  This paper is concerned with risk-sensitive performance analysis for linear
quantum stochastic systems interacting with external bosonic fields. We
consider a cost functional in the form of the exponential moment of the
integral of a quadratic polynomial of the system variables over a bounded time
interval. An integro-differential equation is obtained for the time evolution
of this quadratic-exponential functional, which is compared with the original
quantum risk-sensitive performance criterion employed previously for
measurement-based quantum control and filtering problems. Using multi-point
Gaussian quantum states for the past history of the system variables and their
first four moments, we discuss a quartic approximation of the cost functional
and its infinite-horizon asymptotic behaviour. The computation of the
asymptotic growth rate of this approximation is reduced to solving two
algebraic Lyapunov equations. We also outline further approximations of the
cost functional, based on higher-order cumulants and their growth rates,
together with large deviations estimates. For comparison, an auxiliary
classical Gaussian Markov diffusion process is considered in a complex
Euclidean space which reproduces the quantum system variables at the level of
covariances but has different higher-order moments relevant to the
risk-sensitive criteria. The results of the paper are also demonstrated by a
numerical example and may find applications to coherent quantum risk-sensitive
control problems, where the plant and controller form a fully quantum
closed-loop system, and other settings with nonquadratic cost functionals.
