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

### Weak approximation of killed diffusion. Part II: discrete Euler scheme.

Auteur(s):

Code(s) de Classification MSC:

• 65U05 Numerical methods in probability and statistics
• 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