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The Joint Laplace Transforms for Diffusion Occupation Times

Part of: Markov processes Stochastic processes

Published online by Cambridge University Press:  04 January 2016

Bin Li  and
Xiaowen Zhou
Bin Li*
Affiliation:
University of Iowa
Xiaowen Zhou*
Affiliation:
Concordia University
*
∗ Current address: Department of Statistics and Actuarial Science, University of Waterloo, 200 University Avenue West, Waterloo, Ontario, N2L 3G1, Canada. Email address: [email protected]
∗∗ Postal address: Department of Mathematics and Statistics, Concordia University, 1455 de Maisonneuve Blvd. West, Montreal, Quebec, H3G 1M8, Canada. Email address: [email protected]
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Abstract

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In this paper we adopt the perturbation approach of Landriault, Renaud and Zhou (2011) to find expressions for the joint Laplace transforms of occupation times for time-homogeneous diffusion processes. The expressions are in terms of solutions to the associated differential equations. These Laplace transforms are applied to study ruin-related problems for several classes of diffusion risk processes.

Keywords

Exit problemoccupation timeruin probabilitytime-homogeneous diffusion

MSC classification

Primary: 60J60: Diffusion processes
Secondary: 60G17: Sample path properties
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 45 , Issue 4 , December 2013 , pp. 1049 - 1067
Copyright
© Applied Probability Trust 

References

Albrecher, H. and Lautscham, V. (2013). From ruin to bankruptcy for compound Poisson surplus processes. ASTIN Bull. 43, 213–243.Google Scholar
Albrecher, H., Cheung, E. C. K. and Thonhauser, S. (2011). Randomized observation periods for the compound Poisson risk model dividends. ASTIN Bull. 41, 645–672.Google Scholar
Albrecher, H., Cheung, E. C. K. and Thonhauser, S. (2013). Randomized observation periods for the compound Poisson risk model: the discounted penalty function. Scand. Actuarial J. 2013, 424–452.Google Scholar
Albrecher, H., Gerber, H. U. and Shiu, E. S. W. (2011). The optimal dividend barrier in the gamma-omega model. Europ. Actuarial J. 1, 43–55.Google Scholar
Asmussen, S. and Taksar, M. (1997). Controlled diffusion models for optimal dividend pay-out. Insurance Math. Econom. 20, 1–15.Google Scholar
Borodin, A. N. and Salminen, P. (2002). Handbook of Brownian Motion—Facts and Formulae, 2nd edn. Birkhäuser, Basel.CrossRefGoogle Scholar
Cai, N., Chen, N. and Wan, X. (2010). Occupation times of Jump-diffusion processes with double exponential Jumps and the pricing of options. Math. Operat. Res. 35, 412–437.Google Scholar
Darling, D. A. and Siegert, A. J. F. (1953). The first passage problem for a continuous Markov process. Ann. Math. Statist. 24, 624–639.CrossRefGoogle Scholar
Dos Reis, A. E. (1993). How long is the surplus below zero? Insurance Math. Econom. 12, 23–38.Google Scholar
Feller, W. (1954). Diffusion processes in one dimension. Trans. Amer. Math. Soc. 77, 1–31.Google Scholar
Gerber, H. U. and Shiu, E. S. W. (2006). On optimal dividends: from reflection to refraction. J. Comput. Appl. Math. 186, 4–22.Google Scholar
Gerber, H. U., Shiu, E. S. W. and Yang, H. (2012). The Omega model: from bankruptcy to occupation times in the red. Europ. Actuarial J. 2, 259–272.CrossRefGoogle Scholar
Gīhman, Ī. Ī. and Skorohod, A. V. (1972). Stochastic Differential Equations. Springer, New York.Google Scholar
Kyprianou, A. E., Pardo, J. C. and Pérez, J. L. (2012). Occupation times of refracted Lévy processes. Preprint. Available at http://arxiv.org/abs/1205.0756v1.Google Scholar
Landriault, D., Renaud, J.-F. and Zhou, X. (2011). Occupation times of spectrally negative Lévy processes with applications. Stoch. Process. Appl. 121, 2629–2641.Google Scholar
Li, B., Tang, Q. and Zhou, X. (2013). Liquidation risk in the presence of chapter 7 and chapter 11 of the U.S. bankruptcy code. Submitted.Google Scholar
Loeffen, R. L., Renaud, J.-F. and Zhou, X. (2013). Occupation times of intervals until first passage times for spectrally negative Lévy processes. To appear in Stoch. Process. Appl. Google Scholar
Pitman, J. and Yor, M. (1999). Laplace transforms related to excursions of a one-dimensional diffusion. Bernoulli 5, 249–255.Google Scholar
Pitman, J. and Yor, M. (2003). Hitting, occupation and inverse local times of one-dimensional diffusions: martingale and excursion approaches. Bernoulli 9, 1–24.Google Scholar
Prokhorov, Y. V. and Shiryaev, A. N. (eds) (1998). Probability theory III. Stochastic calculus. Springer, Berlin.Google Scholar
Simpson, D. J. W. and Kuske, R. (2013). The positive occupation time of Brownian motion with two-valued drift and asymptotic dynamics of sliding motion with noise. Preprint. Available at http://arxiv.org/abs/1204.5985v2.Google Scholar
Stroock, D. W. and Varadhan, S. R. S. (1969). Diffusion processes with continuous coefficients. I. Commun. Pure Appl. Math. 22, 345–400.Google Scholar
Yin, C. and Yuen, K. (2013). Some exact Joint laws associated with spectrally negative Lévy processes and applications to insurance risk theory. Preprint. Available at http://arxiv.org/abs1101.0445v3.Google Scholar
Zhang, C. and Wu, R. (2002). Total duration of negative surplus for the compound Poisson process that is perturbed by diffusion. J. Appl. Prob. 39, 517–532.Google Scholar