Hostname: page-component-78c5997874-fbnjt Total loading time: 0 Render date: 2024-11-05T04:22:17.811Z Has data issue: false hasContentIssue false
  • English
  • Français

On the forward algorithm for stopping problems on continuous-time Markov chains

Part of: Markov processes Mathematical finance

Published online by Cambridge University Press:  22 November 2021

Laurent Miclo  and
Stéphane Villeneuve
Laurent Miclo*
Affiliation:
CNRS and Université de Toulouse
Stéphane Villeneuve*
Affiliation:
Université de Toulouse
*
*Postal address: Toulouse School of Economics, 1 Esplanade de l’université, 31080 Toulouse cedex 06, France.
*Postal address: Toulouse School of Economics, 1 Esplanade de l’université, 31080 Toulouse cedex 06, France.
Get access Rights & Permissions [Opens in a new window]

Abstract

We revisit the forward algorithm, developed by Irle, to characterize both the value function and the stopping set for a large class of optimal stopping problems on continuous-time Markov chains. Our objective is to renew interest in this constructive method by showing its usefulness in solving some constrained optimal stopping problems that have emerged recently.

Keywords

Forward algorithmconstrained optimal stoppingMarkov chains

MSC classification

Primary: 60J27: Continuous-time Markov processes on discrete state spaces
Secondary: 60J45: Probabilistic potential theory 60J28: Applications of continuous-time Markov processes on discrete state spaces 90C40: Markov and semi-Markov decision processes 91G60: Numerical methods (including Monte Carlo methods)
Type
Original Article
Information
Journal of Applied Probability , Volume 58 , Issue 4 , December 2021 , pp. 1043 - 1063
Check for updates
Copyright
© The Author(s), 2021. Published by Cambridge University Press on behalf of Applied Probability Trust

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Abergel, F. and Jedidi, A. (2013). A mathematical approach to order book modeling. Int. J. Theor. Appl. Finance 16, 1350025.10.1142/S0219024913500258CrossRefGoogle Scholar
Christensen, S. (2014). A method for pricing American options using semi-infinite linear programming. Math. Finance 24, 156–172.CrossRefGoogle Scholar
Cox, J. C, Ross, S. and Rubinstein, M. (1979). Option pricing: A simplified approach. J. Financial Econom. 7, 229–262.10.1016/0304-405X(79)90015-1CrossRefGoogle Scholar
Crocce, F. and Mordecki, E. (2019). An algorithm to solve optimal stopping problems for one-dimensional diffusions. Preprint, arXiv:1909.10257.Google Scholar
Dayanik, S. and Karatzas, I. (2003). On the optimal stopping problem for one-dimensional diffusions. Stoch. Proc. Appl. 107, 173–212.10.1016/S0304-4149(03)00076-0CrossRefGoogle Scholar
Dixit, A. K. and Pindyck, R. S. (1994). Investment under Uncertainty. Princeton University Press.CrossRefGoogle Scholar
Dupuis, P. and Wang, H. (2002). Optimal stopping with random intervention times. Adv. Appl. Prob. 34, 141–157.10.1239/aap/1019160954CrossRefGoogle Scholar
Egloff, D. and Leippold, M. (2009). The valuation of American options with stochastic time constraints. Appl. Math. Finance 16, 287–305.10.1080/13504860802645706CrossRefGoogle Scholar
Eriksson, B. and Pistorius, M. R. (2015). American option valuation under continuous-time Markov chains. Adv. Appl. Prob. 47, 378–401.CrossRefGoogle Scholar
Ethier, S. N. and Kurtz, T. G. (2005). Markov Processes: Characterization and Convergence. John Wiley, New York.Google Scholar
Helmes, K. and Stockbridge, R. (2010). Construction of the value function and optimal stopping rules in optimal stopping of one-dimensional diffusions. Adv. Appl. Prob. 42, 158–182.CrossRefGoogle Scholar
Irle, A. (2006). A forward algorithm for solving optimal stopping problems. J. Appl. Prob. 43, 102–113.10.1239/jap/1143936246CrossRefGoogle Scholar
Irle, A. (2009). On forward improvement iteration for stopping problems. In Proc. Int. Workshop Sequential Methodologies, Troyes.Google Scholar
Kolodko, A. and Schoenmakers, J. (2006). Iterative construction of the optimal Bermudan stopping time. Finance Stoch. 10, 27–49.10.1007/s00780-005-0168-5CrossRefGoogle Scholar
Kou, S. C. and Kou, S. G. (2003). Modeling growth stocks with birth–death processes. Adv. Appl. Prob. 35, 641–664.10.1239/aap/1059486822CrossRefGoogle Scholar
Lamberton, D. (1998). Error estimates for the binomial approximation of American put options. Ann. Appl. Prob. 8, 206–233.10.1214/aoap/1027961041CrossRefGoogle Scholar
Lempa, J. (2012). Optimal stopping with information constraint. Appl. Math. Optim. 66, 147–173.10.1007/s00245-012-9166-0CrossRefGoogle Scholar
Menaldi, J. L. and Robin, M. (2016). On some optimal stopping problems with constraint. SIAM J. Control Optim. 54, 2650–2671.10.1137/15M1040001CrossRefGoogle Scholar
Mordecki, E. (2002). Optimal stopping and perpetual options for LÉvy processes. Finance Stoch. 6, 473–493.10.1007/s007800200070CrossRefGoogle Scholar
Myneni, R. (1992). The pricing of the American option. Ann. Appl. Prob. 2, 1–23.10.1214/aoap/1177005768CrossRefGoogle Scholar
Peskir, G. and Shiryaev, A. N. (2006). Optimal Stopping and Free-Boundary Problems. Birkhauser, Basel.Google Scholar
Shiryaev, A. N. (1978). Optimal Stopping Rules. Springer, New York.Google Scholar
Snell, L. (1953). Applications of Martingale system theorems. Trans. Amer. Math. Soc. 73, 293–312.10.1090/S0002-9947-1952-0050209-9CrossRefGoogle Scholar
Sonin, I. (1999). The elimination algorithm for the problem of optimal stopping. Math. Methods Operat. Res. 49, 111–123.Google Scholar
Wald, A. (1945). Sequential tests of statistical hypotheses. Ann. Math. Statist. 16, 117–186.10.1214/aoms/1177731118CrossRefGoogle Scholar