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Tools to Estimate the First Passage Time to a Convex Barrier

Part of: Stochastic processes Limit theorems Sequential methods

Published online by Cambridge University Press:  14 July 2016

Ola Hammarlid
Ola Hammarlid*
Affiliation:
Stockholm University
*
∗Postal address: Department of Mathematics, Stockholm University, 106 91 Stockholm, Sweden. Email address: olah@math.su.se
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Abstract

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The first passage time of a random walk to a barrier (constant or concave) is of great importance in many areas, such as insurance, finance, and sequential analysis. Here, we consider a sum of independent, identically distributed random variables and the convex barrier cb(n/c), where c is a scale parameter and n is time. It is shown, using large-deviation techniques, that the limit distribution of the first passage time decays exponentially in c. Under a tilt of measure, which changes the drift, four properties are proved: the limit distribution of the overshoot is distributed as an overshoot over a linear barrier; the stopping time is asymptotically normally distributed when it is properly normalized; the overshoot and the asymptotic normal part are asymptotically independent; and the overshoot over a linear barrier is bounded by an exponentially distributed random variable. The determination of the function that multiplies the exponential part is guided by consideration of these properties.

Keywords

First passage timestopping timelarge deviationrate functionsequential analysisconvex barrier

MSC classification

Secondary: 60F10: Large deviations 60G50: Sums of independent random variables; random walks 60G40: Stopping times; optimal stopping problems; gambling theory 62L10: Sequential analysis
Type
Research Papers
Information
Journal of Applied Probability , Volume 42 , Issue 1 , March 2005 , pp. 61 - 81
Copyright
© Applied Probability Trust 2005 

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