Just as the Riemann–Stieltjes integral can be equivalently defined as a Lebesgue integral with the corresponding Lebesgue–Stieltjes measure, I am looking for the corresponding results for the relationship between Stochastic integrals (both Ito and Stratonovich) and Bochner Integrals (or any other integral taking values in Banach spaces). I'm also looking for results on the corresponding measure defined on the relevant Banach Spaces.
Information pertaining to Stochastic integrals with integration with respect to Brownian motion is more than welcome, but I am looking for results pertaining more to integrals with integration in respect to more general semi-martingales. Any references or results you can forward to me is greatly appreciated. Thanks.