Discussion on the paper "On Simulation and Properties of the Stable Law" by L. Devroye and L. James

We congratulate the authors for the interesting paper. The reading has been really pleasant and instructive. We discuss briefly only some of the interesting results given in Devroye and James "On simulation and properties of the stable law", 2014 with particular attention to evolution problems. The contribution of the results collected in the paper is useful in a more wide class of applications in many areas of applied mathematics

The Normal-Generalised Gamma-Pareto process: A novel pure-jump L\'evy process with flexible tail and jump-activity properties

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