The Annals of Probability

Stochastic area for Brownian motion on the Sierpinski gasket

B. M. Hambly and T. J. Lyons

Full-text: Open access

Abstract

We construct a Lévy stochastic area for Brownian motion on the Sierpinski gasket. The standard approach via Itô integrals fails because this diffusion has sample paths which are far rougher than those of semimartingales. We thus provide an example demonstrating the restrictions of the semimartingale framework. As a consequence of the existence of the area one has a stochastic calculus and can solve stochastic differential equations driven by Brownian motion on the Sierpinski gasket.

Article information

Source
Ann. Probab., Volume 26, Number 1 (1998), 132-148.

Dates
First available in Project Euclid: 31 May 2002

Permanent link to this document
https://projecteuclid.org/euclid.aop/1022855414

Digital Object Identifier
doi:10.1214/aop/1022855414

Mathematical Reviews number (MathSciNet)
MR1617044

Zentralblatt MATH identifier
0936.60073

Subjects
Primary: 60J60: Diffusion processes [See also 58J65]
Secondary: 60J65: Brownian motion [See also 58J65] 60J25: Continuous-time Markov processes on general state spaces

Keywords
Stochastic area differential equations fractals

Citation

Hambly, B. M.; Lyons, T. J. Stochastic area for Brownian motion on the Sierpinski gasket. Ann. Probab. 26 (1998), no. 1, 132--148. doi:10.1214/aop/1022855414. https://projecteuclid.org/euclid.aop/1022855414


Export citation

References

  • [1] Barlow, M. T. and Perkins, E. A. (1988). Brownian motion on the Sierpinski gasket. Probab. Theory Related Fields 79 543-624.
  • [2] L´evy, P. (1948). Processus Stochastiques et Mouvement Brownien. Gauthier-Villars, Paris.
  • [3] Lyons, T. J. (1995). Differential equations driven by rough signals. Rev. Mat. Iberoamericana. To appear.
  • [4] Protter, P. (1977). On the existence, uniqueness, convergence and explosions of solutions of systems of stochastic differential equations. Ann. Probab. 5 243-261.
  • [5] Sipil¨ainen, E.-M. (1993). A pathwise view of solutions of stochastic differential equations Ph.D. dissertation, Univ. Edinburgh.
  • [6] Wong, E. and Zakai, M. (1965). On the relationship between ordinary and stochastic differential equations. Internat. J. Engrg. Sci. 3 213-229.
  • [7] Wong, E. and Zakai, M. (1965). On the convergence of ordinary integrals to stochastic integrals. Ann. Math. Statist. 36 1560-1564.