Electronic Journal of Probability

On random walk simulation of one-dimensional diffusion processes with discontinuous coefficients

Pierre Etoré

Full-text: Open access

PDF File (488 KB)

Abstract

In this paper, we provide a scheme for simulating one-dimensional processes generated by divergence or non-divergence form operators with discontinuous coefficients. We use a space bijection to transform such a process in another one that behaves locally like a Skew Brownian motion. Indeed the behavior of the Skew Brownian motion can easily be approached by an asymmetric random walk.

Article information

Source
Electron. J. Probab., Volume 11 (2006), paper no. 9, 249-275.

Dates
Accepted: 15 March 2006
First available in Project Euclid: 31 May 2016

Permanent link to this document
https://projecteuclid.org/euclid.ejp/1464730544

Digital Object Identifier
doi:10.1214/EJP.v11-311

Mathematical Reviews number (MathSciNet)
MR2217816

Zentralblatt MATH identifier
1112.60061

Subjects
Primary: 60J60: Diffusion processes [See also 58J65]
Secondary: 65C

Keywords
Monte Carlo methods random walk Skew Brownian motion one-dimensional process divergence form operator

Rights
This work is licensed under aCreative Commons Attribution 3.0 License.

Citation

Etoré, Pierre. On random walk simulation of one-dimensional diffusion processes with discontinuous coefficients. Electron. J. Probab. 11 (2006), paper no. 9, 249--275. doi:10.1214/EJP.v11-311. https://projecteuclid.org/euclid.ejp/1464730544


Export citation

References

  • R.F. Bass and Z.-Q. Chen. One-dimensional Stochastic Differential Equations with Singular and Degenerate Coefficients. Sankhya 67 (2005), 19-45.
    Zentralblatt MATH: 1192.60081
  • L. Breiman. Probability. Addison-Wesley Publishing Company, Reading, Mass.-London-Don Mills, Ont. (1968), ix+421 pp.
  • H. Brézis. Analyse fonctionnelle. Masson. (1983), xiv+234 pp.
  • M. Decamps, A. De Schepper and M. Goovaerts. Applications of delta-function pertubation to the pricing of derivative securities. Physica A 342:3-4 (2004), 677-692.
  • S.N. Ethier and T.G. Kurtz. Markov processes. Characterization and convergence. Wiley Series in Probability and Mathematical Statistics: Probability and Mathematical Statistics. John Wiley & Sons, Inc., New York. (1986), x+534pp.
    Zentralblatt MATH: 0592.60049
  • J.M. Harrison and L.A. Shepp. On skew Brownian Motion. The Annals of Probability 9 (1981), 309-313.
    Zentralblatt MATH: 0462.60076
    Digital Object Identifier: doi:10.1214/aop/1176994472
    Project Euclid: euclid.aop/1176994472
  • K. It and H.P. McKean. Diffusion Processes and Their Sample Paths. Springer, Berlin. (1974), xv+321 pp.
  • P.E. Kloeden and E. Platen. Numerical Solution of Stochastic Differential Equations. Springer, Berlin (1992), xxxvi+632 pp.
    Mathematical Reviews (MathSciNet): MR1214374
    Zentralblatt MATH: 0752.60043
  • O.A. Ladyzenskaja, V.A. Solonnikov and N.N. Ural'ceva. Linear and Quasilinear Equations of Parabolic Type. Translations of Mathematical Monographs, American Mathematical Society. 23 (1968), xi+648 pp.
  • O.A. Ladyzenskaja, V.J. Rivkind and N.N. Ural'ceva. The classical solvability of diffraction problems. Trudy Mat. Inst. Steklov 92 (1966), 116-146.
  • A. Lejay. Méthodes probabilistes pour l'homogénéisation des opérateurs sous forme divergence: cas linéaires et semi-linéaires. PhD thesis, Université de Provence, Marseille, France, 2000.
  • A. Lejay and M. Martinez. A scheme for simulating one-dimensional diffusion processes with discontinuous coefficients. Preprint submitted to The Annals of Applied Probability. (2006).
    Mathematical Reviews (MathSciNet): MR2209338
    Digital Object Identifier: doi:10.1214/105051605000000656
    Project Euclid: euclid.aoap/1141654283
  • J.F. Le Gall. One-dimensional stochastic differential equations involving the local time of the unknown process. Springer, Stochastic Analysis and Applications, Lect. Notes in Math. 1095 (1985) 51-82.
  • M. Martinez. Interprtations probabilistes d'oprateurs sous forme divergence et analyse de mthodes numriques probabilistes associes. PhD thesis, Universit 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. Paris 342:1 (2006), 51-56.
  • D. Revuz and M. Yor. Continuous Martingale and Brownian Motion. Springer, Heidelberg (1991), x+533 pp.
    Mathematical Reviews (MathSciNet): MR1083357
    Zentralblatt MATH: 0731.60002
  • D. W. Stroock. Aronson's Estimate for Elliptic Operators in Divergence Form. Springer, Lecture Notes in Mathematics, Seminaire de Probabilités XXII 1321 (1982), 316-347.
  • J.B. Walsh. A Diffusion with a Discontinuous Local Time. Société Mathématique de France, Astérisque 52-53 (1978), 37-45.

The Institute of Mathematical Statistics and the Bernoulli Society

Electronic Journal of Probability

New content alerts

Email RSS ToC RSS Article
  • You have access to this content.
  • You have partial access to this content.
  • You do not have access to this content.
Turn Off MathJax

What is MathJax?

More like this

+ See more