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A Bayesian sequential test for the drift of a fractional Brownian motion

Part of: Stochastic processes Sequential methods

Published online by Cambridge University Press:  03 December 2020

Alexey Muravlev  and
Mikhail Zhitlukhin
Alexey Muravlev*
Affiliation:
Steklov Mathematical Institute of Russian Academy of Sciences
Mikhail Zhitlukhin*
Affiliation:
Steklov Mathematical Institute of Russian Academy of Sciences
*
*Postal address: 8 Gubkina St., Moscow119991, Russia
*Postal address: 8 Gubkina St., Moscow119991, Russia
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Abstract

We consider a fractional Brownian motion with linear drift such that its unknown drift coefficient has a prior normal distribution and construct a sequential test for the hypothesis that the drift is positive versus the alternative that it is negative. We show that the problem of constructing the test reduces to an optimal stopping problem for a standard Brownian motion obtained by a transformation of the fractional Brownian motion. The solution is described as the first exit time from some set, and it is shown that its boundaries satisfy a certain integral equation, which is solved numerically.

Keywords

Sequential testChernoff’s testfractional Brownian motionoptimal stopping

MSC classification

Primary: 62L10: Sequential analysis
Secondary: 62L15: Optimal stopping 60G40: Stopping times; optimal stopping problems; gambling theory
Type
Original Article
Information
Advances in Applied Probability , Volume 52 , Issue 4 , December 2020 , pp. 1308 - 1324
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Copyright
© Applied Probability Trust 2020

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