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Convexity and smoothness of scale functions and de Finetti’s control problem. (English) Zbl 1188.93115
Summary: We continue the recent work of F. Avram, A. E. Kyprianou and M. R. Pistorius [Ann. Appl. Probab. 14, No. 1, 215–238 (2004; Zbl 1042.60023)] and R. L. Loeffen [Ann. Appl. Probab. 18, No. 5, 1669–1680 (2008; Zbl 1152.60344)] by showing that whenever the Lévy measure of a spectrally negative Lévy process has a density which is log-convex then the solution of the associated actuarial control problem of de Finetti is solved by a barrier strategy. Moreover, the level of the barrier can be identified in terms of the scale function of the underlying Lévy process. Our method appeals directly to very recent developments in the theory of potential analysis of subordinators and their application to convexity and smoothness properties of the relevant scale functions.

93E20 Optimal stochastic control
60J99 Markov processes
60G51 Processes with independent increments; Lévy processes
Full Text: DOI
[1] Avram, F., Kyprianou, A.E., Pistorius, M.R.: Exit problems for spectrally negative Lévy processes and applications to (Canadized) Russian options. Ann. Appl. Probab. 14, 215–238 (2004) · Zbl 1042.60023 · doi:10.1214/aoap/1075828052
[2] Avram, F., Palmowski, Z., Pistorius, M.R.: On the optimal dividend problem for a spectrally negative Lévy process. Ann. Appl. Probab. 17, 156–180 (2007) · Zbl 1136.60032 · doi:10.1214/105051606000000709
[3] Azcue, P., Muler, N.: Optimal reinsurance and dividend distribution policies in the Cramér–Lundberg model. Math. Financ. 15, 261–308 (2005) · Zbl 1136.91016 · doi:10.1111/j.0960-1627.2005.00220.x
[4] Bertoin, J.: Lévy Processes. Cambridge University Press, Cambridge (1996) · Zbl 0861.60003
[5] Bertoin, J.: Exponential decay and ergodicity of completely asymmetric Lévy processes in a finite interval. Ann. Appl. Probab. 7, 156–169 (1997) · Zbl 0880.60077 · doi:10.1214/aoap/1034625257
[6] De Finetti, B.: Su un’impostazione alternativa dell teoria collecttiva del rischio. Transactions of the XVth International Congress of Actuaries, vol. 2, pp. 433–443 (1957)
[7] Doney, R.A., Kyprianou, A.E.: Overshoots and undershoots of Lévy processes. Ann. Appl. Probab. 16, 91–106 (2006) · Zbl 1101.60029 · doi:10.1214/105051605000000647
[8] Gerber, H.U.: Entscheidungskriterien für den zusammengesetzten Poisson-Prozess. Schweiz. Ver. Versicher.math. Mitt. 69, 185–228 (1969) · Zbl 0193.20501
[9] Gripenberg, G.: On positive, nonincreasing resolvents of Volterra equations. J. Differ. Equ. 30, 380–390 (1978) · Zbl 0418.45002 · doi:10.1016/0022-0396(78)90007-4
[10] Gripenberg, G.: On Volterra equations of the first kind. Integr. Equ. Oper. Theory 3, 473–488 (1980) · Zbl 0445.45001 · doi:10.1007/BF01702311
[11] Hubalek, F., Kyprianou, A.E.: Old and new examples of scale functions for spectrally negative Lévy processes (2007). arXiv:0801.0393v1 [math.PR] · Zbl 1274.60148
[12] Huzak, M., Perman, M., Šikić, H., Vondraček, Z.: Ruin probabilities and decompositions for general perturbed risk processes. Ann. Appl. Probab. 14, 1378–1397 (2004) · Zbl 1061.60075 · doi:10.1214/105051604000000332
[13] Huzak, M., Perman, M., Šikić, H., Vondraček, Z.: Ruin probabilities for competing claim processes. J. Appl. Probab. 41, 679–690 (2004) · Zbl 1065.60100 · doi:10.1239/jap/1091543418
[14] Klüppelberg, C., Kyprianou, A.E.: On extreme ruinous behaviour of Lévy insurance risk processes. J. Appl. Probab. 43, 594–598 (2006) · Zbl 1118.60071 · doi:10.1239/jap/1152413744
[15] Klüppelberg, C., Kyprianou, A.E., Maller, R.A.: Ruin probabilities and overshoots for general Lévy insurance risk processes. Ann. Appl. Probab. 14, 1766–1801 (2004) · Zbl 1066.60049 · doi:10.1214/105051604000000927
[16] Kyprianou, A.E.: Introductory Lectures on Fluctuations of Lévy Processes with Applications. Springer, Berlin (2006) · Zbl 1104.60001
[17] Kyprianou, A.E., Palmowski, Z.: Distributional study of de Finetti’s dividend problem for a general Lévy insurance risk process. J. Appl. Probab. 44, 349–365 (2007) · Zbl 1137.60047 · doi:10.1239/jap/1183667412
[18] Kyprianou, A.E., V, Rivero.: Special, conjugate and complete scale functions for spectrally negative Lévy processes. Electron. J. Probab. 57, 1672–1701 (2008) · Zbl 1193.60064
[19] Loeffen, R.L.: On optimality of the barrier strategy in de Finetti’s dividend problem for spectrally negative Lévy processes. Ann. Appl. Probab. 18(5), 1669–1680 (2008) · Zbl 1152.60344 · doi:10.1214/07-AAP504
[20] Protter, P.: Stochastic Integration and Differential Equations, 2nd edn. Springer, Berlin (2005)
[21] Renaud, J.-F., Zhou, X.: Distribution of the dividend payments in a general Lévy risk model. J. Appl. Probab. 44, 420–427 (2007) · Zbl 1132.60041 · doi:10.1239/jap/1183667411
[22] Song, R., Vondraček, Z.: Potential theory of special subordinators and subordinate killed stable processes. J. Theor. Probab. 19, 817–847 (2006) · Zbl 1119.60063 · doi:10.1007/s10959-006-0045-y
[23] Song, R., Vondraček, Z.: Some remarks on special subordinators. Rocky M.J. Math. (2007, to appear)
[24] Song, R., Vondraček, Z.: On suprema of Lévy processes and application in risk theory. Ann. Inst. Henri Poincaré 44, 977–986 (2008) · Zbl 1178.60036 · doi:10.1214/07-AIHP142
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