  We develop a new Monte Carlo variance reduction method to estimate the
expectation of two commonly encountered path-dependent functionals:
first-passage times and occupation times of sets. The method is based on a
recursive approximation of the first-passage time probability and expected
occupation time of sets of a Levy bridge process that relies in part on a
randomisation of the time parameter. We establish this recursion for general
Levy processes and derive its explicit form for mixed-exponential
jump-diffusions, a dense subclass (in the sense of weak approximation) of Levy
processes, which includes Brownian motion with drift, Kou's double-exponential
model and hyper-exponential jump-diffusion models. We present a highly accurate
numerical realisation and derive error estimates. By way of illustration the
method is applied to the valuation of range accruals and barrier options under
exponential Levy models and Bates-type stochastic volatility models with
exponential jumps. Compared with standard Monte Carlo methods, we find that the
method is significantly more efficient.
