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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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