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Ekström, Erik

Open this publication in new window or tab >>Sequential testing of a Wiener process with costly observations### Dyrssen, Hannah

### Ekström, Erik

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##### Abstract [en]

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

TAYLOR & FRANCIS INC, 2018
##### Keywords

Brownian motion, hypothesis testing, optimal stopping, sequential analysis
##### National Category

Probability Theory and Statistics
##### Identifiers

urn:nbn:se:uu:diva-351033 (URN)10.1080/07474946.2018.1427973 (DOI)000427076300004 ()
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##### Funder

Swedish Research Council, 622-2008-447]
Available from: 2018-05-18 Created: 2018-05-18 Last updated: 2018-05-18Bibliographically 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, Analysis and Probability Theory.

We consider the sequential testing of two simple hypotheses for the drift of a Brownian motion when each observation of the underlying process is associated with a positive cost. In this setting where continuous monitoring of the underlying process is not feasible, the question is not only whether to stop or to continue at a given observation time but also, if continuing, how to distribute the next observation time. Adopting a Bayesian methodology, we show that the value function can be characterized as the unique fixed point of an associated operator and that it can be constructed using an iterative scheme. Moreover, the optimal sequential distribution of observation times can be described in terms of the fixed point.

Open this publication in new window or tab >>Dynkin Games With Heterogeneous Beliefs### Ekström, Erik

### Glover, Kristoffer

### Leniec, Marta

PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt184_1_j_idt188_some",{id:"formSmash:j_idt184:1:j_idt188:some",widgetVar:"widget_formSmash_j_idt184_1_j_idt188_some",multiple:true}); PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt184_1_j_idt188_otherAuthors",{id:"formSmash:j_idt184:1:j_idt188:otherAuthors",widgetVar:"widget_formSmash_j_idt184_1_j_idt188_otherAuthors",multiple:true}); 2017 (English)In: Journal of Applied Probability, ISSN 0021-9002, E-ISSN 1475-6072, Vol. 54, no 1, p. 236-251Article in journal (Refereed) Published
##### Abstract [en]

##### Keywords

Dynkin game, heterogeneous belief, multiple Nash equilibria, optimal stopping theory
##### National Category

Other Mathematics
##### Identifiers

urn:nbn:se:uu:diva-287358 (URN)10.1017/jpr.2016.97 (DOI)000399075200016 ()
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##### Funder

Swedish Research Council
Available from: 2016-04-24 Created: 2016-04-24 Last updated: 2017-05-15Bibliographically approved

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

Univ Technol Sydney, POB 123, Broadway, NSW 2007, Australia..

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

We study zero-sum optimal stopping games (Dynkin games) between two players who disagree about the underlying model. In a Markovian setting, a verification result is established showing that if a pair of functions can be found that satisfies some natural conditions, then a Nash equilibrium of stopping times is obtained, with the given functions as the corresponding value functions. In general, however, there is no uniqueness of Nash equilibria, and different equilibria give rise to different value functions. As an example, we provide a thorough study of the game version of the American call option under heterogeneous beliefs. Finally, we also study equilibria in randomized stopping times.

Open this publication in new window or tab >>The dividend problem with a finite horizon### De Angelis, Tiziano

### Ekström, Erik

Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Probability Theory.PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt184_2_j_idt188_some",{id:"formSmash:j_idt184:2:j_idt188:some",widgetVar:"widget_formSmash_j_idt184_2_j_idt188_some",multiple:true}); PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt184_2_j_idt188_otherAuthors",{id:"formSmash:j_idt184:2:j_idt188:otherAuthors",widgetVar:"widget_formSmash_j_idt184_2_j_idt188_otherAuthors",multiple:true}); 2017 (English)In: The Annals of Applied Probability, ISSN 1050-5164, E-ISSN 2168-8737, Vol. 27, no 6, p. 3525-3546Article in journal (Refereed) Published
##### Abstract [en]

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

INST MATHEMATICAL STATISTICS, 2017
##### Keywords

The dividend problem, singular control, optimal stopping
##### National Category

Mathematics
##### Identifiers

urn:nbn:se:uu:diva-340290 (URN)10.1214/17-AAP1286 (DOI)000417972700007 ()
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##### Funder

Swedish Research Council
Available from: 2018-02-01 Created: 2018-02-01 Last updated: 2018-02-01Bibliographically approved

Univ Leeds, Sch Math, Leeds LS2 9JT, W Yorkshire, England..

We characterise the value function of the optimal dividend problem with a finite time horizon as the unique classical solution of a suitable Hamilton-Jacobi-Bellman equation. The optimal dividend strategy is realised by a Skorokhod reflection of the fund's value at a time-dependent optimal boundary. Our results are obtained by establishing for the first time a new connection between singular control problems with an absorbing boundary and optimal stopping problems on a diffusion reflected at 0 and created at a rate proportional to its local time.

Open this publication in new window or tab >>Momentum liquidation under partial information### Ekström, Erik

### Vannestål, Martin

Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Probability Theory.PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt184_3_j_idt188_some",{id:"formSmash:j_idt184:3:j_idt188:some",widgetVar:"widget_formSmash_j_idt184_3_j_idt188_some",multiple:true}); PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt184_3_j_idt188_otherAuthors",{id:"formSmash:j_idt184:3:j_idt188:otherAuthors",widgetVar:"widget_formSmash_j_idt184_3_j_idt188_otherAuthors",multiple:true}); 2016 (English)In: Journal of Applied Probability, ISSN 0021-9002, E-ISSN 1475-6072, Vol. 53, no 2, p. 341-359Article in journal (Refereed) Published
##### Abstract [en]

##### Keywords

Momentum trading; stock selling; optimal stopping; quickest detection problem for Brownian motion
##### National Category

Mathematics Probability Theory and Statistics
##### Identifiers

urn:nbn:se:uu:diva-283519 (URN)10.1017/jpr.2016.4 (DOI)000378598700003 ()
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##### Funder

Swedish Research Council
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.

Momentum is the notion that an asset that has performed well in the past will continue to do so for some period. We study the optimal liquidation strategy for a momentum trade in a setting where the drift of the asset drops from a high value to a smaller one at some random change-point. This change-point is not directly observable for the trader, but it is partially observable in the sense that it coincides with one of the jump times of some exogenous Poisson process representing external shocks, and these jump times are assumed to be observable. Comparisons with existing results for momentum trading under incomplete information show that the assumption that the disappearance of the momentum effect is triggered by observable external shocks significantly improves the optimal strategy.

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

### Vaicenavicius, Juozas

PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt184_4_j_idt188_some",{id:"formSmash:j_idt184:4:j_idt188:some",widgetVar:"widget_formSmash_j_idt184_4_j_idt188_some",multiple:true}); PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt184_4_j_idt188_otherAuthors",{id:"formSmash:j_idt184:4:j_idt188:otherAuthors",widgetVar:"widget_formSmash_j_idt184_4_j_idt188_otherAuthors",multiple:true}); 2016 (English)In: SIAM Journal on Financial Mathematics, ISSN 1945-497X, E-ISSN 1945-497XArticle in journal (Refereed) Published
##### Abstract [en]

##### Keywords

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 >>The inverse first-passage problem and optimal stopping### Ekström, Erik

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

Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Probability Theory.PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt184_5_j_idt188_some",{id:"formSmash:j_idt184:5:j_idt188:some",widgetVar:"widget_formSmash_j_idt184_5_j_idt188_some",multiple:true}); PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt184_5_j_idt188_otherAuthors",{id:"formSmash:j_idt184:5:j_idt188:otherAuthors",widgetVar:"widget_formSmash_j_idt184_5_j_idt188_otherAuthors",multiple:true}); 2016 (English)In: The Annals of Applied Probability, ISSN 1050-5164, E-ISSN 2168-8737, Vol. 26, no 5, p. 3154-3177Article in journal (Refereed) Published
##### Abstract [en]

##### Keywords

Inverse first-passage problem, optimal stopping, nonlinear integral equation
##### National Category

Mathematics Probability Theory and Statistics
##### Identifiers

urn:nbn:se:uu:diva-283524 (URN)10.1214/16-AAP1172 (DOI)000386864600017 ()
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##### Funder

Swedish Research CouncilKnut and Alice Wallenberg Foundation
Available from: 2016-04-13 Created: 2016-04-13 Last updated: 2017-11-30Bibliographically approved

Given a survival distribution on the positive half-axis and a Brownian motion, a solution of the inverse first-passage problem consists of a boundary so that the first passage time over the boundary has the given distribution. We show that the solution of the inverse first-passage problem coincides with the solution of a related optimal stopping problem. Consequently, methods from optimal stopping theory may be applied in the study of the inverse first passage problem. We illustrate this with a study of the associated integral equation for the boundary.

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

### Vaicenavicius, Juozas

PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt184_6_j_idt188_some",{id:"formSmash:j_idt184:6:j_idt188:some",widgetVar:"widget_formSmash_j_idt184_6_j_idt188_some",multiple:true}); PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt184_6_j_idt188_otherAuthors",{id:"formSmash:j_idt184:6:j_idt188:otherAuthors",widgetVar:"widget_formSmash_j_idt184_6_j_idt188_otherAuthors",multiple:true}); 2015 (English)In: ESAIM. P&S, ISSN 1292-8100, E-ISSN 1262-3318, Vol. 19, p. 626-648Article in journal (Refereed) Published
##### Abstract [en]

##### Keywords

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 >>Feynman-Kac Theorems for Generalized Diffusions### Ekström, Erik

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

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

Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Probability Theory.PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt184_7_j_idt188_some",{id:"formSmash:j_idt184:7:j_idt188:some",widgetVar:"widget_formSmash_j_idt184_7_j_idt188_some",multiple:true}); PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt184_7_j_idt188_otherAuthors",{id:"formSmash:j_idt184:7:j_idt188:otherAuthors",widgetVar:"widget_formSmash_j_idt184_7_j_idt188_otherAuthors",multiple:true}); 2015 (English)In: Transactions of the American Mathematical Society, ISSN 0002-9947, E-ISSN 1088-6850, Vol. 367, no 11, p. 8051-8070, article id PII S0002-9947(2015)06278-3Article in journal (Refereed) Published
##### Abstract [en]

##### Keywords

Gap diffusions, Feynman-Kac representation theorem, martingales
##### National Category

Mathematics
##### Identifiers

urn:nbn:se:uu:diva-264018 (URN)000360988900018 ()
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##### Funder

Swedish Research CouncilKnut and Alice Wallenberg Foundation
Available from: 2015-10-08 Created: 2015-10-05 Last updated: 2017-12-01Bibliographically approved

We find Feynman-Kac type representation theorems for generalized diffusions. To do this we need to establish existence, uniqueness and regularity results for equations with measure-valued coefficients.

Open this publication in new window or tab >>Short-time implied volatility in exponential Lévy models### Ekström, Erik

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

Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics.PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt184_8_j_idt188_some",{id:"formSmash:j_idt184:8:j_idt188:some",widgetVar:"widget_formSmash_j_idt184_8_j_idt188_some",multiple:true}); PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt184_8_j_idt188_otherAuthors",{id:"formSmash:j_idt184:8:j_idt188:otherAuthors",widgetVar:"widget_formSmash_j_idt184_8_j_idt188_otherAuthors",multiple:true}); 2015 (English)In: International Journal of Theoretical and Applied Finance, ISSN 0219-0249, Vol. 18, no 4, article id 1550025Article in journal (Other academic) Published
##### Abstract [en]

##### Keywords

implied volatility; exponential Levy models; short-time asymptotic behavior.
##### National Category

Mathematics Other Mathematics
##### Identifiers

urn:nbn:se:uu:diva-209221 (URN)10.1142/S0219024915500259 (DOI)000365773000004 ()
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Available from: 2013-10-15 Created: 2013-10-15 Last updated: 2017-12-06Bibliographically approved

We show that a necessary and sufficient condition for the explosion of implied volatility near expiry in exponential Levy models is the existence of jumps towards the strike price in the underlying process. When such jumps do not exist, the implied volatility converges to the volatility of the Gaussian component of the underlying Levy process as the time to maturity tends to zero.Those results are proved by comparing the short-time asymptotics of the Black-Scholes price to the explicit formulas for upper or lower bounds of option prices in exponential Levy models.

Open this publication in new window or tab >>Pricing equations in jump-to-default models### Dyrssen, Hannah

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

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

Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics.PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt184_9_j_idt188_some",{id:"formSmash:j_idt184:9:j_idt188:some",widgetVar:"widget_formSmash_j_idt184_9_j_idt188_some",multiple:true}); PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt184_9_j_idt188_otherAuthors",{id:"formSmash:j_idt184:9:j_idt188:otherAuthors",widgetVar:"widget_formSmash_j_idt184_9_j_idt188_otherAuthors",multiple:true}); 2014 (English)In: International Journal of Theoretical and Applied Finance, ISSN 0219-0249Article in journal (Refereed) Published
##### Abstract [en]

##### National Category

Mathematics
##### Identifiers

urn:nbn:se:uu:diva-283517 (URN)
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Available from: 2016-04-13 Created: 2016-04-13 Last updated: 2017-11-30

We study pricing equations in jump-to-default models, and we provide conditions under which the option price is the unique classical solution, with a special focus on boundary conditions. In particular, we find precise conditions ensuring that the option price at the default boundary coincides with the recovery payment. We also study spatial convexity of the option price, and we explore the connection between preservation of convexity and parameter monotonicity.