Complexity of Banach space valued and parametric stochastic Itô integration

We present a complexity analysis for strong approximation of Banach space valued and parameter dependent scalar stochastic Itô integration, driven by a Wiener process. Both definite and indefinite integration are considered. We analyze the Banach space valued version of the Euler-Maruyama scheme. Based on these results, we define a multilevel algorithm for the parameter dependent stochastic integration problem and show its order optimality for various input classes.