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  • 1.
    Ekström, Erik
    et al.
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics.
    Vaicenavicius, Juozas
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics.
    Bayesian Sequential Testing Of The Drift Of A Brownian Motion2015In: ESAIM. P&S, ISSN 1292-8100, E-ISSN 1262-3318, Vol. 19, p. 626-648Article in journal (Refereed)
    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.

  • 2.
    Ekström, Erik
    et al.
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Applied Mathematics and Statistics.
    Vaicenavicius, Juozas
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Applied Mathematics and Statistics.
    Optimal liquidation of an asset under drift uncertainty2016In: SIAM Journal on Financial Mathematics, ISSN 1945-497X, E-ISSN 1945-497XArticle in journal (Refereed)
    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.

  • 3.
    Hambly, Ben
    et al.
    Radcliffe Observ Quarter, Math Inst, Oxford OX2 6GG, England..
    Vaicenavicius, Juozas
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics.
    The 3/2 Model As A Stochastic Volatility Approximation For A Large-Basket Price-Weighted Index2015In: International Journal of Theoretical and Applied Finance, ISSN 0219-0249, Vol. 18, no 6, article id 1550041Article in journal (Refereed)
    Abstract [en]

    We derive large-basket approximations of a price-weighted index whose component prices follow a single sector jump-diffusion model. As the basket size approaches infinity, a suitable average converges to a Black-Scholes model driven by the common factor process. We extend this by considering the behavior of the residual idiosyncratic noise and show that a version of the 3/2 model emerges as a natural stochastic volatility model approximation. This provides a theoretical justification for its use as a model for jointly pricing index and volatility derivatives.

  • 4.
    Vaicenavicius, Juozas
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics.
    Optimal Sequential Decisions in Hidden-State Models2017Doctoral 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.

    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: 2017-11-30Bibliographically 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, p. 626-648Article 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: 2017-11-30Bibliographically 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
  • 5.
    Vaicenavicius, Juozas
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Probability Theory.
    Optimal Stopping under Drift Uncertainty2015Licentiate thesis, comprehensive summary (Other academic)
1 - 5 of 5
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