On the Joint Distribution of First-passage Time and First-passage Area of Drifted Brownian Motion

On the Joint Distribution of First-passage Time and First-passage Area of Drifted Brownian Motion For drifted Brownian motion X(t) = x-µ t + B t (µ > 0) starting from x > 0, we study the joint distribution of the first-passage time below zero ,t(x), and the first-passage area ,A(x), swept out by X till the time t(x). In particular, we establish differential equations with boundary conditions for the joint moments E[t(x) m A(x) n ], and we present an algorithm to find recursively them, for any m and n. Finally, the expected value of the time average of X till the time t(x) is obtained. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Methodology and Computing in Applied Probability Springer Journals

On the Joint Distribution of First-passage Time and First-passage Area of Drifted Brownian Motion

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Publisher
Springer US
Copyright
Copyright © 2017 by Springer Science+Business Media New York
Subject
Statistics; Statistics, general; Life Sciences, general; Electrical Engineering; Economics, general; Business and Management, general
ISSN
1387-5841
eISSN
1573-7713
D.O.I.
10.1007/s11009-017-9546-7
Publisher site
See Article on Publisher Site

Abstract

For drifted Brownian motion X(t) = x-µ t + B t (µ > 0) starting from x > 0, we study the joint distribution of the first-passage time below zero ,t(x), and the first-passage area ,A(x), swept out by X till the time t(x). In particular, we establish differential equations with boundary conditions for the joint moments E[t(x) m A(x) n ], and we present an algorithm to find recursively them, for any m and n. Finally, the expected value of the time average of X till the time t(x) is obtained.

Journal

Methodology and Computing in Applied ProbabilitySpringer Journals

Published: Jan 25, 2017

References

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