Frechet-valued martingales and stochastic integrals

Abstract

Stochastic processes with values in a separable Frechet space whose a itinuous linear functional are real-valued square integrable martingales are investigated. The coordinate measures on the Frechet space are obtained from cylinder set measures on a Hilbert space that is dense in the Frechet space. Real-valued stochastic integrals are defined from the Frechet-valued martingales using integrands from the topological dual of the aforementioned Hilbert space. An increasing process with values in the self adjoint operators on the Hilbert space plays a fundamental role in the definition of stochastic integrals. For Banach-valued Brownian motion the change of variables formula of K. Ito is generalized. A converse to the construction of the measures on the Frechet space from cylinder set measures on a Hilbert space is also obtained.