Bayesian sequential testing of the drift of a Brownian motion
Ekström, Erik1 ; Vaicenavicius, Juozas1
1 Uppsala University, Box 480, 75106 Uppsala, Sweden
ESAIM: Probability and Statistics, Tome 19 (2015), pp. 626-648.

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.

DOI : 10.1051/ps/2015012
Classification : 62L10, 60G40, 62C10
Mots-clés : Bayesian analysis, sequential hypothesis testing, optimal stopping
Ekström, Erik 1 ; Vaicenavicius, Juozas 1

1 Uppsala University, Box 480, 75106 Uppsala, Sweden 
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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. http://www.numdam.org/articles/10.1051/ps/2015012/

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