Hostname: page-component-cd9895bd7-hc48f Total loading time: 0 Render date: 2024-12-27T11:21:24.441Z Has data issue: false hasContentIssue false
  • English
  • Français

Stochastic processes directed by randomized time

Part of: Stochastic processes Distribution theory - Probability

Published online by Cambridge University Press:  14 July 2016

Mei-Ling Ting Lee  and
G. Alex Whitmore
Mei-Ling Ting Lee*
Affiliation:
Harvard University
G. Alex Whitmore*
Affiliation:
McGill University
*
∗ Present address: Department of Statistics, Harvard University, 1 Oxford St. Cambridge, MA 02138, USA.
∗∗∗ Postal address: Faculty of Management, McGill University, 1001 Sherbrooke St West, Montreal, Quebec H3A 1G5, Canada.
Get access Rights & Permissions [Opens in a new window]

Abstract

The paper investigates stochastic processes directed by a randomized time process. A new family of directing processes called Hougaard processes is introduced. Monotonicity properties preserved under subordination, and dependence among processes directed by a common randomized time are studied. Results for processes subordinated to Poisson and stable processes are presented. Potential applications to shock models and threshold models are also discussed. Only Markov processes are considered.

Keywords

CLUSTERINGDEPENDENCEGAMMAHOUGAARDINVERSE GAUSSIANMARKOVPOISSONSHOCK MODELSSTABLESUBORDINATIONTHRESHOLD MODELSTOTAL POSITIVITY

MSC classification

Primary: 60G07: General theory of processes
Secondary: 60G55: Point processes 60E07: Infinitely divisible distributions; stable distributions
Type
Research Papers
Information
Journal of Applied Probability , Volume 30 , Issue 2 , June 1993 , pp. 302 - 314
Copyright
Copyright © Applied Probability Trust 1993 

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

Akgiray, V. and Booth, G. G. (1988) The stable-law model of stock returns. J. Bus. Econ. Statist. 6, 51–57.Google Scholar
Barlow, R. E. and Proschan, F. (1981) Statistical Theory of Reliability and Life Testing. Holt, Reinhart and Winston, New York.Google Scholar
Bochner, S. (1955) Harmonic Analysis and the Theory of Probability. University of California Press.Google Scholar
Chhikara, R. S. and Folks, J. L. (1989) The Inverse Gaussian Distribution. Marcel Dekker, New York.Google Scholar
Clark, P. K. (1973) A subordinated stochastic process model with finite variance for speculative prices. Econometrica 41, 135–155.CrossRefGoogle Scholar
Cohen, J. W. (1962) Derived Markov chains. Proc. Kon. Ned. Akad. Wetensch. A 65, 55–92.Google Scholar
Esary, J. D., Marshall, A. W. and Proschan, F. (1973) Shock models and wear processes. Ann. Prob. 1, 627–649.Google Scholar
Fama, ?. (1965) The behavior of stock market prices. J. Business 38, 34–105.CrossRefGoogle Scholar
Feller, W. (1971) An Introduction to Probability Theory and Its Applications, Volume II, 2nd edn. Wiley, New York.Google Scholar
Ghosh, M. and Ebrahimi, N. (1982) Shock models leading to increasing failure rate and decreasing mean residual life survival. J. Appl. Prob. 19, 158–166.Google Scholar
Grandell, J. (1976) Doubly Stochastic Poisson Processes. Springer-Verlag, New York.Google Scholar
Griffith, W. and Srinivasan, C. (1985) A note on survival models under a Markov process. J. Appl. Prob. 22, 982–988.CrossRefGoogle Scholar
Hougaard, P. (1986) Survival models for heterogeneous populations derived from stable distributions. Biometrika 73, 387–396.Google Scholar
Karlin, S. (1968) Total Positivity, Vol. I. Stanford Univ. Press, Stanford, California.Google Scholar
Lindgren, G. and Rootzen, H. (1987) Extreme values: theory and technical applications. Scand. J. Statist. 14, 241–279.Google Scholar
Mandelbrot, B. B. (1963) The variation of certain speculative prices. J. Business 36, 394–419.Google Scholar
Mandelbrot, B. B. (1977) The Fractal Geometry of Nature. W. H. Freeman, New York.Google Scholar
Mandelbrot, B. B. and Taylor, H. M. (1967) On the distribution of stable price differences. Operat. Res. 15, 1057–1062.Google Scholar
Mittnik, S. and Rachev, S. T. (1991) Alternative multivariate stable distributions. In Stable Processes and Related Topics, ed. Cambanis, S., Samorodnitsky, G., and Taqqu, M. S., pp. 107–120. Birkhauser, Boston.Google Scholar
Rachev, S. T. and Rüschendorf, L. (1991) On the Cox, Ross and Rubinstein model for option prices. Preprint.Google Scholar
Rachev, S. T. and Samorodnitsky, G. (1991) Option pricing formulae for speculative prices modelled by subordinated stochastic process. Preprint.Google Scholar
Samorodnitsky, G. and Taqqu, M. S. (1992) Stable Random Processes. Preprints.Google Scholar
Stam, A. J. (1965) Derived stochastic processes. Compositio Math. 17, 102–140.Google Scholar
Upton, D. E. and Shannon, D. S. (1979) The stable Paretian distribution, subordinated stochastic processes, and asymptotic lognormality: an empirical investigation. J. Finance 34, 1031–1039.Google Scholar
Wasan, M. T. (1968) On the inverse Gaussian process. Skand. Aktuarietidskr. Part 1–2, 69–96.CrossRefGoogle Scholar
Weron, A. (1983) Stable processes and measures: a survey. Probability Theory on Vector Spaces III, Springer-Verlag, Lublin.Google Scholar
Westerfield, R. (1976) The distribution of common stock price changes: an application of transactions time and subordinated stochastic models. J. Finan. Quant. Anal. 12, 743–765.Google Scholar
Whitmore, G. A. and Lee, M.-L. T. (1992) Inference for Poisson-Hougaard processes. Submitted.Google Scholar