In spirit it is similar to the deterministic PDEs where one needs to use the regularity of the generator eg. its spectrum. The following presentation is from ["Stochastic Equations in Infinite Dimensions"][1]. Let's look at the linear case with additive noise 

$$d X(t) = (AX(t) + f (t))dt + BdW(t),$$

where $A: D(A) \subset H \to H$ and $B : U \to H$ are linear operators and $f$ is an $H-$valued stochastic process. For generator $A$, let $S$ be the semigroup generated by it, then 

$$X(t) = S(t)X_0 + \int_{0}^{t}S(t − s) f (s)ds + \int_{0}^{t} S(t − s)BdW(s),$$

solve the SPDE. Let $W_{A}(t):= \int_{0}^{t} S(t − s)BdW(s)$.

**Removing the White noise** Indeed, to estimate moments as the one you mentioned, I would take a look at "4.6 Basic estimates" that return to the deterministic case. For example

[![enter image description here][2]][2]


So you see here that they transfer moments for the semigroup-formulation to those of the semigroup itself up to some loss in the moments. This is proved for general $\Phi$ and so here you can insert the Holder difference too. Then in "5.3 Continuity of weak solution" they use those estimates to obtain Holder regularity  they show in theorems 5.14,5.15 a Holder regularity for them


**Using the spectrum**  Another way is using the spectral information of the operator A eg. "The case when A is self-adjoint". Using the following bounds

[![enter image description here][4]][4]


they obtain a nice Holder-moments result 
[![enter image description here][5]][5]

and then they apply Kolmogorov result for random fields theorem 3.5.

  [1]: https://www.cambridge.org/core/books/stochastic-equations-in-infinite-dimensions/17CB9F06965F01647D576C62D28049F6
  [2]: https://i.sstatic.net/mk8tV.png
  [3]: https://i.sstatic.net/iJGHa.png
  [4]: https://i.sstatic.net/UDlSF.png
  [5]: https://i.sstatic.net/rBphR.png