Applying the Wiener-Hopf Monte Carlo Simulation Technique for Lévy Processes to Path Functionals

Abstract

In this paper we apply the recently established Wiener-Hopf Monte Carlo simulation technique for Lévy processes from Kuznetsov et al. (2011) to path functionals; in particular, first passage times, overshoots, undershoots, and the last maximum before the passage time. Such functionals have many applications, for instance, in finance (the pricing of exotic options in a Lévy model) and insurance (ruin time, debt at ruin, and related quantities for a Lévy insurance risk process). The technique works for any Lévy process whose running infimum and supremum evaluated at an independent exponential time can be sampled from. This includes classic examples such as stable processes, subclasses of spectrally one-sided Lévy processes, and large new families such as meromorphic Lévy processes. Finally, we present some examples. A particular aspect that is illustrated is that the Wiener-Hopf Monte Carlo simulation technique (provided that it applies) performs much better at approximating first passage times than a 'plain' Monte Carlo simulation technique based on sampling increments of the Lévy process.

Probabilistic solutions to nonlinear fractional differential equations of generalized Caputo and Riemann–Liouville type

“…From the point of view of numerical analysis, this representation can be exploited to obtain numerical solutions to a variety of problems by performing Monte Carlo techniques. Simulation methods have been effectively used for classical differential equations and, in recent years, different methods for evaluating path functionals of Lévy processes have been actively researched (see, e.g., [12]- [13], [23]). Finally, by using the monotonicity of the underlying processes, we obtain explicit solutions in terms of the transition densities of the Markov processes involved.…”