The exit time and the dividend value function for one-dimensional diffusion processes. (English) Zbl 1470.60232

Li, Peng; Yin, Chuancun; Zhou, Ming

Abstr. Appl. Anal. 2013, Article ID 675202, 9 p. (2013).

Summary: We investigate the exit times from an interval for a general one-dimensional time-homogeneous diffusion process and their applications to the dividend problem in risk theory. Specifically, we first use Dynkin’s formula to derive the ordinary differential equations satisfied by the Laplace transform of the exit times. Then, as some examples, we solve the closed-form expression of the Laplace transform of the exit times for several popular diffusions, which are commonly used in modelling of finance and insurance market. Most interestingly, as the applications of the exit times, we create the connect between the dividend value function and the Laplace transform of the exit times. Both the barrier and threshold dividend value function are clearly expressed in terms of the Laplace transform of the exit times.

MSC:

60J70

Applications of Brownian motions and diffusion theory (population genetics, absorption problems, etc.)

References:

[1]	Asmussen, S.; Taksar, M., Controlled diffusion models for optimal dividend pay-out, Insurance, 20, 1, 1-15 (1997) · Zbl 1065.91529
[2]	Cai, J.; Gerber, H. U.; Yang, H., Optimal dividends in an Ornstein-Uhlenbeck type model with credit and debit interest, North American Actuarial Journal, 10, 2, 94-119 (2006) · Zbl 1479.91308
[3]	Fang, Y.; Wu, R., Optimal dividends in the Brownian motion risk model with interest, Journal of Computational and Applied Mathematics, 229, 1, 145-151 (2009) · Zbl 1162.91012
[4]	Yin, C. C.; Wang, H. Q., The first passage time and the dividend value function for one-dimension diffusion processes between two reflecting barriers, International Journal of Stochastic Analysis, 2012 (2012) · Zbl 1260.60164
[5]	Yin, C. C.; Shen, Y.; Wen, Y. Z., Exit problems for jump processes with applications to dividend problems, Journal of Computational and Applied Mathematics, 245, 30-52 (2013) · Zbl 1267.91076
[6]	Ditlevsen, S., A result on the first-passage time of an Ornstein-Uhlenbeck process, Statistics and Probability Letters, 77, 18, 1744-1749 (2007) · Zbl 1133.60314
[7]	Cox, J. C.; Ingersoll, J. E.; Ross, S. A., A theory of the term structure of interest rates, Econometrica, 53, 2, 385-407 (1985) · Zbl 1274.91447
[8]	Madec, Y.; Japhet, C., First passage time problem for a drifted Ornstein-Uhlenbeck process, Mathematical Biosciences, 189, 2, 131-140 (2004) · Zbl 1047.92027
[9]	Wang, L.; Pötzelberger, K., Crossing probabilities for diffusion processes with piecewise continuous boundaries, Methodology and Computing in Applied Probability, 9, 1, 21-40 (2007) · Zbl 1122.60070
[10]	Gerber, H. U.; Shiu, E. S. W., On the time value of ruin, North American Actuarial Journal, 2, 1, 48-78 (1998) · Zbl 1085.62508
[11]	Gerber, H. U.; Shiu, E. S. W., Optimal dividends: analysis with Brownian motion, North American Actuarial Journal, 8, 1, 1-20 (2004) · Zbl 1085.62122
[12]	Wang, C. W.; Yin, C. C.; Li, E. Q., On the classical risk model with credit and debit interests under absolute ruin, Statistics and Probability Letters, 80, 5-6, 427-436 (2010) · Zbl 1183.91078
[13]	Yuen, K. C.; Yin, C., On optimality of the barrier strategy for a general Lévy risk process, Mathematical and Computer Modelling, 53, 9-10, 1700-1707 (2011) · Zbl 1219.91076
[14]	Zhang, S. J.; Jin, J. M., Computation of Special Functions (2011), Nanjing, China: Nanjing University Press, Nanjing, China
[15]	Abramowitz; Stegun, M., Handbook of Mathematical Functions: With Formulas, Graphs, and Mathematical Tables (1972), Washington, DC, USA: United States Department of Commerce, US Government Printing Office, Washington, DC, USA · Zbl 0543.33001
[16]	Chi, Y.; Lin, X. S., On the threshold dividend strategy for a generalized jump-diffusion risk model, Insurance, 48, 3, 326-337 (2011) · Zbl 1218.91072
[17]	Ng, A. C. Y., On a dual model with a dividend threshold, Insurance, 44, 2, 315-324 (2009) · Zbl 1163.91441