On the minimax detection of imperfectly known signal in a white Gaussian noise

Let according to the hypothesis $H_0$ the observed signal $X_t$ is given by the stochastic equation
$$ dX_t=s_t dt+dW_t\qquad s_t\in S\subset L_2 [0, T], $$
where the set $S$ is known and $W_t$ is a Wiener process. Fot the alternative $H_1$ the observed signal $X_t$ is given by equation $dX_t=dW_t$. It is shown that very often instead of the set $S$ one can consider the reduced version of it. Nonasymptotic properties of maximum likelyhood ratio criteria are investigated.