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Optimal Sequential Decisions in Hidden-State Models
Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics.
2017 (English)Doctoral thesis, comprehensive summary (Other academic)
Description
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 [en]
sequential analysis, optimal stopping, optimal liquidation, drift uncertainty, incomplete information, stochastic filtering
National Category
Probability Theory and Statistics
Identifiers
URN: urn:nbn:se:uu:diva-320809ISBN: 978-91-506-2641-4 (print)OAI: oai:DiVA.org:uu-320809DiVA: diva2:1091116
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
List of papers
1. Optimal liquidation of an asset under drift uncertainty
Open this publication in new window or tab >>Optimal liquidation of an asset under drift uncertainty
2016 (English)In: SIAM Journal on Financial Mathematics, ISSN 1945-497X, E-ISSN 1945-497XArticle in journal (Refereed) Published
Abstract [en]

We study a problem of finding an optimal stopping strategy to liquidate an asset with unknown drift. Taking a Bayesian approach, we model the initial beliefs of an individual about the drift by allowing an arbitrary probability distribution to characterize the uncertainty about the drift parameter. Filtering theory is used to describe the evolution of the posterior beliefs about the drift once the price process is being observed. An optimal stopping time is determined as the first passage time of the posterior mean below a monotone boundary, which can be characterized as the unique solution to a nonlinear integral equation. We also study monotonicity properties with respect to the prior distribution and the asset volatility.

Keyword
optimal liquidation, incomplete information, sequential analysis
National Category
Mathematics
Identifiers
urn:nbn:se:uu:diva-283523 (URN)10.1137/15M1033265 (DOI)000391850000013 ()
Available from: 2016-04-13 Created: 2016-04-13 Last updated: 2017-04-26Bibliographically approved
2. Asset liquidation under drift uncertainty and regime-switching volatility
Open this publication in new window or tab >>Asset liquidation under drift uncertainty and regime-switching volatility
(English)Article in journal (Other academic) Submitted
Abstract [en]

Optimal liquidation of an asset with unknown constant drift and stochastic regime-switching volatility is studied. The uncertainty about the drift is represented by an arbitrary probability distribution, the stochastic volatility is modelled by m-state Markov chain. Using filtering theory, an equivalent reformulation of the original problem as a four-dimensional optimal stopping problem is found and then analysed by constructing approximating sequences of three-dimensional optimal stopping problems. An optimal liquidation strategy and various structural properties of the problem are determined. Analysis of the two-point prior case is presented in detail, building on which, an outline of the extension to the general prior case is given.

Keyword
optimal liquidation, model uncertainty, regime-switching volatility, sequential analysis
National Category
Probability Theory and Statistics
Identifiers
urn:nbn:se:uu:diva-320805 (URN)
Available from: 2017-04-25 Created: 2017-04-25 Last updated: 2017-04-26
3. Bayesian Sequential Testing Of The Drift Of A Brownian Motion
Open this publication in new window or tab >>Bayesian Sequential Testing Of The Drift Of A Brownian Motion
2015 (English)In: ESAIM. P&S, ISSN 1292-8100, E-ISSN 1262-3318, Vol. 19, 626-648 p.Article in journal (Refereed) Published
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.

Keyword
Bayesian analysis, sequential hypothesis testing, optimal stopping
National Category
Probability Theory and Statistics
Identifiers
urn:nbn:se:uu:diva-276901 (URN)10.1051/ps/2015012 (DOI)000368218600031 ()
Funder
Swedish Research Council
Available from: 2016-02-16 Created: 2016-02-16 Last updated: 2017-04-26Bibliographically approved
4. Optimal stopping of a Brownian bridge with an unknown pinning point
Open this publication in new window or tab >>Optimal stopping of a Brownian bridge with an unknown pinning point
(English)Article in journal (Other academic) Submitted
National Category
Probability Theory and Statistics
Identifiers
urn:nbn:se:uu:diva-320806 (URN)
Available from: 2017-04-25 Created: 2017-04-25 Last updated: 2017-04-26
5. Wiener disorder detection under disorder magnitude uncertainty
Open this publication in new window or tab >>Wiener disorder detection under disorder magnitude uncertainty
(English)Manuscript (preprint) (Other academic)
National Category
Probability Theory and Statistics
Identifiers
urn:nbn:se:uu:diva-320807 (URN)
Available from: 2017-04-25 Created: 2017-04-25 Last updated: 2017-04-26

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