Total Duration of Negative Surplus for a Diffusion Surplus Process with Stochastic Return on Investments

ABSTRACT

In this paper, we consider a Brownian motion risk model with stochastic return on investments. Using the strong Markov property and exploiting the limitation idea, we derive the Laplace-Stieltjes Transform(LST) of the total duration of negative surplus. In addition, two examples are also present.

In this paper, we consider a Brownian motion risk model with stochastic return on investments. Using the strong Markov property and exploiting the limitation idea, we derive the Laplace-Stieltjes Transform(LST) of the total duration of negative surplus. In addition, two examples are also present.

Cite this paper

H. You and C. Yin, "Total Duration of Negative Surplus for a Diffusion Surplus Process with Stochastic Return on Investments,"*Applied Mathematics*, Vol. 3 No. 11, 2012, pp. 1674-1679. doi: 10.4236/am.2012.311231.

H. You and C. Yin, "Total Duration of Negative Surplus for a Diffusion Surplus Process with Stochastic Return on Investments,"

References

[1] J. Paulsen and H. K. Gjessing, “Optimal Choice of Dividend Barriers for a Risk Process with Stochastic Return on Investments,” Insurance: Mathematics and Economics, Vol. 20, No. 3, 1997, pp. 215-223. doi:10.1016/S0167-6687(97)00011-5

[2] J. Paulsen, “Risk Theory in a Stochastic Economic Environment,” Stochastic Processes and Their Applications, Vol. 46, No. 2, 1993, pp. 327-361. doi:10.1016/0304-4149(93)90010-2

[3] N. Ikeda and S. Watanabe, “Stochastic Differential Equations and Diffusion Processes,” North-Holland Publishing Company, Amsterdam, 1981.

[4] A. D. Egdio dos Reis, “How Long Is the Surplus below Zero?” Insurance: Mathematics and Economics, Vol. 12, No. 1, 1993, pp. 23-38. doi:10.1016/0167-6687(93)90996-3

[5] C. S. Zhang and R. Wu, “Total Duration of Negative Surplus for the Compound Poisson Process That Is Perturbed by Diffusion,” Insurance: Mathematics and Economics, Vol. 39, No. 3, 2002, pp. 517-532.

[6] S. N. Chiu and C. C. Yin, “On Occupation Times for a Risk Process with Reserve-Dependent Premium,” Stochastic Models, Vol. 18, No. 2, 2001, pp. 245-255. doi:10.1081/STM-120004466

[7] J. M. He, R. Wu and H. Y. Zhang, “Total Duration of Negative Surplus for the Risk Model with Debit Interest,” Statistics and Probability Letters, Vol. 79, No. 10, 2009, pp. 1320-1326. doi:10.1016/j.spl.2009.02.005

[8] W. Wang and J. M. He, “Total Duration of Negative Surplus for a Brownian Motion Risk Model with Interest,” Acta Mathematica Sinica, 2012, (Submitted).

[9] L. Breiman, “Probability,” Addison-Wesley, Reading, 1968.

[10] J. Cai, H. U. Gerber and H. L. Yang, “Optimal Dividends in an Ornstein-Uhlenbeck Type Model with Credit and Debit Interest,” North American Actuarial Journal, Vol. 10, No. 2, 2006, pp. 94-119.

[11] M. Abramowitz and I. A. Stegun, “Handbook of Mathematical Functions: With Formulas, Graphs, and Mathematical Tables United States Department of Commerce,” US Government Printing Office, Washington DC, 1972.

[1] J. Paulsen and H. K. Gjessing, “Optimal Choice of Dividend Barriers for a Risk Process with Stochastic Return on Investments,” Insurance: Mathematics and Economics, Vol. 20, No. 3, 1997, pp. 215-223. doi:10.1016/S0167-6687(97)00011-5

[2] J. Paulsen, “Risk Theory in a Stochastic Economic Environment,” Stochastic Processes and Their Applications, Vol. 46, No. 2, 1993, pp. 327-361. doi:10.1016/0304-4149(93)90010-2

[3] N. Ikeda and S. Watanabe, “Stochastic Differential Equations and Diffusion Processes,” North-Holland Publishing Company, Amsterdam, 1981.

[4] A. D. Egdio dos Reis, “How Long Is the Surplus below Zero?” Insurance: Mathematics and Economics, Vol. 12, No. 1, 1993, pp. 23-38. doi:10.1016/0167-6687(93)90996-3

[5] C. S. Zhang and R. Wu, “Total Duration of Negative Surplus for the Compound Poisson Process That Is Perturbed by Diffusion,” Insurance: Mathematics and Economics, Vol. 39, No. 3, 2002, pp. 517-532.

[6] S. N. Chiu and C. C. Yin, “On Occupation Times for a Risk Process with Reserve-Dependent Premium,” Stochastic Models, Vol. 18, No. 2, 2001, pp. 245-255. doi:10.1081/STM-120004466

[7] J. M. He, R. Wu and H. Y. Zhang, “Total Duration of Negative Surplus for the Risk Model with Debit Interest,” Statistics and Probability Letters, Vol. 79, No. 10, 2009, pp. 1320-1326. doi:10.1016/j.spl.2009.02.005

[8] W. Wang and J. M. He, “Total Duration of Negative Surplus for a Brownian Motion Risk Model with Interest,” Acta Mathematica Sinica, 2012, (Submitted).

[9] L. Breiman, “Probability,” Addison-Wesley, Reading, 1968.

[10] J. Cai, H. U. Gerber and H. L. Yang, “Optimal Dividends in an Ornstein-Uhlenbeck Type Model with Credit and Debit Interest,” North American Actuarial Journal, Vol. 10, No. 2, 2006, pp. 94-119.

[11] M. Abramowitz and I. A. Stegun, “Handbook of Mathematical Functions: With Formulas, Graphs, and Mathematical Tables United States Department of Commerce,” US Government Printing Office, Washington DC, 1972.