## Journal of Applied Probability

### On optimal stopping problems for matrix-exponential jump-diffusion processes

#### Abstract

In this paper we consider optimal stopping problems for a general class of reward functions under matrix-exponential jump-diffusion processes. Given an American call-type reward function in this class, following the averaging problem approach (see, for example, Alili and Kyprianou (2005), Kyprianou and Surya (2005), Novikov and Shiryaev (2007), and Surya (2007)), we give an explicit formula for solutions of the corresponding averaging problem. Based on this explicit formula, we obtain the optimal level and the value function for American call-type optimal stopping problems.

#### Article information

Source
J. Appl. Probab., Volume 49, Number 2 (2012), 531-548.

Dates
First available in Project Euclid: 16 June 2012

https://projecteuclid.org/euclid.jap/1339878803

Digital Object Identifier
doi:10.1239/jap/1339878803

Mathematical Reviews number (MathSciNet)
MR2977812

Zentralblatt MATH identifier
1252.60039

#### Citation

Sheu, Yuan-Chung; Tsai, Ming-Yao. On optimal stopping problems for matrix-exponential jump-diffusion processes. J. Appl. Probab. 49 (2012), no. 2, 531--548. doi:10.1239/jap/1339878803. https://projecteuclid.org/euclid.jap/1339878803

#### References

• Alili, L. and Kyprianou, A. E. (2005). Some remarks on first passage of Lévy processes, the American put and pasting principles. Ann. Appl. Prob. 15, 2062–2080.
• Asmussen, S. and Albrecher, H. (2010). Ruin Probabilities, 2nd edn. World Scientific, Hackensack, NJ.
• Asmussen, S., Avram, F. and Pistorius, M. R. (2004). Russian and American put options under exponential phase-type Lévy models. Stoch. Process. Appl. 109, 79–111.
• Boyarchenko, S. I. and Levendorskiǐ, S. Z. (2005). American options: the EPV pricing model. Ann. Finance 1, 267–292.
• Chen, Y.-T. and Sheu, Y.-C. (2009). A note on $r$-balayages of matrix-exponential Lévy processes. Electron. Commun. Prob. 14, 165–175.
• Deligiannidis, G., Le, H. and Utev, S. (2009). Optimal stopping for processes with independent increments, and applications. J. Appl. Prob. 46, 1130–1145.
• Ivanovs, J. (2011). One-sided Markov additive processes and related exit problems. Doctoral Thesis, University of Amsterdam.
• Kyprianou, A. E. and Surya, B. A. (2005). On the Novikov–Shiryaev optimal stopping problems in continuous time. Electron. Commun. Prob. 10, 146–154.
• Lewis, A. L. and Mordecki, E. (2008). Wiener–Hopf factorization for Lévy processes having positive jumps with rational transforms. J. Appl. Prob. 45, 118–134.
• Mordecki, E. (2002). Optimal stopping and perpetual options for Lévy processes. Finance Stoch. 6, 473–493.
• Mordecki, E. and Salminen, P. (2007). Optimal stopping of Hunt and Lévy processes. Stochastics 79, 233–251.
• Novikov, A and Shiryaev, A. (2007). On a solution of the optimal stopping problem for processes with independent increments. Stochastics 79, 393–406.
• Surya, B. A. (2007). An approach for solving perpetual optimal stopping problems driven by Lévy processes. Stochastics 79, 337–361.