Decomposition of Dirichlet processes and its applications. (English) Zbl 0804.60044

Lyons, T. J.; Zhang, T. S.

Ann. Probab. 22, No. 1, 494-524 (1994).

In the general framework of a (local) Dirichlet form \(({\mathcal E},{\mathcal F})\) with associated Markov process \((X_ t)\) the authors define a Stratonovich integral for a forward and backward predictable process against a Dirichlet process, i.e. a process which in general is not a semimartingale. By using a decomposition of the Dirichlet process \(\widetilde {f}(X_ t)- \widetilde{f} (X_ 0)\), \(f\in{\mathcal F}\), into a forward and backward martingale the Stratonovich integral is defined in terms of Itô integrals with respect to the martingales. It turns out that the integral can be obtained as the limit of corresponding Riemann sums and the chain rule holds. As an application the authors prove tightness and convergence results for diffusion processes on an infinite- dimensional state space.

Reviewer: W.Hoh (Erlangen)

MSC:

60H05	Stochastic integrals
31C25	Dirichlet forms
60B12	Limit theorems for vector-valued random variables (infinite-dimensional case)
60J35	Transition functions, generators and resolvents
60B11	Probability theory on linear topological spaces

Keywords:

Markov process; Stratonovich integral; Itô integrals; tightness; convergence results for diffusion processes