This study demonstrates that the boundedness of the \( H^\infty \)-calculus for the negative discrete Laplace operator is independent of the spatial mesh size. Using this result, we deduce the discrete stochastic maximal \( L^p \)-regularity estimate for a spatial semidiscretization. Furthermore, we derive (nearly) sharp error estimates for the semidiscretization under the general spatial \( L^q \)-norms.

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