  This paper is concerned with linear stochastic systems whose output is a
stationary Gaussian random process related by an integral operator to a
standard Wiener process at the input. We consider a performance criterion which
involves the trace of an analytic function of the spectral density of the
output process. This class of "covariance-analytic" cost functionals includes
the usual mean square and risk-sensitive criteria as particular cases. Due to
the presence of the "cost-shaping" analytic function, the performance criterion
is related to higher-order Hardy-Schatten norms of the system transfer
function. These norms have links with the asymptotic properties of cumulants of
finite-horizon quadratic functionals of the system output and satisfy
variational inequalities pertaining to system robustness to statistically
uncertain inputs. In the case of strictly proper finite-dimensional systems,
governed in state space by linear stochastic differential equations, we develop
a method for recursively computing the Hardy-Schatten norms through a recently
proposed technique of rearranging cascaded linear systems, which resembles the
Wick ordering of annihilation and creation operators in quantum mechanics. The
resulting computational procedure involves a recurrence sequence of solutions
to algebraic Lyapunov equations and represents the covariance-analytic cost as
the squared $\mathcal{H}_2$-norm of an auxiliary cascaded system. These results
are also compared with an alternative approach which uses higher-order
derivatives of stabilising solutions of parameter-dependent algebraic Riccati
equations.
