Stochastic area for Brownian motion on the Sierpiński gasket. (English) Zbl 0936.60073

Hambly, B. M.; Lyons, T. J.

Ann. Probab. 26, No. 1, 132-148 (1998).

Summary: We construct a Lévy stochastic area for Brownian motion on the Sierpiński 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 Sierpiński gasket.

MSC:

60J60	Diffusion processes
60J65	Brownian motion
60J25	Continuous-time Markov processes on general state spaces

Keywords:

stochastic area; differential equations; fractals

References:

[1]	Barlow, M. T. and Perkins, E. A. (1988). Brownian motion on the Sierpinski gasket. Probab. Theory Related Fields 79 543-624. · Zbl 0635.60090 · doi:10.1007/BF00318785
[2]	Lévy, P. (1948). Processus Stochastiques et Mouvement Brownien. Gauthier-Villars, Paris. · Zbl 0034.22603
[3]	Lyons, T. J. (1995). Differential equations driven by rough signals. Rev. Mat. Iberoamericana. · Zbl 0923.34056 · doi:10.4171/RMI/240 · eudml:39555
[4]	Protter, P. (1977). On the existence, uniqueness, convergence and explosions of solutions of systems of stochastic differential equations. Ann. Probab. 5 243-261. · Zbl 0363.60044 · doi:10.1214/aop/1176995849
[5]	Sipiläinen, 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. · Zbl 0131.16401 · doi:10.1016/0020-7225(65)90045-5
[7]	Wong, E. and Zakai, M. (1965). On the convergence of ordinary integrals to stochastic integrals. Ann. Math. Statist. 36 1560-1564. · Zbl 0138.11201 · doi:10.1214/aoms/1177699916