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We study a classical Bayesian statistics problem of sequentially testing the sign of the drift of an arithmetic Brownian motion with the 0-1 loss function and a constant cost of observation per unit of time for general prior distributions. The statistical problem is reformulated as an optimal stopping problem with the current conditional probability that the drift is non-negative as the underlying process. The volatility of this conditional probability process is shown to be non-increasing in time, which enables us to prove monotonicity and continuity of the optimal stopping boundaries as well as to characterize them completely in the finite-horizon case as the unique continuous solution to a pair of integral equations. In the infinite-horizon case, the boundaries are shown to solve another pair of integral equations and a convergent approximation scheme for the boundaries is provided. Also, we describe the dependence between the prior distribution and the long-term asymptotic behaviour of the boundaries.
Ekström, Erik 1 ; Vaicenavicius, Juozas 1
@article{PS_2015__19__626_0,
author = {Ekstr\"om, Erik and Vaicenavicius, Juozas},
title = {Bayesian sequential testing of the drift of a {Brownian} motion},
journal = {ESAIM: Probability and Statistics},
pages = {626--648},
publisher = {EDP-Sciences},
volume = {19},
year = {2015},
doi = {10.1051/ps/2015012},
mrnumber = {3433430},
zbl = {1369.62201},
language = {en},
url = {http://geodesic.mathdoc.fr/articles/10.1051/ps/2015012/}
}
TY - JOUR AU - Ekström, Erik AU - Vaicenavicius, Juozas TI - Bayesian sequential testing of the drift of a Brownian motion JO - ESAIM: Probability and Statistics PY - 2015 SP - 626 EP - 648 VL - 19 PB - EDP-Sciences UR - http://geodesic.mathdoc.fr/articles/10.1051/ps/2015012/ DO - 10.1051/ps/2015012 LA - en ID - PS_2015__19__626_0 ER -
%0 Journal Article %A Ekström, Erik %A Vaicenavicius, Juozas %T Bayesian sequential testing of the drift of a Brownian motion %J ESAIM: Probability and Statistics %D 2015 %P 626-648 %V 19 %I EDP-Sciences %U http://geodesic.mathdoc.fr/articles/10.1051/ps/2015012/ %R 10.1051/ps/2015012 %G en %F PS_2015__19__626_0
Ekström, Erik; Vaicenavicius, Juozas. Bayesian sequential testing of the drift of a Brownian motion. ESAIM: Probability and Statistics, Tome 19 (2015), pp. 626-648. doi: 10.1051/ps/2015012
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