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Weak approximation of killed diffusion using Euler schemes. (English) Zbl 1045.60082

Summary: 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 \((\widetilde X_t)_{0\leq t\leq T}\) or by its discrete one \((\widetilde X_{t_i})_{0\leq i \leq N}\), with discretization step \(T/N\). If we set \(\tau :=\inf \{t>0\: X_t \notin D\}\) and \(\widetilde {\tau }_c :=\inf \{t>0\: \widetilde X_t \notin D\}\), we prove that the discretization error \(\mathbb E_x [\mathbf 1_{T < \widetilde {\tau }_c} f(\widetilde X_T)] - \mathbb E_x [\mathbf 1_{T<\tau } f(X_T)]\) can be expanded to the first order in \(N^{-1}\), provided support or regularity conditions on \(f\). For the discrete scheme, if we set \(\widetilde {\tau }_d := \inf \{t_i>0\: \widetilde X_{t_i} \notin D\}\), the error \(\mathbb E_x [\mathbf 1_{T<\widetilde {\tau }_d}f(\widetilde X_T)] - \mathbb E_x [\mathbf 1_{T<t} f(X_t)]\) is of order \(N^{-1/2}\), under analogous assumptions on \(f\). This rate of convergence is actually exact and intrinsic to the problem of discrete killing time.

MSC:

60J60 Diffusion processes
65C30 Numerical solutions to stochastic differential and integral equations
60H35 Computational methods for stochastic equations (aspects of stochastic analysis)
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