Les processus de Dirichlet en tant qu’espace de Banach. (Dirichlet processes as Banach spaces). (French) Zbl 0602.60069

Bertoin, Jean

Stochastics 18, 155-168 (1986).

What is called a Dirichlet process is any sum of a square integrable martingale and a process of vanishing quadratic variation (plus some natural assumptions of adaptivity and continuity), but when the last is defined the partitions of the time interval [0,1] are supposed to be defined by stopping times rather than fixed times. The class \({\mathcal D}\) of all such processes is proved to be a Banach space in which semimartingales are dense. When the process \(X_ t\), \(t\in [0,1]\), and the function F obey certain conditions it is proved that \(F(X_ t)- F(X_ 0)\) is a Dirichlet process and its martingale part is explicitly given. A necessary condition and a sufficient condition is found for a function f so that \(f(B_ t)\) is a Dirichlet process \((B_ t\) is the Brownian motion).

Reviewer: O.Enchev

MSC:

60J60	Diffusion processes
60J65	Brownian motion
60G44	Martingales with continuous parameter

Keywords:

functionals of Brownian motion; Dirichlet process; sum of a square integrable martingale; stopping times

References:

[1]	Yor M., Sur la transformee de Hilbert des temps locaux Browniens, et une extension de la formule d’Ito. Semipaire le Probabilites XVI, Lecture NOtes in Mathematics (1983) · Zbl 0495.60080
[2]	Wang A. T., Annals of Probability 5 (1977)
[3]	Fukushima M., Dirichlet Forms and Markov Processes, (1980) · Zbl 0422.31007
[4]	Follmer H., Lecture Notes in Mathematics 851 (1981)
[5]	Stein E. W., Singular Integral and Differentiability Property of Functions, (1970)