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First passage problems for upwards skip-free random walks via the scale functions paradigm

Part of: Mathematical economics Stochastic processes

Published online by Cambridge University Press:  07 August 2019

Florin Avram  and
Matija Vidmar
Florin Avram*
Affiliation:
University of Pau
Matija Vidmar*
Affiliation:
University of Ljubljana
*
*Postal address: Laboratoire de Mathématiques Appliquées, Université de Pau, Avenue de l’Université, 64012 Pau, France.
**Postal address: Department of Mathematics, Faculty of Mathematics and Physics, University of Ljubljana, Jadranska ulica 21, 1000 Ljubljana, Slovenia.
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Abstract

In this paper we develop the theory of the W and Z scale functions for right-continuous (upwards skip-free) discrete-time, discrete-space random walks, along the lines of the analogous theory for spectrally negative Lévy processes. Notably, we introduce for the first time in this context the one- and two-parameter scale functions Z, which appear for example in the joint deficit at ruin and time of ruin problems of actuarial science. Comparisons are made between the various theories of scale functions as one makes time and/or space continuous.

Keywords

Skip-free Markovian jump processrandom walkscale functionmartingalecompound binomial risk model

MSC classification

Primary: 60G50: Sums of independent random variables; random walks
Secondary: 91B30: Risk theory, insurance
Type
Original Article
Information
Advances in Applied Probability , Volume 51 , Issue 2 , June 2019 , pp. 408 - 424
Copyright
© Applied Probability Trust 2019 

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References

Albrecher, H. and Ivanovs, J. (2017). On the joint distribution of tax payments and capital injections for a Lévy risk model. Prob. Math. Statist. 37, 219–227.Google Scholar
Albrecher, H., Ivanovs, J. and Zhou, X. (2016). Exit identities for Lévy processes observed at Poisson arrival times. Bernoulli 22, 1364–1382.CrossRefGoogle Scholar
Avram, F. and Vidmar, M. (2018). First passage problems for upwards skip-free random walks via the $$\Phi,W,Z$$ paradigm. Preprint. Available at https://arxiv.org/abs/1708.06080v2.Google Scholar
Avram, F., Grahovac, D. and Vardar-Acar, C. (2017). The W, Z scale functions kit for first passage problems of spectrally negative Lévy processes, and applications to the optimization of dividends. Preprint. Available at https://arxiv.org/abs/1706.06841.Google Scholar
Avram, F., Kyprianou, A. and Pistorius, M. R. (2004). Exit problems for spectrally negative Lévy processes and applications to (Canadized) Russian options. Ann. Appl. Prob. 14, 215–238.Google Scholar
Avram, F., Palmowski, Z. and Pistorius, M. R. (2007). On the optimal dividend problem for a spectrally negative Lévy process. Ann. Appl. Prob. 17, 156–180.CrossRefGoogle Scholar
Avram, F., Palmowski, Z. and Pistorius, M. R. (2015). On Gerber–Shiu functions and optimal dividend distribution for a Lévy risk process in the presence of a penalty function. Ann. Appl. Prob. 25, 1868–1935.CrossRefGoogle Scholar
Banderier, C. and Flajolet, P. (2002). Basic analytic combinatorics of directed lattice paths. Theoret. Comput. Sci. 281, 37–80.CrossRefGoogle Scholar
Bertoin, J. (1997). Exponential decay and ergodicity of completely asymmetric Lévy processes in a finite interval. Ann. Appl. Prob. 7, 156–169.Google Scholar
Borovkov, A. A. (2012). Stochastic Processes in Queueing Theory. Springer Science & Business Media.Google Scholar
Brown, M., Peköz, E. A. and Ross, S. M. (2010). Some results for skip-free random walk. Prob. Eng. Inf. Sci. 24, 491–507.CrossRefGoogle Scholar
Cheng, S., Gerber, H. U. and Shiu, E. S. W. (2000). Discounted probabilities and ruin theory in the compound binomial model. Insurance Math. Econom . 26, 239–250.CrossRefGoogle Scholar
Choi, M. C. H. and Patie, P. (2019). Skip-free Markov chains. Trans. Amer. Math. Soc. 371, 7301–7342.CrossRefGoogle Scholar
Consul, P. C. and Famoye, F. (2006). Lagrangian Probability Distributions. Birkhäuser, Boston.Google Scholar
Feller, W. (1971). An Introduction to Probability Theory and its Applications, Vol. II. John Wiley, New York. Google Scholar
Gerber, H. U. (1988). Mathematical fun with ruin theory. Insurance Math. Econom . 7, 15–23.CrossRefGoogle Scholar
Gerber, H. U., Lin, X. S. and Yang, H. (2006). A note on the dividends-penalty identity and the optimal dividend barrier. ASTIN Bull . 36, 489–503.CrossRefGoogle Scholar
Gerber, H. U., Shiu, E. S. W. and Yang, H. (2010). An elementary approach to discrete models of dividend strategies. Insurance Math. Econom . 46, 109–116.CrossRefGoogle Scholar
Ivanovs, J. (2011). One-sided Markov additive processes and related exit problems. Doctroal Thesis, Universiteit van Amsterdam.Google Scholar
Ivanovs, J. and Palmowski, Z. (2012). Occupation densities in solving exit problems for Markov additive processes and their reflections. Stoch. Process. Appl. 122, 3342–3360.CrossRefGoogle Scholar
Kemperman, J. H. B. (1961). The passage problem for a stationary Markov chain. In Statistical Research Monographs, Vol. I, University of Chicago Press.Google Scholar
Kuznetsov, A., Kyprianou, A. E. and Rivero, V. (2012). The theory of scale functions for spectrally negative Lévy processes. In Lévy Matters II. Springer, Heidelberg, pp. 97–186.CrossRefGoogle Scholar
Kyprianou, A. E. (2014). Fluctuations of Lévy Processes with Applications: Introductory Lectures. Springer, Berlin.CrossRefGoogle Scholar
Loeffen, R. L., Renaud, J.-F. and Zhou, X. (2014). Occupation times of intervals until first passage times for spectrally negative Lévy processes. Stoch. Process. Appl. 124, 1408–1435.CrossRefGoogle Scholar
Lundberg, F. (1903). Approximerad framställning av sannolikhetsfunktionen. Doctoral Thesis, Akad. Afhandling. Almqvist och Wiksell, Uppsala.Google Scholar
Marchal, P. (2001). A combinatorial approach to the two-sided exit problem for left-continuous random walks. Combinatorics Prob. Comput . 10, 251–266.CrossRefGoogle Scholar
Mijatovi, A., Vidmar, M. and Jacka, S. (2015). Markov chain approximations to scale functions of Lévy processes. Stoch. Process. Appl. 125, 3932–3957.CrossRefGoogle Scholar
Pistorius, M. (2005). A potential-theoretical review of some exit problems of spectrally negative Lévy processes. In Séminaire de Probabilités XXXVIII, Springer, Berlin, pp. 30–41.CrossRefGoogle Scholar
Quine, M. P. (2004). On the escape probability for a left or right continuous random walk. Ann. Combinatorics 8, 221–223.CrossRefGoogle Scholar
Shiu, E. S. W. (1989). The probability of eventual ruin in the compound binomial model. ASTIN Bull . 19, 179–190.CrossRefGoogle Scholar
Spitzer, F. (2013). Principles of Random Walk. Springer Science and Business Media.Google Scholar
Suprun, V. N. (1976). Problem of destruction and resolvent of a terminating process with independent increments. Ukrainian Math. J . 28, 39–51.CrossRefGoogle Scholar
Takàcs, L. (1977). Combinatorial Methods in the Theory of Stochastic Processes. Robert E. Krieger Publishing Co., Huntington, NY.Google Scholar
Vidmar, M. (2015). Non-random overshoots of Lévy processes. Markov Process. Relat. Fields 21, 39–56.Google Scholar
Vidmar, M. (2018). Fluctuation theory for upwards skip-free Lévy chains. Risks 6, 24pp.CrossRefGoogle Scholar
Willmot, G. E. (1993). Ruin probabilities in the compound binomial model. Insurance Math. Econom . 12, 133–142.CrossRefGoogle Scholar
Xin, G. (2004). The ring of Malcev-Neumann series and the residue theorem. Preprint. Available at https://arxiv.org/abs/math/0405133.Google Scholar