  We consider the estimation of parameters encoded in the measurement record of
a continuously monitored quantum system in the jump unraveling. This unraveling
picture corresponds to a single-shot scenario, where information is
continuously gathered. Here, it is generally difficult to assess the precision
of the estimation procedure via the Fisher Information due to intricate
temporal correlations and memory effects. In this paper we provide a full set
of solutions to this problem. First, for multi-channel renewal processes we
relate the Fisher Information to an underlying Markov chain and derive a easily
computable expression for it. For non-renewal processes, we introduce a new
algorithm that combines two methods: the monitoring operator method for
metrology and the Gillespie algorithm which allows for efficient sampling of a
stochastic form of the Fisher Information along individual quantum
trajectories. We show that this stochastic Fisher Information satisfies useful
properties related to estimation in the single-shot scenario. Finally, we
consider the case where some information is lost in data
compression/post-selection, and provide tools for computing the Fisher
Information in this case. All scenarios are illustrated with instructive
examples from quantum optics and condensed matter.
