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On the ruin probability of the generalised Ornstein–Uhlenbeck process in the cramér case

Part of: Markov processes Mathematical economics Stochastic analysis

Published online by Cambridge University Press:  14 July 2016

Damien Bankovsky ,
Claudia Klüppelberg  and
Ross Maller
Damien Bankovsky
Affiliation:
Australian National University, Mathematical Sciences Institute, Australian National University, Canberra, ACT 0200, Australia. Email address: [email protected]
Claudia Klüppelberg
Affiliation:
Technische Universität München, Center for Mathematical Sciences, and Institute for Advanced Study, Technische Universität München, 85748 Garching, Germany. Email address: [email protected]
Ross Maller
Affiliation:
Australian National University, Mathematical Sciences Institute, and School of Finance and Applied Statistics, Australian National University, Canberra, ACT 0200, Australia. Email address: [email protected]
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Abstract

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For a bivariate Lévy process (ξt,ηt)t≥ 0 and initial value V0 define the generalised Ornstein–Uhlenbeck (GOU) process Vt:=eξt (V0+∫t0 e-ξs-dηs), t≥0, and the associated stochastic integral process Zt:=∫0t e-ξs-dηs, t≥0. Let Tz:=inf{t>0: Vt<0|V0=z} and ψ(z):=P(Tz<∞) for z≥0 be the ruin time and infinite horizon ruin probability of the GOU process. Our results extend previous work of Nyrhinen (2001) and others to give asymptotic estimates for ψ(z) and the distribution of Tz as z→∞, under very general, easily checkable, assumptions, when ξ satisfies a Cramér condition.

Keywords

Exponential functionals of Lévy processesgeneralised Ornstein–Uhlenbeck processruin probabilitystochastic recurrence equation

MSC classification

Primary: 60H30: Applications of stochastic analysis (to PDE, etc.) 60J25: Continuous-time Markov processes on general state spaces 91B30: Risk theory, insurance
Secondary: 60H25: Random operators and equations
Type
Part 1. Risk Theory
Information
Journal of Applied Probability , Volume 48 , Issue A: New Frontiers in Applied Probability (Journal of Applied Probability Special Volume 48A) , August 2011 , pp. 15 - 28
Copyright
Copyright © Applied Probability Trust 2011 

References

[1] Asmussen, S., (1998). Subexponential asymptotics for stochastic processes: extremal behavior, stationary distributions and first passage probabilities. Ann. Appl. Prob. 8, 354–374.CrossRefGoogle Scholar
[2] Bankovsky, D., (2010). Conditions for certain ruin for the generalised Ornstein–Uhlenbeck process and the structure of the upper and lower bounds. Stoch. Process. Appl. 120, 255–280.Google Scholar
[3] Bankovsky, D. and Sly, A., (2009). Exact conditions for no ruin for the generalised Ornstein–Uhlenbeck process. Stoch. Process. Appl. 119, 2544–2562.CrossRefGoogle Scholar
[4] Bertoin, J., Lindner, A. and Maller, R., (2008). On continuity properties of the law of integrals of Lévy processes. In Séminaire de Probabilités XLI} (Lecture Notes Math. 1934, Springer, Berlin, pp. 137–159.Google Scholar
[5] Brokate, M., et al. (2008). On the distribution tail of an integrated risk model: a numerical approach. Insurance Math. Econom. 42, 101–106.Google Scholar
[6] Cai, J., (2004). Ruin probabilities and penalty functions with stochastic rates of interest. Stoch. Process. Appl. 112, 53–78.Google Scholar
[7] Carmona, P., Petit, F. and Yor, M., (1997). On the distribution and asymptotic results for exponential functionals of Lévy processes. In Exponential Functionals and Principal Values Related to Brownian Motion, ed. Yor, M., Rev. Mat. Iberoamericana, Madrid, pp. 73–130.Google Scholar
[8] Carmona, P., Petit, F. and Yor, M., (2001). Exponential functionals of Lévy processes. In Lévy Processes, eds Barndorff-Nielsen, O. E. et al., Birkhäuser, Boston, pp. 41–55.Google Scholar
[9] Cont, R. and Tankov, P., (2004). Financial Modelling with Jump Processes. Chapman and Hall/CRC, Boca Raton, FL.Google Scholar
[10] De Haan, L. and Karandikar, R. L., (1989). Embedding a stochastic difference equation into a continuous-time process. Stoch. Process. Appl. 32, 225–235.Google Scholar
[11] Dufresne, D., (1990). The distribution of a perpetuity, with applications to risk theory and pension funding. Scand. Actuarial J. 1990, 39–79.Google Scholar
[12] Erickson, K. B. and Maller, R. A., (2005). Generalised Ornstein–Uhlenbeck processes and the convergence of Lévy integrals. In Séminaire de Probabilités XXXVIII} (Lecture Notes Math. 1857, Springer, Berlin, pp. 70–94.Google Scholar
[13] Fasen, V., (2009). Extremes of continuous-time processes. In Handbook of Financial Time Series, eds Andersen, T. G. et al., Springer, Berlin, pp. 653–667.Google Scholar
[14] Gjessing, H. K. and Paulsen, J., (1997). Present value distributions with applications to ruin theory and stochastic equations. Stoch. Process. Appl. 71, 123–144.Google Scholar
[15] Goldie, C. M., (1991). Implicit renewal theory and tails of solutions of random equations. Ann. Appl. Prob. 1, 126–166.Google Scholar
[16] Goldie, C. M. and Maller, R. A., (2000). Stability of perpetuities. Ann. Prob. 28, 1195–1218.CrossRefGoogle Scholar
[17] Harrison, J. M., (1977). Ruin problems with compounding assets. Stoch. Process. Appl. 5, 67–79.Google Scholar
[18] Kalashnikov, V. and Norberg, R., (2002). Power tailed ruin probabilities in the presence of risky investments. Stoch. Process. Appl. 98, 211–228.Google Scholar
[19] Klüppelberg, C. and Kostadinova, R., (2008). Integrated insurance risk models with exponential Lévy investment. Insurance Math. Econom. 42, 560–577.CrossRefGoogle Scholar
[20] Klüppelberg, C. and Stadtmüller, U., (1998). Ruin probabilities in the presence of heavy-tails and interest rates. Scand. Actuarial J. 1998, 49–58.Google Scholar
[21] Klüppelberg, C., Lindner, A. and Maller, R., (2004). A continuous time GARCH process driven by a Lévy process: stationarity and second order behaviour. J. Appl. Prob. 41, 601–622.Google Scholar
[22] Konstantinides, D. G. and Mikosch, T., (2005). Large deviations and ruin probabilities for solutions to stochastic recurrence equations with heavy-tailed innovations. Ann. Prob. 33, 1992–2035.CrossRefGoogle Scholar
[23] Lindner, A. and Maller, R. A., (2005). Lévy integrals and the stationarity of generalised Ornstein-Uhlenbeck processes. Stoch. Process. Appl. 115, 1701–1722.CrossRefGoogle Scholar
[24] Maller, R. A., Müller, G. and Szimayer, A., (2009). Ornstein-Uhlenbeck processes and extensions. In Handbook of Financial Time Series, eds Andersen, T. G. et al., Springer, Berlin, pp. 421–438.CrossRefGoogle Scholar
[25] Maulik, K. and Zwart, B., (2006). Tail asymptotics for exponential functionals of Lévy processes. Stoch. Process. Appl. 116, 156–177.Google Scholar
[26] Nyrhinen, H., (2001). Finite and infinite time ruin probabilities in a stochastic economic environment. Stoch. Process. Appl. 92, 265–285.CrossRefGoogle Scholar
[27] Paulsen, J., (1993). Risk theory in a stochastic economic environment. Stoch. Process. Appl. 46, 327–361.Google Scholar
[28] Paulsen, J., (1998). Sharp conditions for certain ruin in a risk process with stochastic return on investments. Stoch. Process. Appl. 75, 135–148.Google Scholar
[29] Paulsen, J., (2002). On Cramér-like asymptotics for risk processes with stochastic return on investments. Ann. Appl. Prob. 12, 1247–1260.Google Scholar
[30] Paulsen, J. and Hove, A., (1999). Markov chain Monte Carlo simulation of the distribution of some perpetuities. Adv. Appl. Prob. 31, 112–134.Google Scholar
[31] Protter, P. E., (2004). Stochastic Integration and Differential Equations, 2nd edn. Springer, Berlin.Google Scholar
[32] Sato, K.-I., (1999). Lévy Processes and Infinitely Divisible Distributions. Cambridge University Press.Google Scholar
[33] Vervaat, W., (1979). On a stochastic difference equation and a representation of nonnegative infinitely divisible random variables. Adv. Appl. Prob. 11, 750–783.Google Scholar
[34] Yin, C. and Chiu, S. N., (2004). A diffusion perturbed risk process with stochastic return on investments. Stoch. Anal. Appl. 22, 341–353.Google Scholar
[35] Yor, M., (2001). Exponential Functionals of Brownian Motion and Related Processes. Springer, Berlin.Google Scholar
[36] Yuen, K. C., Wang, G. and Ng, K. W., (2004). Ruin probabilities for a risk process with stochastic return on investments. Stoch. Process. Appl. 110, 259–274.Google Scholar