Hostname: page-component-cd9895bd7-lnqnp Total loading time: 0 Render date: 2024-12-26T21:59:52.194Z Has data issue: false hasContentIssue false
  • English
  • Français

On Optimal Stopping Problems for Matrix-Exponential Jump-Diffusion Processes

Part of: Markov processes Stochastic processes

Published online by Cambridge University Press:  04 February 2016

Yuan-Chung Sheu  and
Ming-Yao Tsai
Yuan-Chung Sheu*
Affiliation:
National Chiao Tung University
Ming-Yao Tsai*
Affiliation:
National Chiao Tung University
*
∗ Postal address: Department of Applied Mathematics, National Chiao Tung University, Hsinchu, Taiwan.
∗ Postal address: Department of Applied Mathematics, National Chiao Tung University, Hsinchu, Taiwan.
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

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.

Keywords

Optimal stopping problemAmerican call-type reward functionaveraging problemmatrix-exponential distributionjump-diffusion process

MSC classification

Primary: 60G40: Stopping times; optimal stopping problems; gambling theory 60J75: Jump processes 60G51: Processes with independent increments; L'evy processes
Type
Research Article
Information
Journal of Applied Probability , Volume 49 , Issue 2 , June 2012 , pp. 531 - 548
Copyright
© Applied Probability Trust 

Footnotes

Partially supported by NSC grant NSC100-2115-M-009-006, CMMSC, and NCTS, Taiwan.

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.CrossRefGoogle Scholar
Asmussen, S. and Albrecher, H. (2010). Ruin Probabilities, 2nd edn. World Scientific, Hackensack, NJ.Google Scholar
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.Google Scholar
Boyarchenko, S. I. and Levendorskii, S. Z. (2005). American options: the EPV pricing model. Ann. Finance 1, 267–292.Google Scholar
Chen, Y.-T. and Sheu, Y.-C. (2009). A note on r-balayages of matrix-exponential Lévy processes. Electron. Commun. Prob. 14, 165–175.Google Scholar
Deligiannidis, G., Le, H. and Utev, S. (2009). Optimal stopping for processes with independent increments, and applications. J. Appl. Prob. 46, 1130–1145.CrossRefGoogle Scholar
Ivanovs, J. (2011). One-sided Markov additive processes and related exit problems. , University of Amsterdam.Google Scholar
Kyprianou, A. E. and Surya, B. A. (2005). On the Novikov–Shiryaev optimal stopping problems in continuous time. Electron. Commun. Prob. 10, 146–154.CrossRefGoogle Scholar
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.Google Scholar
Mordecki, E. (2002). Optimal stopping and perpetual options for Lévy processes. Finance Stoch. 6, 473–493.Google Scholar
Mordecki, E. and Salminen, P. (2007). Optimal stopping of Hunt and Lévy processes. Stochastics 79, 233–251.Google Scholar
Novikov, A and Shiryaev, A. (2007). On a solution of the optimal stopping problem for processes with independent increments. Stochastics 79, 393–406.CrossRefGoogle Scholar
Surya, B. A. (2007). An approach for solving perpetual optimal stopping problems driven by Lévy processes. Stochastics 79, 337–361.CrossRefGoogle Scholar