ON WEIGHTED L²(Ω)-SPACES, THEIR DUALS AND ITÔ INTEGRATION

Abstract

We construct spaces of smooth and generalized random variables which can be considered as weighted L ²(Ω)-spaces. Itô integration for generalized stochastic processes is defined. The construction follows closely the standard L ²(Ω)-case, except for the norms which are weighted. As an application of our results, we derive a Clark-Ocone formula.

Keywords:

• Generalized random variables
• Itô integration
• Clark-Ocone formula

ACKNOWLEDGMENTS

The author is grateful for inspiring discussions with Bernt øksendal.