# A Donsker theorem to simulate one-dimensional processes with measurable coefficients

ESAIM: Probability and Statistics (2007)

- Volume: 11, page 301-326
- ISSN: 1292-8100

## Access Full Article

top## Abstract

top## How to cite

topÉtoré, Pierre, and Lejay, Antoine. "A Donsker theorem to simulate one-dimensional processes with measurable coefficients." ESAIM: Probability and Statistics 11 (2007): 301-326. <http://eudml.org/doc/250096>.

@article{Étoré2007,

abstract = {
In this paper, we prove a Donsker theorem for one-dimensional processes generated by an operator with measurable
coefficients. We construct a random walk on any grid on the state space, using the transition probabilities of the approximated process, and the conditional average times it spends on each cell of the grid. Indeed we can compute these
quantities by solving some suitable
elliptic PDE problems.
},

author = {Étoré, Pierre, Lejay, Antoine},

journal = {ESAIM: Probability and Statistics},

keywords = {Monte Carlo methods; Donsker theorem; one-dimensional process;
scale function;
divergence form operators; Feynman-Kac formula; elliptic PDE problem.; donsker theorem; scale function; divergence form operators; elliptic PDE problem},

language = {eng},

month = {8},

pages = {301-326},

publisher = {EDP Sciences},

title = {A Donsker theorem to simulate one-dimensional processes with measurable coefficients},

url = {http://eudml.org/doc/250096},

volume = {11},

year = {2007},

}

TY - JOUR

AU - Étoré, Pierre

AU - Lejay, Antoine

TI - A Donsker theorem to simulate one-dimensional processes with measurable coefficients

JO - ESAIM: Probability and Statistics

DA - 2007/8//

PB - EDP Sciences

VL - 11

SP - 301

EP - 326

AB -
In this paper, we prove a Donsker theorem for one-dimensional processes generated by an operator with measurable
coefficients. We construct a random walk on any grid on the state space, using the transition probabilities of the approximated process, and the conditional average times it spends on each cell of the grid. Indeed we can compute these
quantities by solving some suitable
elliptic PDE problems.

LA - eng

KW - Monte Carlo methods; Donsker theorem; one-dimensional process;
scale function;
divergence form operators; Feynman-Kac formula; elliptic PDE problem.; donsker theorem; scale function; divergence form operators; elliptic PDE problem

UR - http://eudml.org/doc/250096

ER -

## References

top- A. Bensoussan, J.L. Lions and G. Papanicolaou, Asymptotic Analysis for Periodic Structures. North-Holland, Amsterdam (1978).
- P. Billingsley, Convergence of Probabilities Measures. John Wiley & Sons (1968).
- L. Breiman, Probability. Addison-Wesley Series in Statistics (1968).
- H. Brezis, Analyse fonctionnelle. Masson (1983).
- M. Decamps, A. De Schepper and M. Goovaerts, Applications of δ-function pertubation to the pricing of derivative securities. Physica A342 (2004) 677–692.
- M. Decamps, M. Goovaerts and W. Schoutens, Self Exciting Threshold Interest Rates Model. Int. J. Theor. Appl. Finance9 (2006) 1093–1122.
- P. Étoré, On random walk simulation of one-dimensional diffusion processes with discontinuous coefficients. Electron. J. Probab.11 (2006) 249–275.
- P. Étoré, Approximation de processus de diffusion à coefficients discontinus en dimension un et applications à la simulation. Ph.D. thesis, Université Henri Poincaré, Nancy, France (2006).
- 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. INRIA research report RR-3673 (1999).
- M. Freidlin and A.D. Wentzell, Necessary and Sufficient Conditions for Weak Convergence of One-Dimensional Markov Processes. Festschrift dedicated to 70th Birthday of Professor E.B. Dynkin, Birkhäuser (1994) 95–109.
- V.V. Jikov, S.M. Kozlov and O.A. Oleinik, Homogenization of Differential Operators and Integral Functionals. Springer (1994).
- P.E. Kloeden and E. Platen, Numerical Solution of Stochastic Differential Equations. Springer, Berlin (1992).
- A. Lejay, Méthodes probabilistes pour l'homogénéisation des opérateurs sous forme divergence: cas linéaires et semi-linéaires. Ph.D. thesis, Université de Provence, Marseille, France (2000).
- A. Lejay, Stochastic Differential Equations Driven by Processes Generated by Divergence Form Operators I: A Wong-Zakai Theorem. ESAIM Probab. Stat.10 (2006) 356–379.
- A. Lejay and M. Martinez, A scheme for simulating one-dimensional diffusion processes with discontinuous coefficients. Annals Appl. Probab.16 (2006) 107–139.
- M. Martinez, Interprétations probabilistes d'opérateurs sous forme divergence et analyse de méthodes numériques probabilistes associées. Ph.D. thesis, Université de Provence, Marseille, France (2004).
- M. Martinez and D. Talay, Discrétisation d'équations différentielles stochastiques unidimensionnelles à générateur sous forme divergence avec coefficient discontinu. C.R. Acad. Sci. Paris342 (2006) 51–56.
- H. Owhadi and L. Zhang, Metric based upscaling. Commun. Pure Appl. Math. (to appear).
- J.M. Ramirez, E.A. Thomann, E.C. Waymire, R. Haggerty and B. Wood, A generalized Taylor-Aris formula and Skew Diffusion. Multiscale Model. Simul.5 (2006) 786–801.
- D. Revuz and M. Yor, Continuous Martingale and Brownian Motion. Springer, Heidelberg (1991).
- A. Rozkosz, Weak convergence of Diffusions Corresponding to Divergence Form Operators. Stochastics Stochastics Rep.57 (1996) 129–157.
- D.W. Stroock, Diffusion semigroups corresponding to uniformly elliptic divergence form operators. Springer, Lecture Notes in Mathematics, Seminaire de Probabilités XXII 1321 (1988) 316–347.
- D.W. Stroock and W. Zheng, Markov chain approximations to symmetric diffusions. Ann. Inst. H. Poincaré Probab. Statist.33 (1997) 619–649.
- V.V. Zhikov, S.M. Kozlov, O.A. Oleinik and K. T'en Ngoan, Averaging and G-convergence of Differential Operators. Russian Math. Survey34 (1979) 69–147.
- V.V. Zhikov, S.M. Kozlov and O.A. Oleinik, G-convergence of Parabolic Operators. Russian Math. Survey36 (1981) 9–60.

## NotesEmbed ?

topTo embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.