Two-Sided First Exit Problem for Jump Diffusion Distribution Processes Having Jumps with a Mixture of Erlang

Yuzhen Wen, Chuancun Yin

School of Mathematical Sciences, Qufu Normal University, Qufu, China.

DOI: 10.4236/am.2013.48153                

Abstract

In this paper, we consider the two-sided first exit problem for jump diffusion processes having jumps with rational Laplace transforms. We investigate the probabilistic property of conditional memorylessness, and drive the joint distribution of the first exit time from an interval and the overshoot over the boundary at the exit time.

Keywords

First Exit Time; Two-Sided Jumps; Jump Diffusion Process; Overshoot

Conflicts of Interest

The authors declare no conflicts of interest.

References

[1]	D. Perry and W. Stadje, “Risk Analysis for a Stochastic Cash Management Model with Two Types of Custom ers,” Insurance: Mathematics and Economics, Vol. 26, No. 1, 2000, pp. 25-36. doi:10.1016/S0167-6687(99)00037-2
[2]	S. G. Kou and H. Wang, “First Passage Times of a Jump Diffusion Process,” Advances in Applied Probability, Vol. 35, No. 2, 2003, pp. 504-531. doi:10.1239/aap/1051201658
[3]	N. Cai, “On First Passage Times of a Hyper-Exponential Jump Diffusion Process,” Operations Research Letters, Vol. 37, No. 2, 2009, pp. 127-134. doi:10.1016/j.orl.2009.01.002
[4]	N. Cai, N. Chen and X. W. Wan, “Pricing Double-Barrier Options under a Flexible Jump Diffusion Model,” Opera tions Research Letters, Vol. 37, No. 3, 2009, pp. 163-167. doi:10.1016/j.orl.2009.02.006
[5]	T. Kadankova and N. Veraverbeke, “On Several Two Bondary Problems for a Particular Class of Lévy Proc esses,” Journal of Theoretical Probability, Vol. 20, No. 4, 2007, pp. 1073-1085. doi:10.1007/s10959-007-0088-8
[6]	S. Fourati, “Explicit Solutions of the Exit Problem for a Class of Lévy Processes; Applications to the Pricing of Double-Barrier Options,” Stochastic Processes and their Applications, Vol. 122, No. 3, 2012, pp. 1034-1067. doi:10.1016/j.spa.2011.09.008
[7]	M. Jacobsen, “The Time to Ruin for a Class of Markov Additive Risk Process with Two-Sided Jumps,” Advances in Applied Probability, Vol. 37, No. 4, 2005, pp. 963-992. doi:10.1239/aap/1134587749
[8]	D. Perry, W. Stadje and S. Zacks, “Contributions to the Theory of First-Exit Times of Some Compound Processes in Queueing Theory,” Queueing Systems, Vol. 33, No. 4, 1999, pp. 369-379. doi:10.1023/A:1019140616021
[9]	N. Cai and S. G. Kou, “Option Pricing under a Mixed-Ex ponential Jump Diffusion Model,” Management Science, Vol. 57, No. 11, 2011, pp. 2067-2081. doi:10.1287/mnsc.1110.1393
[10]	A. L. Lewis and E. Mordecki, “Wiener-Hopf Factoriza tion for Lévy Processes Having Positive Jumps with Ra tional Transforms,” Journal of Applied Probability, Vol. 45, No. 1, 2008, pp. 118-134. doi:10.1239/jap/1208358956
[11]	A. Kuznetsov, “On the Distribution of Exponential Func tionals for Lévy Processes with Jumps of Rational Trans form,” Stochastic Processes and their Applications, Vol. 122, No. 2, 2012, pp. 654-663. doi:10.1016/j.spa.2011.09.007