  We analyse the problem of estimating a scalar parameter $g$ that controls the
Hamiltonian of a quantum system subject to Markovian noise. Specifically, we
place bounds on the growth rate of the quantum Fisher information with respect
to $g$, in terms of the Lindblad operators and the $g$-derivative of the
Hamiltonian $H$. Our new bounds are not only more generally applicable than
those in the literature -- for example, they apply to systems with
time-dependent Hamiltonians and/or Lindblad operators, and to
infinite-dimensional systems such as oscillators -- but are also tighter in the
settings where previous bounds do apply. We derive our bounds directly from the
stochastic master equation describing the system, without needing to discretise
its time evolution. We also use our results to investigate how sensitive a
single detection system can be to signals with different time dependences. We
demonstrate that the sensitivity bandwidth is related to the quantum
fluctuations of $\partial H/\partial g$, illustrating how 'non-classical'
states can enhance the range of signals that a system is sensitive to, even
when they cannot increase its peak sensitivity.
