This paper deals with the spatial and temporal regularity of the unique Hilbert space valued mild solution to a semilinear stochastic parabolic partial differential equation with nonlinear terms that satisfy global Lipschitz conditions and certain linear growth bounds. It is shown that the mild solution has the same optimal regularity properties as the stochastic convolution. The proof is elementary and makes use of existing results on the regularity of the solution, in particular, the Hölder continuity with a non-optimal exponent.
"Optimal regularity for semilinear stochastic partial differential equations with multiplicative noise." Electron. J. Probab. 17 1 - 19, 2012. https://doi.org/10.1214/EJP.v17-2240