Hostname: page-component-586b7cd67f-g8jcs Total loading time: 0 Render date: 2024-11-22T14:54:57.130Z Has data issue: false hasContentIssue false
  • English
  • Français

Euler schemes and half-space approximation for the simulation of diffusion in a domain

Published online by Cambridge University Press:  15 August 2002

Emmanuel Gobet
Emmanuel Gobet*
Affiliation:
École Polytechnique, Centre de Mathématiques Appliquées, 91128 Palaiseau Cedex, France; [email protected].
Get access

Abstract

This paper is concerned with the problem of simulation of (Xt)0≤t≤T , thesolution of a stochastic differential equation constrained by some boundary conditions in a smooth domainD: namely, we consider the case where the boundary ∂D is killing, or where it is instantaneouslyreflecting in an oblique direction. Given N discretization times equally spaced on the interval [0,T],we propose new discretization schemes: they are fully implementable and provide a weak error of orderN -1 under some conditions. The construction of these schemes is based on a natural principle of localapproximation of the domain into a half space, for which efficient simulations are available.

Keywords

Killed diffusionreflected diffusiondiscretization schemesrates of convergenceweakapproximationboundary value problems for parabolic PDE.
Type
Research Article
Information
ESAIM: Probability and Statistics , Volume 5 , 2001 , pp. 261 - 297
Copyright
© EDP Sciences, SMAI, 2001

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Baldi, P., Exact asymptotics for the probability of exit from a domain and applications to simulation. Ann. Probab. 23 (1995) 1644-1670. CrossRef
P. Baldi, L. Caramellino and M.G. Iovino, Pricing complex barrier options with general features using sharp large deviation estimates, edited by Niederreiter, Harald et al., Monte-Carlo and quasi-Monte-Carlo methods 1998, in Proc. of a conference held at the Claremont Graduate University. Claremont, CA, USA, June 22-26, 1998. Springer, Berlin (2000) 149-162.
Bally, V. and Talay, D., The law of the Euler scheme for stochastic differential equations: I. Convergence rate of the distribution function. Probab. Theory Related Fields 104-1 (1996) 43-60. CrossRef
M. Bossy, E. Gobet and D. Talay, Computation of the invariant law of a reflected diffusion process (in preparation).
Cattiaux, P., Hypoellipticité et hypoellipticité partielle pour les diffusions avec une condition frontière. Ann. Inst. H. Poincaré Probab. Statist. 22 (1986) 67-112.
Cattiaux, P., Régularité au bord pour les densités et les densités conditionnelles d'une diffusion réfléchie hypoelliptique. Stochastics 20 (1987) 309-340. CrossRef
Constantini, C., Pacchiarotti, B. and Sartoretto, F., Numerical approximation for functionnals of reflecting diffusion processes. SIAM J. Appl. Math. 58 (1998) 73-102.
O. Faugeras, F. Clément, R. Deriche, R. Keriven, T. Papadopoulo, J. Roberts, T. Viéville, F. Devernay, J. Gomes, G. Hermosillo, P. Kornprobst and D. Lingrand, The inverse EEG and MEG problems: The adjoint state approach. I. The continuous case. Rapport de recherche INRIA No. 3673 (1999).
M. Freidlin, Functional integration and partial differential equations. Ann. of Math. Stud. Princeton University Press (1985).
D. Gilbarg and N.S. Trudinger, Elliptic partial differential equations of second order. Springer Verlag (1977).
E. Gobet, Schémas d'Euler pour diffusion tuée. Application aux options barrière, Ph.D. Thesis. Université Denis Diderot Paris 7 (1998).
Gobet, E., Euler schemes for the weak approximation of killed diffusion. Stochastic Process. Appl. 87 (2000) 167-197. CrossRef
Gobet, E., Efficient schemes for the weak approximation of reflected diffusions. Monte Carlo Methods Appl. 7 (2001) 193-202. Monte Carlo and probabilistic methods for partial differential equations. Monte Carlo (2000). CrossRef
Hausenblas, E., A numerical scheme using excursion theory for simulating stochastic differential equations with reflection and local time at a boundary. Monte Carlo Methods Appl. 6 (2000) 81-103.
Kanagawa, S. and Saisho, Y., Strong approximation of reflecting Brownian motion using penalty method and its application to computer simulation. Monte Carlo Methods Appl. 6 (2000) 105-114. CrossRef
S. Kusuoka and D. Stroock, Applications of the Malliavin calculus I, edited by K. Itô, Stochastic Analysis, in Proc. Taniguchi Internatl. Symp. Katata and Kyoto 1982. Kinokuniya, Tokyo (1984) 271-306.
O.A. Ladyzenskaja, V.A. Solonnikov and N.N. Ural'ceva, Linear and quasi-linear equations of parabolic type. Amer. Math. Soc., Providence, Transl. Math. Monogr. 23 (1968).
Lépingle, D., Un schéma d'Euler pour équations différentielles stochastiques réfléchies. C. R. Acad. Sci. Paris Sér. I Math. 316 (1993) 601-605.
Lépingle, D., Euler scheme for reflected stochastic differential equations. Math. Comput. Simulation 38 (1995) 119-126. CrossRef
Lions, P.L. and Sznitman, A.S., Stochastic differential equations with reflecting boundary conditions. Comm. Pure Appl. Math. 37 (1984) 511-537. CrossRef
Y. Liu, Numerical approaches to reflected diffusion processes. Technical Report (1993).
Menaldi, J.L., Stochastic variational inequality for reflected diffusion. Indiana Univ. Math. J. 32 (1983) 733-744. CrossRef
Milshtein, G.N., Application of the numerical integration of stochastic equations for the solution of boundary value problems with Neumann boundary conditions. Theory Probab. Appl. 41 (1996) 170-177.
C. Miranda, Partial differential equations of elliptic type. Springer, New York (1970).
Pardoux, E. and Williams, R.J., Symmetric reflected diffusions. Ann. Inst. H. Poincaré Probab. Statist. 30 (1994) 13-62.
Pettersson, R., Approximations for stochastic differential equations with reflecting convex boundaries. Stochastic Process. Appl. 59 (1995) 295-308. CrossRef
Pettersson, R., Penalization schemes for reflecting stochastic differential equations. Bernoulli 3 (1997) 403-414. CrossRef
D. Revuz and M. Yor, Continuous martingales and Brownian motion, 2nd Ed. Springer, Berlin, Grundlehren Math. Wiss. 293 (1994).
Saisho, Y., Stochastic differential equations for multi-dimensional domain with reflecting boundary. Probab. Theory Related Fields 74 (1987) 455-477. CrossRef
Serrin, J., The problem of Dirichlet for quasilinear elliptic differential equations with many independent variables. Philos. Trans. Roy. Soc. London Ser. A 264 (1969) 413-496. CrossRef
Slominski, L., On approximation of solutions of multidimensional SDEs with reflecting boundary conditions. Stochastic Process. Appl. 50 (1994) 197-219. CrossRef
Talay, D. and Tubaro, L., Expansion of the global error for numerical schemes solving stochastic differential equations. Stochastic Anal. Appl. 8-4 (1990) 94-120.
R.J. Williams, Semimartingale reflecting Brownian motions in the orthant, Stochastic networks. Springer, New York (1995) 125-137.