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Exponential convergence rate of ruin probabilities for level-dependent Lévy-driven risk processes

Part of: Stochastic analysis Mathematical economics Markov processes

Published online by Cambridge University Press:  11 December 2019

Pierre-Olivier Goffard  and
Andrey Sarantsev
Pierre-Olivier Goffard*
Affiliation:
Université Lyon 1
Andrey Sarantsev*
Affiliation:
University of Nevada, Reno
*
*Postal address: LSAF EA2429, Université Lyon 1, 50 Avenue Tony Garnier, Institut de Science Financière et d’Assurances, F-69007 Lyon, France. Email address: pierre.olivier.goffard@gmail.com
**Postal address: Department of Mathematics and Statistics, University of Nevada, Reno, 1664 North Virginia Street, NV 89557, USA. Email address: asarantsev@unr.edu
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Abstract

We find explicit estimates for the exponential rate of long-term convergence for the ruin probability in a level-dependent Lévy-driven risk model, as time goes to infinity. Siegmund duality allows us to reduce the problem to long-term convergence of a reflected jump-diffusion to its stationary distribution, which is handled via Lyapunov functions.

Keywords

Ruin probabilityuniform ergodicityLyapunov functionstochastically ordered processSiegmund duality

MSC classification

Primary: 91B30: Risk theory, insurance
Secondary: 60H10: Stochastic ordinary differential equations 60J60: Diffusion processes 60J75: Jump processes
Type
Research Papers
Information
Journal of Applied Probability , Volume 56 , Issue 4 , December 2019 , pp. 1244 - 1268
Copyright
© Applied Probability Trust 2019 

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References

Appelbaum, D (2009). Lévy Processes and Stochastic Calculus, 2nd edn (Cambridge Studies Adv. Math. 116). Cambridge University Press.CrossRefGoogle Scholar
Asmussen, S. (1984). Approximations for the probability of ruin within finite time. Scand. Actuarial J. 1984 (1), 31–57.CrossRefGoogle Scholar
Asmussen, S. and Albrecher, H. (2008). Ruin Probabilities (Adv. Ser. Statist. Sci. Appl. Prob. 14). World Scientific.Google Scholar
Asmussen, S. and Teugels, J. L. (1996). Convergence rates for M/G/1 queues and ruin problems with heavy tails. J. Appl. Prob. 33 (4), 1181–1190.CrossRefGoogle Scholar
Bakry, D., Cattiaux, P. and Guillin, A. (2008). Rate of convergence of ergodic continuous Markov processes: Lyapunov versus Poincaré. J. Funct. Anal. 254 (3), 727–759.CrossRefGoogle Scholar
Davies, P. L. (1986). Rates of convergence to the stationary distribution for k-dimensional diffusion processes. J. Appl. Prob. 23 (2), 370–384.CrossRefGoogle Scholar
Down, D., Meyn, S. P. and Tweedie, R. L. (1995). Exponential and uniform ergodicity for Markov processes. Ann. Prob. 23 (4), 1671–1691.CrossRefGoogle Scholar
Dufresne, F. and Gerber, H. (1991). Risk theory for the compound Poisson process that is perturbed by diffusion. Insurance Math. Econom. 10 (1), 51–59.CrossRefGoogle Scholar
Furrer, H. (1998). Risk processes perturbed by α-stable Lévy motion. Scand. Actuarial J. 1, 59–74.CrossRefGoogle Scholar
Ichiba, T. and Sarantsev, A. (2019). Convergence and stationary distributions for Walsh diffusions. Bernoulli 25 (4A), 2439–2478.CrossRefGoogle Scholar
Jansen, S. and Kurt, N. (2014). On the notion(s) of duality for Markov processes. Prob. Surv. 11, 59–120.CrossRefGoogle Scholar
Kamae, T., Krengel, U. and Brien, G. L. O. (1977). Stochastic inequalities on partially ordered spaces. Ann. Prob. 5 (6), 899–912.CrossRefGoogle Scholar
Karatzas, I. and Shreve, S. E. (1998). Brownian Motion and Stochastic Calculus, 2nd edn (Graduate Texts Math. 113). Springer.CrossRefGoogle Scholar
Khasminskii, R. (2012). Stochastic Stability of Differential Equations (Stoch. Model. Appl. Prob. 66). Springer.CrossRefGoogle Scholar
Kolokoltsev, V. (2011). Stochastic monotonicity and duality for one-dimensional Markov processes. Math. Notes 89 (5), 652–660.CrossRefGoogle Scholar
Kolokoltsev, V. and Lee, R. X. (2013). Stochastic duality of Markov processes: a study via generators. Stoch. Anal. Appl. 31 (6), 992–1023.CrossRefGoogle Scholar
Lévy, P. (1948). Processes Stochastiques et Mouvement Brownien. Gauthier-Villars.Google Scholar
Lund, R. B. and Tweedie, R. L. (1996). Geometric convergence rates for stochastically ordered Markov chains. Math. Operat. Res. 21 (1), 182–194.CrossRefGoogle Scholar
Lund, R. B., Meyn, S. P. and Tweedie, R. L. (1996). Computable exponential convergence rates for stochastically ordered Markov processes. Ann. Appl. Prob. 6 (1), 218–237.Google Scholar
Meyn, S. P. and Tweedie, R. L. (1993). Stability of Markovian processes II: continuous-time processes and sampled chains. Adv. Appl. Prob. 25 (3), 487–517.CrossRefGoogle Scholar
Meyn, S. P. and Tweedie, R. L. (1993). Stability of Markovian processes III: Foster–Lyapunov criteria for continuous-time processes. Adv. Appl. Prob. 25 (3), 518–548.CrossRefGoogle Scholar
Meyn, S. P. and Tweedie, R. L. (1994). Computable bounds for geometric convergence rates of Markov chains. Ann. Appl. Prob. 4 (4), 981–1011.CrossRefGoogle Scholar
Michna, Z., Palmowski, Z. and Pistorius, M. (2015). The distribution of the supremum for spectrally asymmetric Lévy processes. Electron. Commun. Prob. 20 (24), 1–20.CrossRefGoogle Scholar
Morales, M. and Schoutens, W. (2003). A risk model Driven by Lévy processes. Appl. Stoch. Models Business Industry 19 (2), 147–167.CrossRefGoogle Scholar
Prabhu, N. U. (1961). On the ruin problem of collective risk theory. Ann. Math. Statist. 4 (4), 981–1011.Google Scholar
Roberts, G. O. and Rosenthal, J. S. (1996). Quantitative bounds for convergence rates of continuous-time Markov processes. Electron. J. Prob. 1 (9), 1–21.CrossRefGoogle Scholar
Roberts, G. O. and Tweedie, R. L. (1999). Bounds on regeneration times and convergence rates for Markov chains. Stoch. Process. Appl. 80 (2), 211–229.CrossRefGoogle Scholar
Roberts, G. O. and Tweedie, R. L. (2000). Rates of convergence of stochastically monotone and continuous-time Markov models. J. Appl. Prob. 37 (2), 359–373.CrossRefGoogle Scholar
Sarantsev, A. (2016). Explicit rates of exponential convergence for reflected jump-diffusions on the half-line. ALEA Lat. Amer. J. Prob. Math. Statist. 13 (2), 1069–1093.Google Scholar
Sarantsev, A. (2017). Reflected Brownian motion in a convex polyhedral cone: tail estimates for the stationary distribution. J. Theoret. Prob. 30 (3), 1200–1223.CrossRefGoogle Scholar
Sato, K.-I. (2013). Lévy Processes and Infinitely Divisible Distributions (Cambridge Studies Adv. Math. 68). Cambridge University Press.Google Scholar
Siegmund, D. (1976). The equivalence of absorbing and reflecting barrier problems for stochastically monotone Markov processes. Ann. Prob. 4 (6), 914–924.CrossRefGoogle Scholar
Sigman, K. and Ryan, R. (2000). Continuous-time monotone stochastic recursions and duality. Adv. Appl. Prob. 32 (2), 426–445.CrossRefGoogle Scholar
Sturm, A. and Swart, J. M. (2018). Pathwise duals of monotone and additive Markov processes. J. Theoret. Prob. 31 (2), 932–983.CrossRefGoogle Scholar
Tang, Q. (2005). The finite-time ruin probability of the compound Poisson model with constant interest force. J. Appl. Prob. 42 (3), 608–619.CrossRefGoogle Scholar
Whitt, W. (2001). The reflection map with discontinuities. Math. Operat. Res. 26 (3), 447–484.CrossRefGoogle Scholar
Zhao, P. (2018). Siegmund duality for continuous time Markov chains on $\mathbb Z_+^d$ . Acta Math. Sinica 34 (9), 1460–1472.CrossRefGoogle Scholar