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Vaicenavicius, Juozas

Open this publication in new window or tab >>Optimal Sequential Decisions in Hidden-State Models### Vaicenavicius, Juozas

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##### Abstract [en]

##### Place, publisher, year, edition, pages

Uppsala: Department of Mathematics, Uppsala University, 2017. p. 26
##### 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

### Pham, Huyên

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##### Supervisors

### Ekström, Erik

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Available from: 2017-05-18 Created: 2017-04-26 Last updated: 2017-05-18

Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics.

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.

Université Paris Diderot (Paris 7).

Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Applied Mathematics and Statistics.

Open this publication in new window or tab >>Optimal liquidation of an asset under drift uncertainty### Ekström, Erik

### Vaicenavicius, Juozas

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##### Abstract [en]

##### Keyword

optimal liquidation, incomplete information, sequential analysis
##### National Category

Mathematics
##### Identifiers

urn:nbn:se:uu:diva-283523 (URN)10.1137/15M1033265 (DOI)000391850000013 ()
#####

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Available from: 2016-04-13 Created: 2016-04-13 Last updated: 2017-11-30Bibliographically approved

Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Applied Mathematics and Statistics.

Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Applied Mathematics and Statistics.

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.

Open this publication in new window or tab >>Bayesian Sequential Testing Of The Drift Of A Brownian Motion### Ekström, Erik

### Vaicenavicius, Juozas

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##### Abstract [en]

##### 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 ()
#####

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##### Funder

Swedish Research Council
Available from: 2016-02-16 Created: 2016-02-16 Last updated: 2017-11-30Bibliographically approved

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.

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.

Open this publication in new window or tab >>Optimal Stopping under Drift Uncertainty### Vaicenavicius, Juozas

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##### Publisher

p. 62
##### Series

U.U.D.M. report / Uppsala University, Department of Mathematics, ISSN 1101-3591 ; 2015:1
##### National Category

Mathematical Analysis
##### Research subject

Mathematics
##### Identifiers

urn:nbn:se:uu:diva-251743 (URN)
##### Presentation

2015-05-13, Ångströmslabratoriet, Uppsala, 13:15
##### Opponent

### Gapeev, Pavel

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##### Supervisors

### Ekström, Erik

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Available from: 2015-04-27 Created: 2015-04-23 Last updated: 2015-04-27Bibliographically approved

Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Probability Theory.

London School of Economics.

Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Probability Theory.

Open this publication in new window or tab >>The 3/2 Model As A Stochastic Volatility Approximation For A Large-Basket Price-Weighted Index### Hambly, Ben

### Vaicenavicius, Juozas

Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics.PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt179_4_j_idt183_some",{id:"formSmash:j_idt179:4:j_idt183:some",widgetVar:"widget_formSmash_j_idt179_4_j_idt183_some",multiple:true}); PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt179_4_j_idt183_otherAuthors",{id:"formSmash:j_idt179:4:j_idt183:otherAuthors",widgetVar:"widget_formSmash_j_idt179_4_j_idt183_otherAuthors",multiple:true}); 2015 (English)In: International Journal of Theoretical and Applied Finance, ISSN 0219-0249, Vol. 18, no 6, article id 1550041Article in journal (Refereed) Published
##### Abstract [en]

##### Keyword

Index models, stochastic volatility models, large portfolio limit, diffusion approximation, volatility derivatives
##### National Category

Computational Mathematics
##### Identifiers

urn:nbn:se:uu:diva-271056 (URN)10.1142/S0219024915500417 (DOI)000365773200006 ()
#####

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Available from: 2016-01-05 Created: 2016-01-05 Last updated: 2017-12-01Bibliographically approved

Radcliffe Observ Quarter, Math Inst, Oxford OX2 6GG, England..

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.