Articlebronze2008

Wiener-Hopf Factorization for Lévy Processes Having Positive Jumps with Rational Transforms

96 citations0/0 files storedJournal of Applied Probability

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Probability and Risk Models Probability and Statistical Research Stochastic processes and financial applications

Abstract

We show that the positive Wiener-Hopf factor of a Lévy process with positive jumps having a rational Fourier transform is a rational function itself, expressed in terms of the parameters of the jump distribution and the roots of an associated equation. Based on this, we give the closed form of the ruin probability for a Lévy process, with completely arbitrary negatively distributed jumps, and finite intensity positive jumps with a distribution characterized by a rational Fourier transform. We also obtain results for the ladder process and its Laplace exponent. A key role is played by the analytic properties of the characteristic exponent of the process and by a Baxter-Donsker-type formula for the positive factor that we derive.

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References

2003Russian and American put options under exponential phase-type Lévy models 2002Non-Gaussian Merton-Black-Scholes Theory 2002Optimal stopping and perpetual options for Lévy processes 2002Non-Gaussian Merton-Black-Scholes Theory 2004Ruin probabilities and overshoots for general Lévy insurance risk processes 2004Ruin Probabilities and Overshoots for General Lévy Insurance Risk Processes 1964The First Passage Time of a Level and the Behavior at Infinity for a Class of Processes with Independent Increments

Cited by

2012Meromorphic Lévy processes and their fluctuation identities 2009VALUATION OF CONTINUOUSLY MONITORED DOUBLE BARRIER OPTIONS AND RELATED SECURITIES 2011On the distribution of exponential functionals for Lévy processes with jumps of rational transform

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