No abstract

We investigate Bochner integrabilities of generalized Wiener functionals. We further formulate an Itô formula for a diffusion in a distributional setting, and apply it to investigate differentiability-index s and integrabilityindex p 2 for which the Bochner integral belongs to D s p .T 0 δ y (X t )dt as a Bochner integral in the space of generalized Wiener functional. We remark here the Bochner integrability seems nontrivial when y = X 0 , since δ y (X t ) no longer makes sense at t = 0. On the other hand, the local time is usually formulated as a classical Wiener functional. Hence, once the Bochner integrability is proved, a "smoothing effect" should occur in the Bochner integral T 0 δ y (X t )dt, i.e., the differentiability-index for T 0 δ y (X t )dt, should be greater than that of δ y (X t ).

We introduce a category Prob of probability spaces whose objects are all probability spaces and whose arrows correspond to measurable functions satisfying an absolutely continuous requirement. We can consider a Prob-arrow as an evolving direction of information. We introduce a contravariant functor E from Prob to Set, the category of sets. The functor E provides conditional expectations along arrows in Prob, which are generalizations of the classical conditional expectations. For a Prob-arrow f − , we introduce two concepts f − -measurability and f − -independence and investigate their interaction with conditional expectations along f − . We also show that the completion of probability spaces is naturally formulated as an endofunctor of Prob.