  An important family of stochastic processes arising in many areas of applied
probability is the class of L\'evy processes. Generally, such processes are not
simulatable especially for those with infinite activity. In practice, it is
common to approximate them by truncating the jumps at some cut-off size
$\varepsilon$ ($\varepsilon\searrow 0$). This procedure leads us to consider a
simulatable compound Poisson process. This paper first introduces, for this
setting, the statistical Romberg method to improve the complexity of the
classical Monte Carlo one. Roughly speaking, we use many sample paths with a
coarse cut-off $\varepsilon^{\beta},$ $\beta\in(0,1)$, and few additional
sample paths with a fine cut-off $\varepsilon$. Central limit theorems of
Lindeberg-Feller type for both Monte Carlo and statistical Romberg method for
the inferred errors depending on the parameter $\varepsilon$ are proved. This
leads to an accurate description of the optimal choice of parameters with
explicit limit variances. Afterwards, the authors propose a stochastic
approximation method of finding the optimal measure change by Esscher transform
for L\'evy processes with Monte Carlo and statistical Romberg importance
sampling variance reduction. Furthermore, we develop new adaptive Monte Carlo
and statistical Romberg algorithms and prove the associated central limit
theorems. Finally, numerical simulations are processed to illustrate the
efficiency of the adaptive statistical Romberg method that reduces at the same
time the variance and the computational effort associated to the effective
computation of option prices when the underlying asset process follows an
exponential pure jump CGMY model.
