  Pure-jump L\'evy processes are popular classes of stochastic processes which
have found many applications in finance, statistics or machine learning. In
this paper, we propose a novel family of self-decomposable L\'evy processes
where one can control separately the tail behavior and the jump activity of the
process, via two different parameters. Crucially, we show that one can sample
exactly increments of this process, at any time scale; this allows the
implementation of likelihood-free Markov chain Monte Carlo algorithms for
(asymptotically) exact posterior inference. We use this novel process in
L\'evy-based stochastic volatility models to predict the returns of stock
market data, and show that the proposed class of models leads to superior
predictive performances compared to classical alternatives.
