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

  • Adrien Brandejsky (a1), Benoîte De Saporta (a2) and François Dufour (a1)

Abstract

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.

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Copyright

Corresponding author

Postal address: INRIA Bordeaux Sud-Ouest, CQFD Team, 351 cours de la Libération, F-33405 Talence, France.
∗∗ Email address: dufour@math.u-bordeaux1.fr

References

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[1] Bally, V. and Pagès, G. (2003). A quantization algorithm for solving multi-dimensional discrete-time optimal stopping problems. Bernoulli 9, 10031049.
[2] Bally, V., Pagès, G. and Printemps, J. (2005). A quantization tree method for pricing and hedging multidimensional American options. Math. Finance 15, 119168.
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[4] 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, 13971403.
[5] Davis, M. H. A. (1993). Markov Models and Optimization (Monogr. Statist. Appl. Prob. 49). Chapman & Hall, London.
[6] De Saporta, B., Dufour, F. and Gonzalez, K. (2010). Numerical method for optimal stopping of piecewise deterministic Markov processes. Ann. Appl. Prob. 20, 16071637.
[7] Gray, R. M. and Neuhoff, D. L. (1998). Quantization. IEEE Trans. Inf. Theory 44, 23252383.
[8] 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, 516530.
[9] Lasserre, J.-B. and Prieto-Rumeau, T. (2004). SDP vs. LP relaxations for the moment approach in some performance evaluation problems. Stoch. Models 20, 439456.
[10] 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. 253297.

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