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Optimizing dividends and limited capital injections via exponential approximations I
Virtualconference
Authors : Avram, Florin (Author of the conference)
CIRM (Publisher )
60G40 - Stopping times; optimal stopping problems; gambling theory
60J35 - Transition functions, generators and resolvents
60J75 - Jump processes
Additional resources :
https://www.cirm-math.com/uploads/2/6/6/0/26605521/avram_cirm.pdf
CIRM (Publisher )
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Abstract :
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
MSC Codes : 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
Additional resources :
https://www.cirm-math.com/uploads/2/6/6/0/26605521/avram_cirm.pdf
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Information on the Video
Film maker : Hennenfent, GuillaumeLanguage : English Available date : 02/11/2020 Conference Date : 02/11/2020 Subseries : Research School arXiv category : Mathematical Finance ; Optimization and Control Mathematical Area(s) : Probability & Statistics Format : MP4 (.mp4) - HD Video Time : 1:09:45 Targeted Audience : Researchers Download : https://videos.cirm-math.fr/2020-10-28_Avram.mp4 
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Information on the Event
Event Title : Jean-Morlet Chair 2020 - Research School: Quasi-Monte Carlo Methods and Applications / Chaire Jean-Morlet 2020 - Ecole: Méthode de quasi-Monte-Carlo et applicationsEvent Organizers : Rivat, Joël ; Thonhauser, Stefan ; Tichy, Robert Dates : 02/11/2020 - 07/11/2020 Event Year : 2020 Event URL : https://www.chairejeanmorlet.com/2255.html
Citation Data
DOI : 10.24350/CIRM.V.19663803Cite this video as: 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 |
See Also
- [Virtualconference] PDMPs in risk theory and QMC integration III / Author of the conference Thonhauser, Stefan.
- [Virtualconference] On the estimation of conditional quantiles - lecture 3 / Author of the conference Maume-Deschamps, Véronique.
- [Virtualconference] On the estimation of conditional quantiles - lecture 2 / Author of the conference Maume-Deschamps, Véronique.
- [Virtualconference] Stochastic differential equations / Author of the conference Leobacher, Gunther.
- [Virtualconference] Number sequences for simulation - lecture 2 / Author of the conference Ökten, Giray.
- [Virtualconference] PDMPs and Integrals PDMPs in risk theory and QMC integration II / Author of the conference Thonhauser, Stefan.
- [Virtualconference] Derivative pricing, simulation from non-uniform distributions - lecture 3 / Author of the conference Ökten, Giray.
- [Multi angle] Quasi-Monte Carlo methods and applications: introduction / Author of the conference Tichy, Robert.
- [Virtualconference] Risk theory: models and objectives PDMPs in risk theory and QMC integration I / Author of the conference Thonhauser, Stefan.
- [Virtualconference] The coordinate sampler: a non-reversible Gibbs-like MCMC sampler / Author of the conference Robert, Christian P.
- [Virtualconference] On the estimation of conditional quantiles - lecture 1 / Author of the conference Maume-Deschamps, Véronique.
- [Virtualconference] Brief introduction of Quasi-Monte Carlo Methods and their Applications / Author of the conference Leobacher, Gunther.
- [Virtualconference] Two concrete FinTech applications of QMC / Author of the conference Larcher, Gerhard.
- [Virtualconference] Number sequences for simulation - lecture 1 / Author of the conference Ökten, Giray.
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