Exact temporal variations and parameter estimation for space--time fractional stochastic heat equations driven by multiplicative noise

We establish exact limits for the normalized temporal power variations of every fixed real order $\gamma \ge 2$ for the mild solution to a one-dimensional space–time fractional stochastic heat equation driven by multiplicative space–time white noise. The results include the exact normalized quadratic variation and the critical power variation of order $2\alpha \mathrm{/}(\alpha -\beta )$ as special cases. We also obtain the corresponding spatially averaged temporal variations. As applications, we construct consistent estimators for the drift parameter and for the ratio $\beta \mathrm{/}\alpha$.