Bayesian sequential testing of the drift of a Brownian motion

Abstract

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.

Anscombe’s model for sequential clinical trials revisited

“…Note that this is not a new approach in these kinds of problems, but corresponds to standard methodology in filtering theory, see Bain and Crisan (2009, Section 3.3). For example, Ekström and Vaicenavicius (2015) make use of essentially the same technique when analysing a sequential testing problem of the drift of a Brownian motion. If a change of measure is defined via…”

“…Now, our interest here is not in providing the precise technical conditions under which a solution may be obtained via the free-boundary method and integral equations. Detailed discussions are provided for a long list of similar problems elsewhere in the literature, see Peškir and Shiryaev (2006) for an overview or Ekström and Vaicenavicius (2015) for a more recent result. Rather, we wish to illustrate how the general approach may be used to solve Anscombe's classical problem and various extensions of it.…”

Bayesian sequential composite hypothesis testing in discrete time

“…set-ups where the unknown parameters can take only two possible values. In [23], a hypothesis testing problem for a case with three possible drifts is examined, and in [10] a composite hypothesis problem for the drift of a Wiener process is studied with a general prior distribution. Moreover, [9] study a sequential estimation problem for a Wiener process in the same set-up.…”

“…Moreover, [9] study a sequential estimation problem for a Wiener process in the same set-up. Key to the analysis in [10] and [9] is the choice of appropriate variables. In fact, in [10] it is shown that if instead of the observation process one uses the conditional probability Π as state variable, then the corresponding continuation region is shrinking in time; a similar result holds for sequential least-square estimation if one uses the conditional expectation as state variable.…”

Sequential testing of a Wiener process with costly observations

“…Utilizing the connection between optimal stopping problems and free-boundary problems, Shiryaev (1969Shiryaev ( , 1978 provides an explicit solution of the hypothesis testing problem when the dri can take only two di erent values. Notable recent contributions include the extension to the nite horizon hypothesis testing problem (Gapeev and Peskir, 2004), the characterization of the solution to the original Cherno problem in terms of an associated integral equation (Zhitlukhin and Muravlev, 2013), a study of the case with three hypotheses (Zhitlukhin and Shiryaev, 2011), and a study of the case with general prior distributions (Ekström and Vaicenavicius, 2015). Along a related line of research, various authors have extended the problem to include more general underlying processes.…”