One of the cores in modern probability theory is the stochastic integral introduced by K.
Ito in the 1940s. Due to the randomness and the irregularity of typical stochastic
integrators (such as the Wiener process) one can not follow a classical approach as in
calculus to define the stochastic integral.
For Hilbert spaces Ito's theory of stochastic integration in finite
dimensions can be generalised. There are several even quite early attempts to tackle
stochastic integration in more general spaces such as Banach spaces but none of them
provides the generality and powerful tool as the theory in Hilbert spaces.
In this talk, we begin with introducing the stochastic integral in Hilbert spaces based
on the classical theory and with explaining the restriction of this approach to Hilbert
spaces. We tackle the problem of stochastic integration in Banach spaces by introducing
a stochastic version of a Pettis integral. In the case of a Wiener process as an integrator,
the stochastic Pettis integrability of a function is related to the extensively studied class of
$\gamma$-radonifying operators. Surprisingly, it turns out that for more general integrators
which are non-Gaussian and discontinuous (Levy processes) such a relation can still be
established but with another subclass of radonifying operators.