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A symmetrized Euler scheme for an efficient approximation of reflected diffusions

Part of: Parabolic equations and systems Probability theory and stochastic processes

Published online by Cambridge University Press:  14 July 2016

Mireille Bossy ,
Emmanuel Gobet  and
Denis Talay
Mireille Bossy*
Affiliation:
INRIA, Sophia-Antipolis
Emmanuel Gobet*
Affiliation:
École Polytechnique, Palaiseau
Denis Talay*
Affiliation:
INRIA, Sophia-Antipolis
*
∗ Postal address: INRIA Unité de Recherche de Sophia-Antipolis, 2004, Route des Lucioles, BP 93, 06902 Sophia-Antipolis Cedex, France
∗ Postal address: CMAP, École Polytechnique, 91128 Palaiseau Cedex, France. Email address: gobet@cmapx.polytechnique.fr
∗ Postal address: INRIA Unité de Recherche de Sophia-Antipolis, 2004, Route des Lucioles, BP 93, 06902 Sophia-Antipolis Cedex, France
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Abstract

In this article, we analyse the error induced by the Euler scheme combined with a symmetry procedure near the boundary for the simulation of diffusion processes with an oblique reflection on a smooth boundary. This procedure is easy to implement and, in addition, accurate: indeed, we prove that it yields a weak rate of convergence of order 1 with respect to the time-discretization step.

Keywords

Reflected diffusiondiscretization schemeweak approximationNeumann boundary condition for parabolic PDE

MSC classification

Secondary: 35K20: Initial-boundary value problems for second-order parabolic equations 60-08: Computational methods (not classified at a more specific level)
Type
Research Papers
Information
Journal of Applied Probability , Volume 41 , Issue 3 , September 2004 , pp. 877 - 889
Copyright
Copyright © Applied Probability Trust 2004 

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References

Bally, V., and Talay, D. (1996). The law of the Euler scheme for stochastic differential equations. I. Convergence rate of the distribution function. Prob. Theory Relat. Fields 104, 43–60.10.1007/BF01303802Google Scholar
Bensoussan, A., and Lions, J.-L. (1984). Impulse Control and Quasivariational Inequalities. Gauthier-Villars, Montrouge.Google Scholar
Clerc, M. et al.(2002). Comparison of BEM and FEM methods for the E/MEG problem. In Proc. BIOMAG 2002 (Jena, August 2002). Available at http://biomag2002.uni-jena.de/.Google Scholar
Clerc, M. et al.(2002). The fast multipole method for the direct E/MEG problem. In Proc. IEEE Internat. Symp. Biomed. Imaging (Piscataway, NJ, July 2002).10.1109/ISBI.2002.1029438Google Scholar
Costantini, C., Pacchiarotti, B., and Sartoretto, F. (1998). Numerical approximation for functionals of reflecting diffusion processes. SIAM J. Appl. Math. 58, 73–102.Google Scholar
Faugeras, O. et al.(1999). The inverse EEG and MEG problems: the adjoint state approach. I. The continuous case. Res. Rep. 3673, INRIA, Sophia Antipolis. Available at http://www.inria.fr/rrrt/.Google Scholar
Freidlin, M. (1985). Functional Integration and Partial Differential Equations (Ann. Math. Studies 109). Princeton University Press.Google Scholar
Gobet, E. (2000). Euler schemes for the weak approximation of killed diffusion. Stoch. Process. Appl. 87, 167–197.10.1016/S0304-4149(99)00109-XGoogle Scholar
Gobet, E. (2001). Efficient schemes for the weak approximation of reflected diffusions. Monte Carlo Meth. Appl. 7, 193–202.10.1515/mcma.2001.7.1-2.193Google Scholar
Gobet, E. (2001). Euler schemes and half-space approximation for the simulation of diffusions in a domain. ESAIM Prob. Statist. 5, 261–297.10.1051/ps:2001112Google Scholar
Kanagawa, S., and Saisho, S. (2000). Strong approximation of reflecting Brownian motion using penalty method and its application to computer simulation. Monte Carlo Meth. Appl. 6, 105–114.10.1515/mcma.2000.6.2.105Google Scholar
Kybic, J. et al.(2003). Integral formulations for the EEG problem. Res. Rep. RR-4735, INRIA, Sophia Antipolis. Available at http://www.inria.fr/rrrt/.Google Scholar
Ladyzenskaja, O. A., Solonnikov, V. A. and Uralceva, N. N. (1967). Linear and Quasilinear Equations of Parabolic Type (Translations Math. Monogr. 23). American Mathematical Society, Providence, RI.Google Scholar
Lépingle, D. (1993). Un schéma d'Euler pour équations différentielles stochastiques réfléchies. C. R. Acad. Sci. Paris Sér. I Math. 316, 601–605.Google Scholar
Lépingle, D. (1995). Euler scheme for reflected stochastic differential equations. Math. Comput. Simul. 38, 119–126.10.1016/0378-4754(93)E0074-FGoogle Scholar
Lions, P., and Sznitman, A. (1984). Stochastic differential equations with reflecting boundary conditions. Commun. Pure Appl. Math. 37, 511–537.10.1002/cpa.3160370408Google Scholar
Makarov, R. (2002). Combined estimates of the Monte Carlo method for the third boundary value problem for a parabolic-type equation. Russian J. Numer. Anal. Math. Model. 17, 547–558.10.1515/rnam-2002-0605Google Scholar
Menaldi, J. (1983). Stochastic variational inequality for reflected diffusion. Indiana Univ. Math. J. 32, 733–744.10.1512/iumj.1983.32.32048Google Scholar
Milshtein, G. (1996). Application of the numerical integration of stochastic equations for the solution of boundary value problems with Neumann boundary conditions. Theory Prob. Appl. 41, 170–177.Google Scholar
Pettersson, R. (1995). Approximations for stochastic differential equations with reflecting convex boundaries. Stoch. Process. Appl. 59, 295–308.10.1016/0304-4149(95)00040-EGoogle Scholar
Pettersson, R. (1997). Penalization schemes for reflecting stochastic differential equations. Bernoulli 3, 403–414.10.2307/3318456Google Scholar
Revuz, D., and Yor, M. (1994). Continuous Martingales and Brownian Motion (Grundlehren Math. Wissensch. 293), 2nd edn. Springer, Berlin.Google Scholar
Saisho, Y. (1987). Stochastic differential equations for multidimensional domain with reflecting boundary. Prob. Theory Relat. Fields 74, 455–477.10.1007/BF00699100Google Scholar
Slomiński, L. (1994). On approximation of solutions of multidimensional SDEs with reflecting boundary conditions. Stoch. Process. Appl. 50, 197–219.10.1016/0304-4149(94)90118-XGoogle Scholar
Talay, D., and Tubaro, L. (1990). Expansion of the global error for numerical schemes solving stochastic differential equations. Stoch. Anal. Appl. 8, 94–120.10.1080/07362999008809220Google Scholar