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 I: continuous 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 $(X_t)_{0\leq t \leq T}$ 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