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The optimal dividend problem in the dual model. (English) Zbl 1303.91187
Summary: We study de Finetti’s optimal dividend problem, also known as the optimal harvesting problem, in the dual model. In this model, the firm value is affected both by continuous fluctuations and by upward directed jumps. We use a fixed point method to show that the solution of the optimal dividend problem with jumps can be obtained as the limit of a sequence of stochastic control problems for a diffusion. In each problem, the optimal dividend strategy is of barrier type, and the rate of convergence of the barrier and the corresponding value function is exponential.

MSC:
91G50 Corporate finance (dividends, real options, etc.)
91B30 Risk theory, insurance (MSC2010)
93E20 Optimal stochastic control
60G51 Processes with independent increments; Lévy processes
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References:
[1] Alvarez, L. H. R. and Rakkolainen, T. A. (2009). Optimal payout policy in presence of downside risk. Math. Meth. Operat. Res. 69, 27-58. · Zbl 1189.90104
[2] Asmussen, S. and Taksar, M. (1997). Controlled diffusion models for optimal dividend pay-out. Insurance Math. Econom. 20, 1-15. · Zbl 1065.91529
[3] Avanzi, B., Gerber, H. U. and Shiu, E. S. W. (2007). Optimal dividends in the dual model. Insurance Math. Econom. 41, 111-123. · Zbl 1131.91026
[4] Avanzi, B., Shen, J. and Wong, B. (2011). Optimal dividends and capital injections in the dual model with diffusion. ASTIN Bull. 41, 611-644. · Zbl 1242.91089
[5] 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. · Zbl 1136.60032
[6] Bayraktar, E. and Egami, M. (2008). Optimizing venture capital investments in a jump diffusion model. Math. Meth. Operat. Res. 67, 21-42. · Zbl 1151.91049
[7] Bayraktar, E. and Xing, H. (2009). Pricing American options for jump diffusions by iterating optimal stopping problems for diffusions. Math. Meth. Operat. Res. 70, 505-525. · Zbl 1178.91189
[8] Bayraktar, E., Kyprianou, A. E. and Yamazaki, K. (2013). On optimal dividends in the dual model. ASTIN Bull. 43, 359-372. · Zbl 1283.91192
[9] Dai, H., Liu, Z. and Luan, N. (2010). Optimal dividend strategies in a dual model with capital injections. Math. Meth. Operat. Res. 72, 129-143. · Zbl 1194.91188
[10] Davis, M. H. A. (1993). Markov Models and Optimization (Monogr. Statist. Appl. Prob. 49 ). Chapman & Hall, London. · Zbl 0780.60002
[11] Dayanik, S., Poor, H. V. and Sezer, S. O. (2008). Multisource Bayesian sequential change detection. Ann. Appl. Prob. 18, 552-590. · Zbl 1133.62062
[12] Gugerli, U. S. (1986). Optimal stopping of a piecewise-deterministic Markov process. Stochastics 19, 221-236. · Zbl 0611.60039
[13] Loeffen, R. L. (2008). On optimality of the barrier strategy in de Finetti’s dividend problem for spectrally negative Lévy processes. Ann. Appl. Prob. 18, 1669-1680. · Zbl 1152.60344
[14] Loeffen, R. L. and Renaud, J.-F. (2010). De Finetti’s optimal dividends problem with an affine penalty function at ruin. Insurance Math. Econom. 46, 98-108. · Zbl 1231.91212
[15] Zhanblan-Pike, M. and Shiryaev, A. N. (1995). Optimization of the flow of dividends. Russian Math. Surveys 50, 257-277. · Zbl 0878.90014
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