An alternative approach to stochastic integration in Banach spaces

Abstract

In his 2019 article, Kalinichenko proposed an alternative way of doing stochastic integration in general separable Banach spaces [12].This way circumvents the usual UMD assumption on our separable Banach space X, and instead imposes a strict condition on the integrating process $\Phi : (0,T)\times\Omega\to \Lll(H,X)$. Namely, we require the existence of an $X$-valued Gaussian $g$ such that almost surely for all $x^*\in X^*$,
\[ \int_0^T \|\Phi(t,\omega)^* x^*\|_H^2 \ dt \leq \EE\langle g,x^*\rangle^2. \]
Most notably, this approach works in any separable Banach space. In this thesis we will take a closer look at the proofs used by Kalinichenko, and place his article in the context of the known theory on stochastic analysis in Banach spaces. We will compare the approach both to the UMD and martingale type 2 situation, and discuss the advantages and disadvantages of either strategy.

Moreover, we will compare the conditions imposed in [12] on the stochastic process to the condition of -radonification as assumed in the UMD case [25].