Asymptotics for the First Passage Times of Lévy Processes and Random Walks

We study the exact asymptotics for the distribution of the first time, τ x , a Lévy process X t crosses a fixed negative level -x. We prove that ℙ{τ x >t} ~V(x) ℙ{X t ≥0}/t as t→∞ for a certain function V(x). Using known results for the large deviations of random walks, we obtain asymptotics for ℙ{τ x >t} explicitly in both light- and heavy-tailed cases.

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

Journal of Applied Probability ◽

10.1239/jap/1429282611 ◽

2015 ◽

Vol 52 (1) ◽

pp. 129-148 ◽

Cited By ~ 2

Author(s):

Albert Ferreiro-Castilla ◽

Kees van Schaik

Keyword(s):

Monte Carlo Simulation ◽

Monte Carlo ◽

Lévy Processes ◽

Levy Processes ◽

Simulation Technique ◽

Levy Process ◽

First Passage Times ◽

First Passage ◽

Monte Carlo Simulation Technique ◽

Passage Times

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.

Download Full-text

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

Journal of Applied Probability ◽

10.1017/s0021900200012249 ◽

2015 ◽

Vol 52 (01) ◽

pp. 129-148 ◽

Cited By ~ 1

Author(s):

Albert Ferreiro-Castilla ◽

Kees van Schaik

Keyword(s):

Monte Carlo Simulation ◽

Monte Carlo ◽

Lévy Processes ◽

Levy Processes ◽

Simulation Technique ◽

Levy Process ◽

First Passage Times ◽

First Passage ◽

Monte Carlo Simulation Technique ◽

Passage Times

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.

Download Full-text