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Large-deviation expressions for the distribution of first-passage coordinates

Part of: Stochastic processes Limit theorems

Published online by Cambridge University Press:  01 July 2016

P. Whittle
P. Whittle*
Affiliation:
University of Cambridge
*
* Postal address: Statistical Laboratory, 16 Mill Lane, Cambridge CB2 1SB, UK.
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Abstract

We consider the distribution of the free coordinates of a time-homogeneous Markov process at the time of its first passage into a prescribed stopping set. This calculation (for an uncontrolled process) is of interest because under some circumstances it enables one to calculate the optimal control for a related controlled process. Scaling assumptions are made which allow the application of large deviation techniques. However, the first-order evaluation obtained by these techniques is often too crude to be useful, and the second-order correction term must be calculated. An expression for this correction term as an integral over time is obtained in Equation (20). The integration can be performed in some cases to yield the conclusions of Theorems 1 and 2, expressed in Equations (7) and (9). Theorem 1 gives the probability density of the state vector (to the required degree of approximation) at a prescribed time for a class of processes we may reasonably term linear. Theorem 2 evaluates (without any assumption of linearity) the ratio of this density to the probability density of the coordinates under general stopping rules.

Keywords

STOPPING TIMEOPTIMAL CONTROL

MSC classification

Primary: 60G40: Stopping times; optimal stopping problems; gambling theory
Secondary: 60F10: Large deviations
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 27 , Issue 3 , September 1995 , pp. 692 - 710
Copyright
Copyright © Applied Probability Trust 1995 

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References

Azencott, R. (1982) Formule de Taylor stochastique et développement asymptotique d'integrales de Feynman. Séminaire de Probabilités XVI. Lecture Notes in Mathematics 921, pp. 237–284. Springer-Verlag, Berlin.Google Scholar
Azencott, R. (1984) Densité des diffusions en temps petits; développements asymptotiques. Séminaire de Probabilitiés XVIII. Lecture notes in Mathematics 1059, pp. 402–498, Springer-Verlag, Berlin.Google Scholar
Bartlett, M. S. (1949) Some evolutionary stochastic processes. J. R. Statist. Soc. B 11, 211–229.Google Scholar
Ben Arous, G. (1988) Méthodes de Laplace et de la phase stationaire sur l'espace de Wiener. Stochastics 25, 125–153.Google Scholar
Daniels, H. E. (1969) The minimum of a stationary Markov process superimposed on a U-shaped trend. J. Appl. Prob. 6, 399–408.CrossRefGoogle Scholar
Day, M. V. (1987) Recent progress on the small parameter exit problem. Stochastics 26, 121–150.Google Scholar
Day, M. V. (1990) Some phenomena of the characteristic boundary exit problem. In Diffusion Processes and Related Problems in Analysis, Vol. 1, ed. Pinsky, M. A.. Birkhäuser, Boston.Google Scholar
Day, M. V. (1992) Conditional exits for small noise diffusions with characteristic boundary. Ann. Prob. 20, 1385–1419.CrossRefGoogle Scholar
Donsker, M. and Varadhan, S. (1976) Asymptotic evaluation of Markov process expectations for large time (III). Comm. Pure Appl. Math. 29, 389–461.Google Scholar
Durbin, J. (1985) The fist-passage density of a continuous Gaussian process to a general boundary. J. Appl. Prob. 22, 99–122.Google Scholar
Durbin, J. (1992) The first-passage density of the Brownian motion process to a curved boundary. J. Appl. Prob. 29, 291–304.Google Scholar
Feynman, R. P. and Hibbs, A. R. (1965) Quantum Mechanics and Path Integrals. McGraw-Hill, New York.Google Scholar
Fleming, W. H. and James, M. R. (1992) Asymptotic series and exit probabilities. Ann. Prob. 20, 1369–1384.Google Scholar
Friedlin, M. I. and Wentzell, A. D. (1984) Random Perturbations of Dynamical Systems. Springer-Verlag, New York. (Russian original published in 1979 by Nauka, Moscow.) Google Scholar
Gutzwiller, M. G. (1990) Chaos in Classical and Quantum Mechanics. Springer-Verlag, Berlin.Google Scholar
Morette, V. (1951) On the definition and approximation of Feynman's path integrals. Phys. Rev. 81, 848–852.CrossRefGoogle Scholar
Moyal, J. (1949) Stochastic processes and statistical physics. J. R. Statist. Soc. B 11, 150–210.Google Scholar
Stroock, D. (1984) An Introduction to the Theory of Large Deviations. Springer-Verlag, Berlin.Google Scholar
Stroock, D. and Varadhan, S. (1979) Multidimensional Diffusion Processes. Springer-Verlag, Berlin.Google Scholar
Van Vleck, J. H. (1982) Proc. Natl. Acad. Sci. USA 14, 178.Google Scholar
Vanderbei, R. J. and Weiss, A. (1988) Large Deviations and Their Application to Computer and Communications Systems. Circulated unpublished notes, AT&T Bell Laboratories.Google Scholar
Varadhan, S. (1984) Large Deviations and Applications. SIAM, Philadelphia.Google Scholar
Whittle, P. (1982) Optimisation over Time, Vol. 1. Wiley, Chichester.Google Scholar
Whittle, P. (1990) A risk-sensitive maximum principle. Syst. Contr. Lett. 15, 183–192.Google Scholar
Whittle, P. (1991a) Likelihood and cost as path integrals. J. R. Statist. Soc. B 53, 505–529.Google Scholar
Whittle, P. (1991b) A risk-sensitive maximum principle; the case of imperfect observation. IEEE Trans. Automat. Control 36, 793–801.Google Scholar
Whittle, P. (1996) Optimal Control: Basics and Beyond. Wiley, Chichester.Google Scholar
Whittle, P. and Gait, P. (1970) Reduction of a class of stochastic control problems. J. Inst. Math. Appl. 6, 131–140.Google Scholar