Exact approximation rate of killed hypoelliptic diffusions using the discrete Euler scheme

• Published in: Stochastic processes and their applications Vol. 112; no. 2; pp. 201 - 223
• Main Authors: Gobet, Emmanuel, Menozzi, Stéphane
• Format: Journal Article
• Language: English
• Published: Amsterdam Elsevier B.V 01-08-2004; Elsevier Science; Elsevier
• Series: Stochastic Processes and their Applications
• Subjects: Discrete exit time; Exact sciences and technology; Killed processes; Markov processes; Mathematics; Overshoot above the boundary; Probability; Probability and statistics; Probability theory and stochastic processes; Sciences and techniques of general use; Stochastic analysis; Stochastic processes; Weak approximation; Weak approximation Killed processes Discrete exit time Overshoot above the boundary; Killed processes; Overshoot above the boundary; 60-08; 60J60; 65Cxx; Discrete exit time; Weak approximation; Lower bound; Error estimation; Bias; Boundary condition; Stochastic process; Euler scheme; Convergence; Numerical approximation; Upper bound; Discrete time; Diffusion process; Application; Biased estimation
• ISSN: 0304-4149; 1879-209X
• DOI: 10.1016/j.spa.2004.03.002

We are interested in approximating a multidimensional hypoelliptic diffusion process (
X
t
)
t⩾0
killed when it leaves a smooth domain
D. When a discrete Euler scheme with time step
h is used, we prove under a noncharacteristic boundary condition that the weak error is upper bounded by
C
1
h
, generalizing the result obtained by Gobet in (Stoch. Proc. Appl. 87 (2000) 167) for the uniformly elliptic case. We also obtain a lower bound with the same rate
h
, thus proving that the order of convergence is exactly 1/2.
This provides a theoretical explanation of the well-known bias that we can numerically observe in that kind of procedure.