$,
when the diffusion is approximated by its discrete Euler scheme $(\tX_{t_i})_{0\leq
i\leq N}$, with discretization step $T/N$. If we set $\tau:=\inf\{t>0:X_t\notin D\}$
and $\tt:=\inf\{t_i>0:\tX_{t_i}\notin D\}$, we prove that the discretization error
$\E_x\left[\1_{T<\tt}\;f(\tX_T)\right]-\E_x\left[\1_{T<\tau}\;f(X_T)\right]$
is of order $N^{-1/2}$, 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. The rate of convergence $N^{-1/2}$ is exact and is intrinsic
to the problem of discrete killing time. In the first part of this work, we have
studied the weak approximation using a continuous Euler scheme: under some
conditions, it enables us to achieve the rate $N^{-1}$.

**Mots Clés:** *weak approximation ; killed diffusion ; Euler scheme ; Malliavin calculus ; Ito's formula ; orthogonal projection ; local time on the
boundary.*

**Date:** 1999-05-04

**Prépublication numéro:** *PMA-502*