We use an extended theory of integral that generalizes the integration of vector valued functions with respect to non-negative, monotonic, countably subadditive set functions, in order to introduce a new approach to stochastic integral. With such an approach, we will explore the possible extension of the theory of stochastic integration to the more general setting of integrable processes taking values in normed vector spaces. We show that our approach makes applications possible to stochastic processes that are not necessarily square integrable, nor even measurable. Such an extension generally consolidates the typical and classical results obtained for the standard scalar case.

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