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We study the non-autonomous stochastic Cauchy problem on a real Banach space E, $dU\left(t\right)=A\left(t\right)U\left(t\right)dt+B\left(t\right)d{W}_H\left(t\right)$, t ∈ [0,T], U(0) = u₀. Here, $W_H$ is a cylindrical Brownian motion on a real separable Hilbert space H, ${\left(B\left(t\right)\right)}_{t\in [0,T]}$ are closed and densely defined operators from a constant domain (B) ⊂ H into E, ${\left(A\left(t\right)\right)}_{t\in [0,T]}$ denotes the generator of an evolution family on E, and u₀ ∈ E. In the first part, we study existence of weak and mild solutions by methods of van Neerven and Weis. Then we use a well-known factorisation method in the setting of evolution...