  Large deviations for additive path functionals of stochastic processes have
attracted significant research interest, in particular in the context of
stochastic particle systems and statistical physics. Efficient numerical
`cloning' algorithms have been developed to estimate the scaled cumulant
generating function, based on importance sampling via cloning of rare event
trajectories. So far, attempts to study the convergence properties of these
algorithms in continuous time have only led to partial results for particular
cases. Adapting previous results from the literature of particle filters and
sequential Monte Carlo methods, we establish a first comprehensive and fully
rigorous approach to bound systematic and random errors of cloning algorithms
in continuous time. To this end we develop a method to compare different
algorithms for particular classes of observables, based on the martingale
characterization of stochastic processes. Our results apply to a large class of
jump processes on compact state space, and do not involve any time
discretization in contrast to previous approaches. This provides a robust and
rigorous framework that can also be used to evaluate and improve the efficiency
of algorithms.
