On the optimal detection of a signal in an arbitrary noise

A criterion
$$ \max_{f\in\Gamma}\frac{\mathbf M[f(\xi+s)-f(\xi)]}{\sqrt{\mathbf Df(\xi)}}\eqno(*) $$
is considered for the choice of the best functional distinguishing the processes $\xi(t)$ and $\eta(t)=\xi(t)+s(t)$, $\xi(t)$ being an arbitrary noise process and $s(t)$ the signal to be detected. Here $\Gamma$ is a class of functional on a function space $X$ which contains almost all sample paths of $\xi(t)$ and $\eta(t)$.
In the class $L^2(\xi)$ of functionals $f$ with $\mathbf Mf^2(\xi)<\infty$ the likelihood ratio, if it exists and belongs to this class, is shown to be the best but its calculation is usually a rather difficult problem. We consider criterions that are connected with the main linear and quadratic terms in $s$ of (*). We reduce the problem of choice of the best functional in the class of integral polynomials of a definite order to that of solving a system of integral equations.