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

### Discrete approximations of killed Itô processes

Auteur(s):

Code(s) de Classification MSC:

• 65Cxx Probabilistic methods, simulation and stochastic differential equations {For theoretical aspects, see 68U20 and 60H35}
• 60J60 Diffusion processes [See also 58J65]
• 60-08 Computational methods (not classified at a more specific level) [See also 65C50]

Résumé: We are interested in approximating a multidimensional Itô process $(X_t)_{t\geq 0}$ killed when it leaves a smooth domain $D$: when the exit time is discretized along a regular mesh with time step $h$, we prove under a non characteristic boundary condition, that the discretization error is bounded from above by $C_1\sqrt h$, extending a previous result Gobet(\textit{Stoch. Proc. App. 2000}) obtained in the Markovian case under uniform ellipticity assumptions. In the case of hypoelliptic diffusion processes and when a discrete Euler scheme is additionally used as an approximation of $X$, we prove that the upper bound for the weak error is still valid and that a lower bound with the same rate $\sqrt h$ holds true, thus proving that the order of convergence is exactly $\frac 12$. This provides a theoretical explanation of the well-known bias that we can numerically observe in that kind of procedure.

Mots Clés: Weak approximation ; Killed processes ; Discrete exit time ; Overshoot above the boundary

Date: 2003-03-13

Prépublication numéro: PMA-807