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Numerical Methods for the Exit Time of a Piecewise-Deterministic Markov Process

Part of: Probabilistic methods, simulation and stochastic differential equations Special processes Markov processes

Published online by Cambridge University Press:  04 January 2016

Adrien Brandejsky ,
Benoîte De Saporta  and
François Dufour
Adrien Brandejsky*
Affiliation:
Université Bordeaux, IMB and INRIA Bordeaux Sud-Ouest
Benoîte De Saporta*
Affiliation:
Université Bordeaux, Gretha and INRIA Bordeaux Sud-Ouest
François Dufour*
Affiliation:
Université Bordeaux, IMB and INRIA Bordeaux Sud-Ouest
*
∗ Postal address: INRIA Bordeaux Sud-Ouest, CQFD Team, 351 cours de la Libération, F-33405 Talence, France.
∗ Postal address: INRIA Bordeaux Sud-Ouest, CQFD Team, 351 cours de la Libération, F-33405 Talence, France.
∗ Postal address: INRIA Bordeaux Sud-Ouest, CQFD Team, 351 cours de la Libération, F-33405 Talence, France.
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Abstract

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We present a numerical method to compute the survival function and the moments of the exit time for a piecewise-deterministic Markov process (PDMP). Our approach is based on the quantization of an underlying discrete-time Markov chain related to the PDMP. The approximation we propose is easily computable and is even flexible with respect to the exit time we consider. We prove the convergence of the algorithm and obtain bounds for the rate of convergence in the case of the moments. We give an academic example and a model from the reliability field to illustrate the results of the paper.

Keywords

Exit timepiecewise-deterministic Markov processquantizationnumerical method

MSC classification

Primary: 60J25: Continuous-time Markov processes on general state spaces 65C20: Models, numerical methods
Secondary: 60K10: Applications (reliability, demand theory, etc.)
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 44 , Issue 1 , March 2012 , pp. 196 - 225
Copyright
© Applied Probability Trust 

References

Bally, V. and Pagès, G. (2003). A quantization algorithm for solving multi-dimensional discrete-time optimal stopping problems. Bernoulli 9, 1003–1049.CrossRefGoogle Scholar
Bally, V., Pagès, G. and Printemps, J. (2005). A quantization tree method for pricing and hedging multidimensional American options. Math. Finance 15, 119–168.Google Scholar
Bouton, C. and Pagès, G. (1997). About the multidimensional competitive learning vector quantization algorithm with constant gain. Ann. Appl. Prob. 7, 679–710.Google Scholar
Chiquet, J. and Limnios, N. (2008). A method to compute the transition function of a piecewise deterministic Markov process with application to reliability. Statist. Prob. Lett. 78, 1397–1403.Google Scholar
Davis, M. H. A. (1993). Markov Models and Optimization (Monogr. Statist. Appl. Prob. 49). Chapman & Hall, London.CrossRefGoogle Scholar
De Saporta, B., Dufour, F. and Gonzalez, K. (2010). Numerical method for optimal stopping of piecewise deterministic Markov processes. Ann. Appl. Prob. 20, 1607–1637.CrossRefGoogle Scholar
Gray, R. M. and Neuhoff, D. L. (1998). Quantization. IEEE Trans. Inf. Theory 44, 2325–2383.Google Scholar
Helmes, K., Röhl, S. and Stockbridge, R. H. (2001). Computing moments of the exit time distribution for Markov processes by linear programming. Operat. Res. 49, 516–530.Google Scholar
Lasserre, J.-B. and Prieto-Rumeau, T. (2004). SDP vs. LP relaxations for the moment approach in some performance evaluation problems. Stoch. Models 20, 439–456.Google Scholar
Pagès, G., Pham, H. and Printemps, J. (2004). Optimal quantization methods and applications to numerical problems in finance. In Handbook of Computational and Numerical Methods in Finance, Birkhäuser, Boston, MA, pp. 253–297.Google Scholar