On the Reflected Geometric Brownian Motion with Two Barriers

Lidong Zhang, Ziping Du

.

DOI: 10.4236/iim.2010.23034                

Abstract

In this paper, we are concerned with Re?ected Geometric Brownian Motion (RGBM) with two barriers. And the stationary distribution of RGBM is derived by Markovian in?nitesimal Generator method. Consequently the ?rst passage time of RGBM is also discussed.

Keywords

Geometric Brownian Motion, Stationary Distribution, First Passage Time

Conflicts of Interest

The authors declare no conflicts of interest.

References

[1]	J. M. Harrison, “Brownian motion and stochastic flow systems,” John Wiley & Sons Ltd., New York, 1986.
[2]	X. Y. Xing, W. Zhang and Y. J. Wang, “The Stationary Distributions of Two Classes of Reflected Ornstein-Uhlenbeck Processes,” Journal of Applied Probability, Vol. 46, No. 3, 2009, pp. 709-720.
[3]	S. Asmussen and O. Kella, “A Multi-Dimensional Martingale for Markov Additive Process and its Applications,” Advances in Applied Probability, Vol. 32, No. 2, 2000, pp. 376-393.
[4]	S. Either and T. Kurtz, “Markov Processes: Characterization and Convergences,” John Wiley Sons, New York, 1986.
[5]	L. J. Bo, L. D. Zhang and Y. J. Wang, “On the First Passage Times of Reflected O-U Processes with Two-Sided Barriers,” Queueing Systems, Vol. 54, No. 4, December 2006, pp. 313-316.
[6]	V. Linetsky, “On the Transition Densities for Reflected Diffusions,” Advances in Applied Probability, Vol. 37, No. 2, 2005, pp. 435-460.
[7]	L. Rabehasaina and B. Sericola, “A Second-Order Markov-Modulated Fluid Queue with Linear Service Rate,” Journal of Applied Probability, Vol. 41, No. 3, 2004, pp. 758-777.