uu.seUppsala University Publications
Change search
CiteExportLink to record
Permanent link

Direct link
Cite
Citation style
  • apa
  • ieee
  • modern-language-association
  • vancouver
  • Other style
More styles
Language
  • de-DE
  • en-GB
  • en-US
  • fi-FI
  • nn-NO
  • nn-NB
  • sv-SE
  • Other locale
More languages
Output format
  • html
  • text
  • asciidoc
  • rtf
Bayesian Sequential Testing Of The Drift Of A Brownian Motion
Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics.
Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics.
2015 (English)In: ESAIM. P&S, ISSN 1292-8100, E-ISSN 1262-3318, Vol. 19, 626-648 p.Article in journal (Refereed) Published
Resource type
Text
Abstract [en]

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.

Place, publisher, year, edition, pages
2015. Vol. 19, 626-648 p.
Keyword [en]
Bayesian analysis, sequential hypothesis testing, optimal stopping
National Category
Probability Theory and Statistics
Identifiers
URN: urn:nbn:se:uu:diva-276901DOI: 10.1051/ps/2015012ISI: 000368218600031OAI: oai:DiVA.org:uu-276901DiVA: diva2:903659
Funder
Swedish Research Council
Available from: 2016-02-16 Created: 2016-02-16 Last updated: 2017-11-30Bibliographically approved
In thesis
1. Optimal Sequential Decisions in Hidden-State Models
Open this publication in new window or tab >>Optimal Sequential Decisions in Hidden-State Models
2017 (English)Doctoral thesis, comprehensive summary (Other academic)
Abstract [en]

This doctoral thesis consists of five research articles on the general topic of optimal decision making under uncertainty in a Bayesian framework. The papers are preceded by three introductory chapters.

Papers I and II are dedicated to the problem of finding an optimal stopping strategy to liquidate an asset with unknown drift. In Paper I, the price is modelled by the classical Black-Scholes model with unknown drift. The first passage time of the posterior mean below a monotone boundary is shown to be optimal. The boundary is characterised as the unique solution to a nonlinear integral equation. Paper II solves the same optimal liquidation problem, but in a more general model with stochastic regime-switching volatility. An optimal liquidation strategy and various structural properties of the problem are determined.

In Paper III, the 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 is studied from a Bayesian perspective. Optimal decision strategies for arbitrary prior distributions are determined and investigated. The strategies consist of two monotone stopping boundaries, which we characterise in terms of integral equations.

In Paper IV, the problem of stopping a Brownian bridge with an unknown pinning point to maximise the expected value at the stopping time is studied. Besides a few general properties established, structural properties of an optimal strategy are shown to be sensitive to the prior. A general condition for a one-sided optimal stopping region is provided.

Paper V deals with the problem of detecting a drift change of a Brownian motion under various extensions of the classical Wiener disorder problem. Monotonicity properties of the solution with respect to various model parameters are studied. Also, effects of a possible misspecification of the underlying model are explored.

Place, publisher, year, edition, pages
Uppsala: Department of Mathematics, Uppsala University, 2017. 26 p.
Series
Uppsala Dissertations in Mathematics, ISSN 1401-2049 ; 101
Keyword
sequential analysis, optimal stopping, optimal liquidation, drift uncertainty, incomplete information, stochastic filtering
National Category
Probability Theory and Statistics
Identifiers
urn:nbn:se:uu:diva-320809 (URN)978-91-506-2641-4 (ISBN)
Public defence
2017-06-09, 80101, Ångströmlaboratoriet, Lägerhyddsvägen 1, Uppsala, 13:00 (English)
Opponent
Supervisors
Available from: 2017-05-18 Created: 2017-04-26 Last updated: 2017-05-18

Open Access in DiVA

No full text

Other links

Publisher's full text

Authority records BETA

Ekström, ErikVaicenavicius, Juozas

Search in DiVA

By author/editor
Ekström, ErikVaicenavicius, Juozas
By organisation
Department of Mathematics
In the same journal
ESAIM. P&S
Probability Theory and Statistics

Search outside of DiVA

GoogleGoogle Scholar

doi
urn-nbn

Altmetric score

doi
urn-nbn
Total: 246 hits
CiteExportLink to record
Permanent link

Direct link
Cite
Citation style
  • apa
  • ieee
  • modern-language-association
  • vancouver
  • Other style
More styles
Language
  • de-DE
  • en-GB
  • en-US
  • fi-FI
  • nn-NO
  • nn-NB
  • sv-SE
  • Other locale
More languages
Output format
  • html
  • text
  • asciidoc
  • rtf