Abstract: An inverse problem of a linear stochastic evolution equation with parameters was solved via the maximum likelihood estimation. All the measurements given contain vector-valued Wiener processes. The likelihood function and the minimizing functional were obtained based on the Cameron-Martin-Girsanov theorem. First, it was  proved that the solution of the equation and the functional are continuously Fréchet differentiable. Second,  the necessary condition of the existence for the minimizer and the gradient operator of the functional were given. Final, the example shows that our results are sharp in some sense.

Key words: maximum likelihood estimation, stochastic evolution equation, Wiener process, gradient operator