Hambly, B. M.; Lyons, T. J. Stochastic area for Brownian motion on the Sierpiński gasket. (English) Zbl 0936.60073 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. Cited in 16 Documents MSC: 60J60 Diffusion processes 60J65 Brownian motion 60J25 Continuous-time Markov processes on general state spaces Keywords:stochastic area; differential equations; fractals PDF BibTeX XML Cite \textit{B. M. Hambly} and \textit{T. J. Lyons}, Ann. Probab. 26, No. 1, 132--148 (1998; Zbl 0936.60073) Full Text: DOI OpenURL 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 [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 [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 [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 [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 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.