We study space-time Hölder regularity of the solutions of the linear stochastic Cauchy problem dU t = AU t dt + dW t t ∈ 0 T U 0 = 0 where A is the generator of an analytic semigroup on a Banach space E and W is an E-valued Brownian motion. When −A admits a -bounded H -calculus, the solution is shown to have maximal regularity in the sense that U has a modification with paths in L 2 0 T −A 1 2. The results are applied to prove optimal and maximal Hölder space-time regularity for second-order parabolic stochastic partial differential equations.

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