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Sample quantiles of additive renewal reward processes

Part of: Markov processes Special processes

Published online by Cambridge University Press:  14 July 2016

Angelos Dassios
Angelos Dassios*
Affiliation:
London School of Economics
*
∗ Postal address: Department of Statistics, London School of Economics, Houghton Street, London WC2A 2AE, UK.
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Abstract

The distribution of the sample quantiles of random processes is important for the pricing of some of the so-called financial ‘look-back' options. In this paper a representation of the distribution of the α-quantile of an additive renewal reward process is obtained as the sum of the supremum and the infimum of two rescaled independent copies of the process. This representation has already been proved for processes with stationary and independent increments. As an example, the distribution of the α-quantile of a randomly observed Brownian motion is obtained.

Keywords

SAMPLE QUANTILESLOOK-BACK FINANCIAL OPTIONSMARKED RENEWAL PROCESSES

MSC classification

Primary: 60K15: Markov renewal processes, semi-Markov processes 60J75: Jump processes

Information

Type
Research Papers
Information
Journal of Applied Probability , Volume 33 , Issue 4 , December 1996 , pp. 1018 - 1032
Copyright
Copyright © Applied Probability Trust 1996 

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References

[1] Akahori, J. (1995) Some formulae for a new type of path-dependent option. Ann. Appl. Prob. 5, 383–388.CrossRefGoogle Scholar
[2] Breiman, L. (1968) Probability. Addison-Wesley, New York.Google Scholar
[3] Dassios, A. (1995) The distribution of the quantiles of a Brownian motion with drift and the pricing of related path-dependent options. Ann. Appl. Prob. 5, 389–398.CrossRefGoogle Scholar
[4] Dassios, A. (1995) Sample quantiles of stochastic processes with stationary and independent increments. Ann. Appl. Prob. (to appear).CrossRefGoogle Scholar
[5] Dassios, A. and Embrechts, P. (1989) Martingales and insurance risk. Comnun. Statist. Stoch. Models 5, 181–217.CrossRefGoogle Scholar
[6] Davis, M. H. A. (1984) Piecewise-deterministic Markov processes: a general class of non-diffusion stochastic models (with discussion). J. R. Statist. Soc. B 46, 353–388.Google Scholar
[7] Davis, M. H. A. (1993) Markov Models and Optimization. Chapman and Hall, London.CrossRefGoogle Scholar
[8] Embrechts, P., Rogers, L. C. G. and Yor, M. (1995) A proof of Dassios' representation of the a-quantile of Brownian motion with drift. Ann. Appl. Prob. 5, 757–767.CrossRefGoogle Scholar
[9] Embrechts, P. and Samorodnitsky, G. (1994) Sample quantiles of heavy tailed stochastic processes. Preprint. ETH, Zürich.Google Scholar
[10] Feller, W. (1966) An Introduction to Probability Theory and its Applications. Vol. II. Wiley, New York.Google Scholar
[11] Kennedy, D. P. (1993) Exact ruin probabilities and the evaluation of program trading in financial markets. Math. Finance 3, 55–63.CrossRefGoogle Scholar
[12] Miura, R. (1992) A note on a look-back option based on order statistics. Hitosubashi J. Commerce Management 27, 15–28.Google Scholar
[13] Ross, S. M. (1983) Stochastic Processes. Wiley, New York.Google Scholar
[14] Wendel, J. G. (1960) Order statistics of partial sums. Ann. Math. Statist. 31, 1034–1044.CrossRefGoogle Scholar
[15] Yor, M. (1995) The distribution of Brownian quantiles. J. Appl. Prob. 32, 405–416.CrossRefGoogle Scholar