Summary: We formulate standard and multilevel Monte Carlo methods for the \(k\)th moment \(\mathbb{M}_\varepsilon^k [\xi]\) of a Banach space valued random variable \(\xi : \Omega \to E\), interpreted as an element of the \(k\)-fold injective tensor product space \(\otimes_\varepsilon^k E\). For the standard Monte Carlo estimator of \(\mathbb{M}_\varepsilon^k [\xi]\), we prove the \(k\)-independent convergence rate \(1 - \frac{ 1}{ p}\) in the \(L_q(\Omega; \otimes_\varepsilon^k E)\)-norm, provided that (i) \(\xi \in L_{kq}(\Omega; E)\) and (ii) \(q \in [p, \infty)\), where \(p \in [1,2]\) is the Rademacher type of \(E\). By using the fact that Rademacher averages are dominated by Gaussian sums combined with a version of Slepian’s inequality for Gaussian processes due to Fernique, we moreover derive corresponding results for multilevel Monte Carlo methods, including a rigorous error estimate in the \(L_q(\Omega; \otimes_\varepsilon^k E)\)-norm and the optimization of the computational cost for a given accuracy. Whenever the type of the Banach space \(E\) is \(p = 2\), our findings coincide with known results for Hilbert space valued random variables.
We illustrate the abstract results by three model problems: second-order elliptic PDEs with random forcing or random coefficient, and stochastic evolution equations. In these cases, the solution processes naturally take values in non-Hilbertian Banach spaces. Further applications, where physical modeling constraints impose a setting in Banach spaces of type \(p < 2\), are indicated.