We consider a compound Poisson insurance risk model perturbed by diffusion with stochastic return on investment and debit interest. If the initial surplus is nonnegative, then the insurance company can invest this surplus in a risky asset and risk-free asset based on a fixed proportion. Otherwise, if the surplus is negative, then the insurance company can get the business loan. The integrodifferential equations for the function generating moments of the values of cumulative dividends are obtained for the barrier and threshold dividend strategies, respectively. The expected dividend value is obtained in the closed form in the case where the claim amount is exponentially distributed.
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References
S. Asmussen, Ruin Probabilities, World Scientific, Singapore (2000).
S. Asmussen and M. Taksar, “Controlled diffusion models for optimal dividend pay-out,” Insurance Math. Econom., 20, 1–15 (1997).
H. Bühlmann, Mathematical Methods in Risk Theory, Springer, Heidelberg (1970).
J. Cai, “On the time value of absolute ruin with debit interest,” Adv. Appl. Probab., 39, 343–359 (2007).
S. N. Chiu and C. C. Yin, “The time of ruin, the surplus prior to ruin, and the deficit at ruin for the classical risk process perturbed by diffusion,” Insurance Math. Econom., 33, No. 1, 59–66 (2003).
B. de Finetti, “Su un’ impostazione alternativa dell teoria collettiva del rischio,” Trans. XVth Intern. Congr. Actuar., 2, 433–443 (1957).
F. Dufresne and H. U. Gerber, “Risk theory for the compound Poisson process that perturbed by diffusion,” Insurance Math. Econom., 10, 51–59 (1991).
H. U. Gerber, “An extension of the renewal equation and its application in the collective theory of risk,” Scand. Actuar. J., 205–210 (1970).
H. U. Gerber, “Der Einfluss von Zins auf die Ruinwahrscheinlichkeit,” Mitt. Ver. schweiz. Versicherungsmath., 63–70 (1971).
H. U. Gerber and S.W. Shiu, “On optimal dividend strategies in the compound Poisson model,” N. Am. Actuar. J., 10, 76–93 (2006).
H. U. Gerber and H. L. Yang, “Absolute ruin probabilities in a jump-diffusion risk model with investment,” N. Am. Actuar. J., 11, No. 3, 159–169 (2008).
M. Jeanblanc-Picqu´e and A. N. Shiryaev, “Optimization of the flow of dividends,” Russ. Math. Surveys, 20, 257–277 (1995).
J. Z. Li, “Asymptotics in a time-dependent renewal risk model with stochastic return,” J. Math. Anal. Appl., 387, 1009–1023 (2012).
Y. H. Lu and R. Wu, “The differentiability of dividends function on jump-diffusion risk process with a barrier dividend strategy,” Front. Math. China, 9, No. 5, 1073–1088 (2014).
Y. H. Lu, R. Wu, and R. Xu, “The joint distributions of some actuarial diagnostics for the jump-diffusion risk process,” Acta Math. Sci. Ser. B, Eng. Ed., 30, No. 3, 664–676 (2010).
J. Paulsen, “Risk theory in a stochastic economic environment,” Stochast. Proc. Appl., 46, 327–361 (1993).
J. Paulsen, “Sharp conditions for certain ruin in a risk process with stochastic return on investments,” Stochast. Proc. Appl., 75, 135–148 (1998).
J. Paulsen, “Ruin models with investment income,” Probab. Surv., 5, 416–434 (2008).
J. Paulsen and H. K. Gjessing, “Optimal choice of dividend barriers for a risk process with stochastic return on investments,” Insurance Math. Econ., 20, 215–223 (1997).
N. Wan, “Dividend payments with a threshold strategy in the compound Poisson risk model perturbed by diffusion,” Insur. Math. Econ., 40, 509–523 (2007).
C. C. Yin and K. C. Yuen, “Optimal dividend problems for a jump-diffusion model with capital injections and proportional transaction costs,” J. Ind. Manag. Optim., 11, No. 4, 1247–1262 (2015).
C. C. Yin and Y. Z. Wen, “An extension of Paulsen–Gjessing’s risk model with stochastic return on investments,” Insurance Math. Econom., 52, 496–476 (2013).
K. C. Yuen, Y. H. Lu, and R. Wu, “The compound Poisson process perturbed by a diffusion with a threshold dividend strategy,” Appl. Stoch. Models Bus. Ind., 25, No. 1, 73–93 (2009).
J. Zhu and H. L. Yang, “Estimates for the absolute ruin probability in the compound Poisson risk model with credit and debit interest,” J. Appl. Probab., 45, No. 3, 818–830 (2008).
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Published in Ukrains’kyi Matematychnyi Zhurnal, Vol. 71, No. 5, pp. 631–644, May, 2019.
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Lu, Y.H., Li, Y.F. Dividend Payments in a Perturbed Compound Poisson Model with Stochastic Investment and Debit Interest. Ukr Math J 71, 718–734 (2019). https://doi.org/10.1007/s11253-019-01680-1
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DOI: https://doi.org/10.1007/s11253-019-01680-1