×

zbMATH — the first resource for mathematics

Optimal dividends and capital injections for a spectrally positive Lévy process. (English) Zbl 1362.93171
Summary: This paper investigates an optimal dividend and capital injection problem for a spectrally positive Lévy process, where the dividend rate is restricted. Both the ruin penalty and the costs from the transactions of capital injection are considered. The objective is to maximize the total value of the expected discounted dividends, the penalized discounted capital injections before ruin, and the expected discounted ruin penalty. By the fluctuation theory of Lévy processes, the optimal dividend and capital injection strategy is obtained. We also find that the optimal return function can be expressed in terms of the scale functions of Lévy processes. Besides, a series of numerical examples are provided to illustrate our consults.

MSC:
93E20 Optimal stochastic control
60G51 Processes with independent increments; Lévy processes
91G80 Financial applications of other theories
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] B. Avanzi, Optimal dividends in the dual model,, Insurance: Mathematics and Economics, 41, 111, (2007) · Zbl 1131.91026
[2] B. Avanzi, Optimal dividends in the dual model with diffusion,, Astin Bulletin, 38, 653, (2008) · Zbl 1274.91463
[3] B. Avanzi, Optimal dividends and capital injections in the dual model with diffusion,, ASTIN Bulletin, 41, 611, (2011) · Zbl 1242.91089
[4] E. Bayraktar, On optimal dividends in the dual model,, ASTIN Bulletin, 43, 359, (2013) · Zbl 1283.91192
[5] E. Bayraktar, Optimal dividends in the dual model under transaction costs,, Insurance: Mathematics and Economics, 54, 133, (2014) · Zbl 1294.91071
[6] J. Bertoin, <em>Lévy Processes</em>,, Cambridge Tracts in Mathematics, (1996) · Zbl 0861.60003
[7] T. Chan, Smoothness of scale functions for spectrally negative Lévy processes,, Probability Theory and Related Fields, 150, 129, (2011) · Zbl 1259.60050
[8] M. Egami, Phase-type fitting of scale functions for spectrally negative Lévy process,, Journal of Computational and Applied Mathematics, 264, 1, (2014) · Zbl 1291.60094
[9] W. Fleming, <em>Controlled Markov Processes and Viscosity Solutions</em>,, \(2^{nd}\) edition, (2006) · Zbl 1105.60005
[10] A. Kuznetsov, The theory of scale functions for spectrally negative Lévy processes,, Lévy Matters II, 97, (2013) · Zbl 1261.60047
[11] A. E. Kyprianou, <em>Introductory Lectures on Fluctuations of Lévy Processes with Applications</em>,, Universitext, (2006) · Zbl 1104.60001
[12] Z. Liang, Dividends and reinsurance under a penalty for ruin,, Insurance: Mathematics and Economics, 50, 437, (2012) · Zbl 1236.91086
[13] X. Peng, Optimal dividend and equity issuance problem with proportional and fixed transaction costs,, Insurance: Mathematics and Economics, 51, 576, (2012) · Zbl 1285.91065
[14] N. Scheer, Optimal dividend strategies in a cramér-lundberg model with capital injections and administration costs,, European Actuarial Journal, 1, 57, (2011) · Zbl 1222.91026
[15] D. Yao, Optimal risk and dividend control problem with fixed costs and salvage value: Variance premium principle,, Economic Modelling, 37, 53, (2014)
[16] D. Yao, Optimal dividend and capital injection problem in the dual model with proportional and fixed transaction costs,, European Journal of Operational Research, 211, 568, (2011) · Zbl 1237.91143
[17] D. Yao, Optimal dividend and capital injection strategy with fixed costs and restricted dividend rate for a dual model,, Journal of Industrial and Management Optimization, 10, 1235, (2014) · Zbl 1281.93108
[18] C. Yin, On the optimal dividend problem for a spectrally positive Lévy process., ASTIN Bulletin, 635, (2014)
[19] Y. Zhao, Optimal dividends and capital injections in the dual model with a random time horizon,, Journal of Optimization Theory and Applications, 167, 272, (2014) · Zbl 1341.49021
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.