English
Optimizing dividends and limited capital injections via exponential approximations I
Virtualconference
Auteurs : Avram, Florin (Auteur de la Conférence)
CIRM (Editeur )
60G40 - Stopping times; optimal stopping problems; gambling theory
60J35 - Transition functions, generators and resolvents
60J75 - Jump processes
Ressources complémentaires :
https://www.cirm-math.com/uploads/2/6/6/0/26605521/avram_cirm.pdf
CIRM (Editeur )
Loading the player...
|
Résumé :
The recent papers Gajek-Kucinsky (2017), Avram-Goreac-LiWu (2020) investigated the control problem of optimizing dividends when limiting capital injections by bankruptcy is taken into consideration. The first paper works under the spectrally negative Levy model; the second works under the Cramer-Lundberg model with exponential jumps, where the results are considerably more explicit.
The first talk extends, exploiting the W-Z scale functions, results of Gajek-Kucinsky (2017) to the case when a final penalty is taken into consideration as well. This requires the introduction of new scale and Gerber-Shiu functions.
The second talk illustrates the fact that quite reasonable approximations of the general problem may be obtained using the exponential particular case studied in Avram-Goreac-LiWu (2020). We start by experimenting with de Vylder type approximations for the scale function $W_q(x)$; this amounts essentially to replacing our process by one with exponential jumps and cleverly crafted parameters based on the first three moments of the claims. We show that very good approximations may be obtained for two fundamental objects of interest: the growth exponent $\Phi_q$ of the scale function $W_q(x)$, and the (last) global minimum of $W_q'(x)$, which is fundamental in the de Finetti barrier problem. Turning then to the dividends and limited capital injections problem, we show that a new exponential approximation specific to this problem achieves very good results: it consists in plugging into the objective function for exponential claims the exact "non-exponential ingredients" (scale functions and, survival and mean functions) of our non-exponential examples.
Keywords : Lokka-Zervos-type alternative; optimal dividends; capital injections; buffered eflection; Cramer-Lundberg model; absolutely continuous supersolutions; scale functions
Codes MSC : The first talk extends, exploiting the W-Z scale functions, results of Gajek-Kucinsky (2017) to the case when a final penalty is taken into consideration as well. This requires the introduction of new scale and Gerber-Shiu functions.
The second talk illustrates the fact that quite reasonable approximations of the general problem may be obtained using the exponential particular case studied in Avram-Goreac-LiWu (2020). We start by experimenting with de Vylder type approximations for the scale function $W_q(x)$; this amounts essentially to replacing our process by one with exponential jumps and cleverly crafted parameters based on the first three moments of the claims. We show that very good approximations may be obtained for two fundamental objects of interest: the growth exponent $\Phi_q$ of the scale function $W_q(x)$, and the (last) global minimum of $W_q'(x)$, which is fundamental in the de Finetti barrier problem. Turning then to the dividends and limited capital injections problem, we show that a new exponential approximation specific to this problem achieves very good results: it consists in plugging into the objective function for exponential claims the exact "non-exponential ingredients" (scale functions and, survival and mean functions) of our non-exponential examples.
Keywords : Lokka-Zervos-type alternative; optimal dividends; capital injections; buffered eflection; Cramer-Lundberg model; absolutely continuous supersolutions; scale functions
60G40 - Stopping times; optimal stopping problems; gambling theory
60J35 - Transition functions, generators and resolvents
60J75 - Jump processes
Ressources complémentaires :
https://www.cirm-math.com/uploads/2/6/6/0/26605521/avram_cirm.pdf
|
Informations sur la Vidéo
Réalisateur : Hennenfent, GuillaumeLangue : Anglais Date de publication : 02/11/2020 Date de captation : 02/11/2020 Sous collection : Research School arXiv category : Mathematical Finance ; Optimization and Control Domaine : Probability & Statistics Format : MP4 (.mp4) - HD Durée : 1:09:45 Audience : Researchers Download : https://videos.cirm-math.fr/2020-10-28_Avram.mp4 
|
Informations sur la Rencontre
Nom de la rencontre : Jean-Morlet Chair 2020 - Research School: Quasi-Monte Carlo Methods and Applications / Chaire Jean-Morlet 2020 - Ecole: Méthode de quasi-Monte-Carlo et applicationsOrganisateurs de la rencontre : Rivat, Joël ; Thonhauser, Stefan ; Tichy, Robert Dates : 02/11/2020 - 07/11/2020 Année de la rencontre : 2020 URL Congrès : https://www.chairejeanmorlet.com/2255.html
Données de citation
DOI : 10.24350/CIRM.V.19663803Citer cette vidéo: Avram, Florin (2020). Optimizing dividends and limited capital injections via exponential approximations I. CIRM. Audiovisual resource. doi:10.24350/CIRM.V.19663803 URI : http://dx.doi.org/10.24350/CIRM.V.19663803 |
Voir aussi
- [Virtualconference] PDMPs in risk theory and QMC integration III / Auteur de la Conférence Thonhauser, Stefan.
- [Virtualconference] On the estimation of conditional quantiles - lecture 3 / Auteur de la Conférence Maume-Deschamps, Véronique.
- [Virtualconference] On the estimation of conditional quantiles - lecture 2 / Auteur de la Conférence Maume-Deschamps, Véronique.
- [Virtualconference] Stochastic differential equations / Auteur de la Conférence Leobacher, Gunther.
- [Virtualconference] Number sequences for simulation - lecture 2 / Auteur de la Conférence Ökten, Giray.
- [Virtualconference] PDMPs and Integrals PDMPs in risk theory and QMC integration II / Auteur de la Conférence Thonhauser, Stefan.
- [Virtualconference] Derivative pricing, simulation from non-uniform distributions - lecture 3 / Auteur de la Conférence Ökten, Giray.
- [Multi angle] Quasi-Monte Carlo methods and applications: introduction / Auteur de la Conférence Tichy, Robert.
- [Virtualconference] Risk theory: models and objectives PDMPs in risk theory and QMC integration I / Auteur de la Conférence Thonhauser, Stefan.
- [Virtualconference] The coordinate sampler: a non-reversible Gibbs-like MCMC sampler / Auteur de la Conférence Robert, Christian P.
- [Virtualconference] On the estimation of conditional quantiles - lecture 1 / Auteur de la Conférence Maume-Deschamps, Véronique.
- [Virtualconference] Brief introduction of Quasi-Monte Carlo Methods and their Applications / Auteur de la Conférence Leobacher, Gunther.
- [Virtualconference] Two concrete FinTech applications of QMC / Auteur de la Conférence Larcher, Gerhard.
- [Virtualconference] Number sequences for simulation - lecture 1 / Auteur de la Conférence Ökten, Giray.
Bibliographie
- AVRAM, Florin, CHEDOM, D. Fotso, et HORVÁTH, András. On moments based Padé approximations of ruin probabilities. Journal of computational and applied mathematics, 2011, vol. 235, no 10, p. 3215-3228. - https://doi.org/10.1016/j.cam.2011.01.008
- AVRAM, Florin, GOREAC, Dan, LI, Juan, et al. Equity Cost-Induced Dichotomy for Optimal Dividends in the Cramér-Lundberg Model. 2020. - https://hal.archives-ouvertes.fr/hal-02912757/
- AVRAM, Florin, GRAHOVAC, Danijel, et VARDAR-ACAR, Ceren. The $ W, Z $ scale functions kit for first passage problems of spectrally negative Levy processes, and applications to the optimization of dividends. arXiv preprint arXiv:1706.06841, 2017. - https://arxiv.org/abs/1706.06841
- AVRAM, Florin, HORVÁTH, Andras, PROVOST, Serge, et al. On the Padé and Laguerre–Tricomi–Weeks Moments Based Approximations of the Scale Function W and of the Optimal Dividends Barrier for Spectrally Negative Lévy Risk Processes. Risks, 2019, vol. 7, no 4, p. 121. - https://doi.org/10.3390/risks7040121
- AVRAM, Florin, KYPRIANOU, Andreas E., PISTORIUS, Martijn R., et al. Exit problems for spectrally negative Lévy processes and applications to (Canadized) Russian options. The Annals of Applied Probability, 2004, vol. 14, no 1, p. 215-238. - http://dx.doi.org/10.1214/aoap/1075828052
- AVRAM, Florin et MINCA, Andreea. Steps towards a management toolkit for central branch risk networks, using rational approximations and matrix scale functions. Modern trends in controlled stochastic processes: theory and applications, 2015, p. 263. -
- AVRAM, Florin et MINCA, Andreea. On the central management of risk networks. Advances in Applied Probability, 2017, vol. 49, no 1, p. 221-237. -
- AVRAM, Florin, PALMOWSKI, Zbigniew, PISTORIUS, Martijn R., et al. On Gerber–Shiu functions and optimal dividend distribution for a Lévy risk process in the presence of a penalty function. The Annals of Applied Probability, 2015, vol. 25, no 4, p. 1868-1935. - http://dx.doi.org/10.1214/14-AAP1038
- TAYLOR, S. J. LÉVY PROCESSES (Cambridge Tracts in Mathematics 121) By Jean Bertoin: 265 pp.,£ 35.00, ISBN 0 521 56243 0 (Cambridge University Press, 1996). Bulletin of the London Mathematical Society, 1998, vol. 30, no 2, p. 196-223. -
- DUMITRESCU, Bogdan, ŞICLERU, Bogdan C., et AVRAM, Florin. Modeling probability densities with sums of exponentials via polynomial approximation. Journal of Computational and Applied Mathematics, 2016, vol. 292, p. 513-525. - https://doi.org/10.1016/j.cam.2015.07.032
- DE VYLDER, Fl. A practical solution to the problem of ultimate ruin probability. Scandinavian Actuarial Journal, 1978, vol. 1978, no 2, p. 114-119. - https://doi.org/10.1080/03461238.1978.10419484
- GAJEK, Lesław et KUCIŃSKI, Łukasz. Complete discounted cash flow valuation. Insurance: Mathematics and Economics, 2017, vol. 73, p. 1-19. - https://doi.org/10.1016/j.insmatheco.2016.12.004
- IVANOVS, Jevgenijs et PALMOWSKI, Zbigniew. Occupation densities in solving exit problems for Markov additive processes and their reflections. Stochastic Processes and their Applications, 2012, vol. 122, no 9, p. 3342-3360. - https://doi.org/10.1016/j.spa.2012.05.016
- KUSNETZOV, A., KYPRIANOU, Andreas E., et RIVERO, Victor. The Theory of Scale Functions for Spectrally Negative Lévy Processes, Lévy Matters II. Springer Lecture Notes in Mathematics, 2013. - http://dx.doi.org/10.1007/978-3-642-31407-0
- KULENKO, Natalie et SCHMIDLI, Hanspeter. Optimal dividend strategies in a Cramér–Lundberg model with capital injections. Insurance: Mathematics and Economics, 2008, vol. 43, no 2, p. 270-278. - https://doi.org/10.1016/j.insmatheco.2008.05.013
- KYPRIANOU, Andreas E. Fluctuations of Lévy processes with applications: Introductory Lectures. Springer Science & Business Media, 2014. - http://dx.doi.org/10.1007/978-3-642-37632-0
- PISTORIUS, Martijn R. On exit and ergodicity of the spectrally one-sided Lévy process reflected at its infimum. Journal of Theoretical Probability, 2004, vol. 17, no 1, p. 183-220. - https://doi.org/10.1023/B:JOTP.0000020481.14371.37
- SUPRUN, V. N. Problem of desteuction and resolvent of a terminating process with independent increments. Ukrainian Mathematical Journal, 1976, vol. 28, no 1, p. 39-51. - https://doi.org/10.1007/BF01559226
