Euler schemes and half-space approximation for the simulation of diffusion in a domain
ESAIM: Probability and Statistics, Tome 5 (2001), pp. 261-297

This paper is concerned with the problem of simulation of (X t ) 0≤t≤T , the solution of a stochastic differential equation constrained by some boundary conditions in a smooth domain D: namely, we consider the case where the boundary ∂D is killing, or where it is instantaneously reflecting 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 order N -1 under some conditions. The construction of these schemes is based on a natural principle of local approximation of the domain into a half space, for which efficient simulations are available.

Classification : 35K20, 60-08, 60J60, 65Cxx
Keywords: killed diffusion, reflected diffusion, discretization schemes, rates of convergence, weak approximation, boundary value problems for parabolic PDE 
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     author = {Gobet, Emmanuel},
     title = {Euler schemes and half-space approximation for the simulation of diffusion in a domain},
     journal = {ESAIM: Probability and Statistics},
     pages = {261--297},
     year = {2001},
     publisher = {EDP Sciences},
     volume = {5},
     mrnumber = {1889160},
     zbl = {0998.60081},
     language = {en},
     url = {https://www.numdam.org/item/PS_2001__5__261_0/}
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Gobet, Emmanuel. Euler schemes and half-space approximation for the simulation of diffusion in a domain. ESAIM: Probability and Statistics, Tome 5 (2001), pp. 261-297. https://www.numdam.org/item/PS_2001__5__261_0/

[1] P. Baldi, Exact asymptotics for the probability of exit from a domain and applications to simulation. Ann. Probab. 23 (1995) 1644-1670. | Zbl | MR

[2] 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. | Zbl | MR

[3] V. Bally and D. Talay, 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. | Zbl | MR

[4] M. Bossy, E. Gobet and D. Talay, Computation of the invariant law of a reflected diffusion process (in preparation).

[5] P. Cattiaux, Hypoellipticité et hypoellipticité partielle pour les diffusions avec une condition frontière. Ann. Inst. H. Poincaré Probab. Statist. 22 (1986) 67-112. | Zbl | MR | Numdam

[6] P. Cattiaux, Régularité au bord pour les densités et les densités conditionnelles d'une diffusion réfléchie hypoelliptique. Stochastics 20 (1987) 309-340. | Zbl

[7] C. Constantini, B. Pacchiarotti and F. Sartoretto, Numerical approximation for functionnals of reflecting diffusion processes. SIAM J. Appl. Math. 58 (1998) 73-102. | Zbl | MR

[8] 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).

[9] M. Freidlin, Functional integration and partial differential equations. Ann. of Math. Stud. Princeton University Press (1985). | Zbl | MR

[10] D. Gilbarg and N.S. Trudinger, Elliptic partial differential equations of second order. Springer Verlag (1977). | Zbl | MR

[11] E. Gobet, Schémas d'Euler pour diffusion tuée. Application aux options barrière, Ph.D. Thesis. Université Denis Diderot Paris 7 (1998).

[12] E. Gobet, Euler schemes for the weak approximation of killed diffusion. Stochastic Process. Appl. 87 (2000) 167-197. | Zbl | MR

[13] E. Gobet, 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). | Zbl | MR

[14] E. Hausenblas, 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. | Zbl | MR

[15] S. Kanagawa and Y. Saisho, Strong approximation of reflecting Brownian motion using penalty method and its application to computer simulation. Monte Carlo Methods Appl. 6 (2000) 105-114. | Zbl | MR

[16] 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. | Zbl | MR

[17] 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). | Zbl

[18] D. Lépingle, Un schéma d'Euler pour équations différentielles stochastiques réfléchies. C. R. Acad. Sci. Paris Sér. I Math. 316 (1993) 601-605. | Zbl

[19] D. Lépingle, Euler scheme for reflected stochastic differential equations. Math. Comput. Simulation 38 (1995) 119-126. | Zbl | MR

[20] P.L. Lions and A.S. Sznitman, Stochastic differential equations with reflecting boundary conditions. Comm. Pure Appl. Math. 37 (1984) 511-537. | Zbl | MR

[21] Y. Liu, Numerical approaches to reflected diffusion processes. Technical Report (1993).

[22] J.L. Menaldi, Stochastic variational inequality for reflected diffusion. Indiana Univ. Math. J. 32 (1983) 733-744. | Zbl | MR

[23] G.N. Milshtein, 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. | Zbl | MR

[24] C. Miranda, Partial differential equations of elliptic type. Springer, New York (1970). | Zbl | MR

[25] E. Pardoux and R.J. Williams, Symmetric reflected diffusions. Ann. Inst. H. Poincaré Probab. Statist. 30 (1994) 13-62. | Zbl | MR | Numdam

[26] R. Pettersson, Approximations for stochastic differential equations with reflecting convex boundaries. Stochastic Process. Appl. 59 (1995) 295-308. | Zbl | MR

[27] R. Pettersson, Penalization schemes for reflecting stochastic differential equations. Bernoulli 3 (1997) 403-414. | Zbl | MR

[28] D. Revuz and M. Yor, Continuous martingales and Brownian motion, 2nd Ed. Springer, Berlin, Grundlehren Math. Wiss. 293 (1994). | Zbl | MR

[29] Y. Saisho, Stochastic differential equations for multi-dimensional domain with reflecting boundary. Probab. Theory Related Fields 74 (1987) 455-477. | Zbl | MR

[30] J. Serrin, The problem of Dirichlet for quasilinear elliptic differential equations with many independent variables. Philos. Trans. Roy. Soc. London Ser. A 264 (1969) 413-496. | Zbl | MR

[31] L. Slomiński, On approximation of solutions of multidimensional SDEs with reflecting boundary conditions. Stochastic Process. Appl. 50 (1994) 197-219. | Zbl | MR

[32] D. Talay and L. Tubaro, Expansion of the global error for numerical schemes solving stochastic differential equations. Stochastic Anal. Appl. 8-4 (1990) 94-120. | Zbl | MR

[33] R.J. Williams, Semimartingale reflecting Brownian motions in the orthant, Stochastic networks. Springer, New York (1995) 125-137. | Zbl | MR