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

  • Pierre-Olivier Goffard (a1) and Andrey Sarantsev (a2)

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.

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Corresponding author

*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

References

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[1] Appelbaum, D (2009). Lévy Processes and Stochastic Calculus, 2nd edn (Cambridge Studies Adv. Math. 116). Cambridge University Press.
[2] Asmussen, S. (1984). Approximations for the probability of ruin within finite time. Scand. Actuarial J. 1984 (1), 3157.
[3] Asmussen, S. and Albrecher, H. (2008). Ruin Probabilities (Adv. Ser. Statist. Sci. Appl. Prob. 14). World Scientific.
[4] 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), 11811190.
[5] Bakry, D., Cattiaux, P. and Guillin, A. (2008). Rate of convergence of ergodic continuous Markov processes: Lyapunov versus Poincaré. J. Funct. Anal. 254 (3), 727759.
[6] Davies, P. L. (1986). Rates of convergence to the stationary distribution for k-dimensional diffusion processes. J. Appl. Prob. 23 (2), 370384.
[7] Down, D., Meyn, S. P. and Tweedie, R. L. (1995). Exponential and uniform ergodicity for Markov processes. Ann. Prob. 23 (4), 16711691.
[8] Dufresne, F. and Gerber, H. (1991). Risk theory for the compound Poisson process that is perturbed by diffusion. Insurance Math. Econom. 10 (1), 5159.
[9] Furrer, H. (1998). Risk processes perturbed by α-stable Lévy motion. Scand. Actuarial J. 1, 5974.
[10] Ichiba, T. and Sarantsev, A. (2019). Convergence and stationary distributions for Walsh diffusions. Bernoulli 25 (4A), 24392478.
[11] Jansen, S. and Kurt, N. (2014). On the notion(s) of duality for Markov processes. Prob. Surv. 11, 59120.
[12] Kamae, T., Krengel, U. and Brien, G. L. O. (1977). Stochastic inequalities on partially ordered spaces. Ann. Prob. 5 (6), 899912.
[13] Karatzas, I. and Shreve, S. E. (1998). Brownian Motion and Stochastic Calculus, 2nd edn (Graduate Texts Math. 113). Springer.
[14] Khasminskii, R. (2012). Stochastic Stability of Differential Equations (Stoch. Model. Appl. Prob. 66). Springer.
[15] Kolokoltsev, V. (2011). Stochastic monotonicity and duality for one-dimensional Markov processes. Math. Notes 89 (5), 652660.
[16] Kolokoltsev, V. and Lee, R. X. (2013). Stochastic duality of Markov processes: a study via generators. Stoch. Anal. Appl. 31 (6), 9921023.
[17] Lévy, P. (1948). Processes Stochastiques et Mouvement Brownien. Gauthier-Villars.
[18] Lund, R. B. and Tweedie, R. L. (1996). Geometric convergence rates for stochastically ordered Markov chains. Math. Operat. Res. 21 (1), 182194.
[19] 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), 218237.
[20] Meyn, S. P. and Tweedie, R. L. (1993). Stability of Markovian processes II: continuous-time processes and sampled chains. Adv. Appl. Prob. 25 (3), 487517.
[21] 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), 518548.
[22] Meyn, S. P. and Tweedie, R. L. (1994). Computable bounds for geometric convergence rates of Markov chains. Ann. Appl. Prob. 4 (4), 9811011.
[23] Michna, Z., Palmowski, Z. and Pistorius, M. (2015). The distribution of the supremum for spectrally asymmetric Lévy processes. Electron. Commun. Prob. 20 (24), 120.
[24] Morales, M. and Schoutens, W. (2003). A risk model Driven by Lévy processes. Appl. Stoch. Models Business Industry 19 (2), 147167.
[25] Prabhu, N. U. (1961). On the ruin problem of collective risk theory. Ann. Math. Statist. 4 (4), 9811011.
[26] Roberts, G. O. and Rosenthal, J. S. (1996). Quantitative bounds for convergence rates of continuous-time Markov processes. Electron. J. Prob. 1 (9), 121.
[27] Roberts, G. O. and Tweedie, R. L. (1999). Bounds on regeneration times and convergence rates for Markov chains. Stoch. Process. Appl. 80 (2), 211229.
[28] Roberts, G. O. and Tweedie, R. L. (2000). Rates of convergence of stochastically monotone and continuous-time Markov models. J. Appl. Prob. 37 (2), 359373.
[29] 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), 10691093.
[30] Sarantsev, A. (2017). Reflected Brownian motion in a convex polyhedral cone: tail estimates for the stationary distribution. J. Theoret. Prob. 30 (3), 12001223.
[31] Sato, K.-I. (2013). Lévy Processes and Infinitely Divisible Distributions (Cambridge Studies Adv. Math. 68). Cambridge University Press.
[32] Siegmund, D. (1976). The equivalence of absorbing and reflecting barrier problems for stochastically monotone Markov processes. Ann. Prob. 4 (6), 914924.
[33] Sigman, K. and Ryan, R. (2000). Continuous-time monotone stochastic recursions and duality. Adv. Appl. Prob. 32 (2), 426445.
[34] Sturm, A. and Swart, J. M. (2018). Pathwise duals of monotone and additive Markov processes. J. Theoret. Prob. 31 (2), 932983.
[35] Tang, Q. (2005). The finite-time ruin probability of the compound Poisson model with constant interest force. J. Appl. Prob. 42 (3), 608619.
[36] Whitt, W. (2001). The reflection map with discontinuities. Math. Operat. Res. 26 (3), 447484.
[37] Zhao, P. (2018). Siegmund duality for continuous time Markov chains on $\mathbb Z_+^d$ . Acta Math. Sinica 34 (9), 14601472.

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

  • Pierre-Olivier Goffard (a1) and Andrey Sarantsev (a2)

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