Limits of Attainable Accuracy for Transmission of a Parameter over a White Gaussian Channel

Assume that an unknown parameter $\theta\in[0,1]$, while a modulating signal $S_t(\theta)$ is transmitted over a channel with white Gaussian noise
$$ dX_t=S_t(\theta)dt+dW_t,\quad t\in T,\quad \theta\in\Theta=[0,1], $$
where $S_t(\theta)$ satisfies only the energy constraint
$$ \int^1_0\|S_t(\theta)\|^2d\theta\leqslant A. $$
New upper and lower bounds are obtained for the minimum possible mean $\alpha$-power risk
$$ e_\alpha(A)=\inf_{S,\hat{\theta}}e_\alpha(S,\hat{\theta}) $$
In particular, for $\alpha\geqslant3$ the exponents of the upper and lower bounds coincide.