scholarly journals Small noise asymptotics and first passage times of integrated Ornstein–Uhlenbeck processes driven by $\alpha$-stable Lévy processes

Bernoulli ◽  
2014 ◽  
Vol 20 (1) ◽  
pp. 265-281 ◽  
Author(s):  
Robert Hintze ◽  
Ilya Pavlyukevich
Keyword(s):  
Lévy Processes ◽  
Levy Processes ◽  
First Passage Times ◽  
First Passage ◽  
Small Noise ◽  
Small Noise Asymptotics ◽  
Passage Times

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

2015 ◽  
Vol 52 (1) ◽  
pp. 129-148 ◽  
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.

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

2015 ◽  
Vol 52 (01) ◽  
pp. 129-148 ◽  
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.

scholarly journals Exponential moments of first passage times and related quantities for Lévy processes

2015 ◽  
Vol 288 (17-18) ◽  
pp. 1921-1938 ◽  
Author(s):  
Frank Aurzada ◽  
Alexander Iksanov ◽  
Matthias Meiners
Keyword(s):  
Lévy Processes ◽  
Levy Processes ◽  
First Passage Times ◽  
First Passage ◽  
Exponential Moments ◽  
Passage Times

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

2013 ◽  
Vol 50 (1) ◽  
pp. 64-84 ◽  
Author(s):  
Denis Denisov ◽  
Vsevolod Shneer
Keyword(s):  
Large Deviations ◽  
Random Walks ◽  
Levy Processes ◽  
Levy Process ◽  
First Passage Times ◽  
Exact Asymptotics ◽  
First Passage ◽  
Heavy Tailed ◽  
First Time ◽  
Passage Times

We study the exact asymptotics for the distribution of the first time, τx, a Lévy process Xt crosses a fixed negative level -x. We prove that ℙ{τx >t} ~V(x) ℙ{Xt≥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{τx>t} explicitly in both light- and heavy-tailed cases.

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

2013 ◽  
Vol 50 (01) ◽  
pp. 64-84 ◽  
Author(s):  
Denis Denisov ◽  
Vsevolod Shneer
Keyword(s):  
Large Deviations ◽  
Random Walks ◽  
Levy Processes ◽  
Levy Process ◽  
First Passage Times ◽  
Exact Asymptotics ◽  
First Passage ◽  
Heavy Tailed ◽  
First Time ◽  
Passage Times

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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