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Some Extensions of Fractional Ornstein-Uhlenbeck Model: Arbitrage and Other Applications
Stockholm University, Faculty of Social Sciences, Department of Statistics.
2017 (English)Doctoral thesis, comprehensive summary (Other academic)
Abstract [en]

This doctoral thesis endeavors to extend probability and statistical models using stochastic differential equations. The described models capture essential features from data that are not explained by classical diffusion models driven by Brownian motion.

New results obtained by the author are presented in five articles. These are divided into two parts. The first part involves three articles on statistical inference and simulation of a family of processes related to fractional Brownian motion and Ornstein-Uhlenbeck process, the so-called fractional Ornstein-Uhlenbeck process of the second kind (fOU2). In two of the articles, we show how to simulate fOU2 by means of circulant embedding method and memoryless transformations. In the other one, we construct a least squares consistent estimator of the drift parameter and prove the central limit theorem using techniques from Stochastic Calculus for Gaussian processes and Malliavin Calculus.

The second phase of my research consists of two articles about jump market models and arbitrage portfolio strategies for an insider trader. One of the articles describes two arbitrage free markets according to their risk neutral valuation formula and an arbitrage strategy by switching the markets. The key aspect is the difference in volatility between the markets. Statistical evidence of this situation is shown from a sequential data set. In the other one, we analyze the arbitrage strategies of an strong insider in a pure jump Markov chain financial market by means of a likelihood process. This is constructed in an enlarged filtration using Itô calculus and general theory of stochastic processes.

Abstract [sv]

Föreliggande doktorsavhandling strävar efter att utöka sannolikhetsbaserade och statistiska modeller med stokastiska differentialekvationer. De beskrivna modellerna fångar väsentliga egenskaper i data som inte förklaras av klassiska diffusionsmodeller för brownsk rörelse. 

Nya resultat, som författaren har härlett, presenteras i fem uppsatser. De är ordnade i två delar. Del 1 innehåller tre uppsatser om statistisk inferens och simulering av en familj av stokastiska processer som är relaterade till fraktionell brownsk rörelse och Ornstein-Uhlenbeckprocessen, så kallade andra ordningens fraktionella Ornstein-Uhlenbeckprocesser (fOU2). I två av uppsatserna visar vi hur vi kan simulera fOU2-processer med hjälp av cyklisk inbäddning och minneslös transformering. I den tredje uppsatsen konstruerar vi en minsta-kvadratestimator som ger konsistent skattning av driftparametern och bevisar centrala gränsvärdessatsen med tekniker från statistisk analys för gaussiska processer och malliavinsk analys. 

Del 2 av min forskning består av två uppsatser om marknadsmodeller med plötsliga hopp och portföljstrategier med arbitrage för en insiderhandlare. En av uppsatserna beskriver två arbitragefria marknader med riskneutrala värderingsformeln och en arbitragestrategi som består i växla mellan marknaderna. Den väsentliga komponenten är skillnaden mellan marknadernas volatilitet. Statistisk evidens i den här situationen visas utifrån ett sekventiellt datamaterial. I den andra uppsatsen analyserar vi arbitragestrategier hos en insiderhandlare i en finansiell marknad som förändrar sig enligt en Markovkedja där alla förändringar i tillstånd består av plötsliga hopp. Det gör vi med en likelihoodprocess. Vi konstruerar detta med utökad filtrering med hjälp av Itôanalys och allmän teori för stokastiska processer.

Place, publisher, year, edition, pages
Stockholm: Department of Statistics, Stockholm University , 2017. , p. 54
Keyword [en]
fractional Ornstein-Uhlenbeck process, insider information, simulation embedding method, jump times, least-squares estimator, likelihood process, Ito calculus, Malliavin calculus, stochastic calculus
National Category
Probability Theory and Statistics
Research subject
Statistics
Identifiers
URN: urn:nbn:se:su:diva-147437ISBN: 978-91-7649-994-8 (print)ISBN: 978-91-7649-995-5 (electronic)OAI: oai:DiVA.org:su-147437DiVA, id: diva2:1145085
Public defence
2017-11-10, William-Olssonsalen, Geovetenskapens hus, Svante Arrhenius väg 14, Stockholm, 13:00 (English)
Opponent
Supervisors
Note

At the time of the doctoral defense, the following papers were unpublished and had a status as follows: Paper 4: Manuscript. Paper 5: Manuscript.

Available from: 2017-10-18 Created: 2017-09-28 Last updated: 2017-10-16Bibliographically approved
List of papers
1. Empirical Evidence on arbitrage by changing the stock exchange
Open this publication in new window or tab >>Empirical Evidence on arbitrage by changing the stock exchange
2009 (English)In: Advances and Applications in Statistics, ISSN 0972-3617, Vol. 12, no 2, p. 223-233Article in journal (Refereed) Published
Keyword
arbitrage, Black-Scholes, option-pricing, relisting, volatility
National Category
Probability Theory and Statistics
Research subject
Statistics
Identifiers
urn:nbn:se:su:diva-147431 (URN)
Available from: 2017-09-27 Created: 2017-09-27 Last updated: 2017-09-29Bibliographically approved
2. Initial Enlargement in a Markov Chain Market Model
Open this publication in new window or tab >>Initial Enlargement in a Markov Chain Market Model
2011 (English)In: Stochastics and Dynamics, ISSN 0219-4937, Vol. 11, no 2-3, p. 389-413Article in journal (Refereed) Published
Abstract [en]

Enlargement of filtrations is a classical topic in the general theory of stochastic processes. This theory has been applied to stochastic finance in order to analyze models with insider information. In this paper we study initial enlargement in a Markov chain market model, introduced by Norberg. In the enlarged filtration, several things can happen: some of the jumps times can be accessible or predictable, but in the original filtration all the jumps times are totally inaccessible. But even if the jumps times change to accessible or predictable, the insider does not necessarily have arbitrage possibilities. Read More: http://www.worldscientific.com/doi/abs/10.1142/S021949371100336X

Keyword
Markov chain market model, initial enlargement, jump times, insider information
National Category
Probability Theory and Statistics
Research subject
Statistics
Identifiers
urn:nbn:se:su:diva-147432 (URN)10.1142/S021949371100336X (DOI)
Available from: 2017-09-27 Created: 2017-09-27 Last updated: 2017-09-29Bibliographically approved
3. Drift parameter estimation for fractional Ornstein-Uhlenbeck process of the second kind
Open this publication in new window or tab >>Drift parameter estimation for fractional Ornstein-Uhlenbeck process of the second kind
2015 (English)In: Statistics (Berlin), ISSN 0233-1888, E-ISSN 1029-4910, Vol. 49, no 1, p. 1-18Article in journal (Refereed) Published
Abstract [en]

The fractional Ornstein-Uhlenbeck process of the second kind (fOU(2)) is the solution of the Langevin equation <inline-graphic xmlns:xlink=http://www.w3.org/1999/xlink xlink:href=gsta_a_863888_ilm0001.gif></inline-graphic> with driving noise <inline-graphic xmlns:xlink=http://www.w3.org/1999/xlink xlink:href=gsta_a_863888_ilm0002.gif></inline-graphic> where B is a fractional Brownian motion with Hurst parameter H(0, 1). In this article, in the case H>1/2, we prove that the least-squares estimator <inline-graphic xmlns:xlink=http://www.w3.org/1999/xlink xlink:href=gsta_a_863888_ilm0003.gif></inline-graphic> introduced in [Hu Y, Nualart D. Parameter estimation for fractional Ornstein-Uhlenbeck processes. Stat. Probab. Lett. 2010;80(11-12):1030-1038], provides a consistent estimator. Moreover, using central limit theorem for multiple Wiener integrals, we prove asymptotic normality of the estimator valid for the whole range H(1/2, 1).

Keyword
fractional Ornstein-Uhlenbeck processes, Malliavin calculus, Langevin equation, least-squares estimator, 60G22, 60H07, 62F12
National Category
Mathematics
Research subject
Statistics
Identifiers
urn:nbn:se:su:diva-114748 (URN)10.1080/02331888.2013.863888 (DOI)000348852300001 ()
Note

AuthorCount:2;

Available from: 2015-03-12 Created: 2015-03-09 Last updated: 2017-09-29Bibliographically approved
4. Simulation of fractional Ornstein-Uhlenbeck of the second kind by Circulant Embedding method
Open this publication in new window or tab >>Simulation of fractional Ornstein-Uhlenbeck of the second kind by Circulant Embedding method
2017 (English)Conference paper, Oral presentation only (Refereed)
National Category
Probability Theory and Statistics
Research subject
Statistics
Identifiers
urn:nbn:se:su:diva-146853 (URN)
Conference
SPAS2017 International Conference on Stochastic Processes and Algebraic Structures – From Theory Towards Applications, Västerås and Stockholm, Sweden, October 4–6, 2017
Available from: 2017-09-13 Created: 2017-09-13 Last updated: 2017-11-28
5. Simulations-based Study of Covariance Structure for Fractional Ornstein-Uhlenbeck process of the Second Kind
Open this publication in new window or tab >>Simulations-based Study of Covariance Structure for Fractional Ornstein-Uhlenbeck process of the Second Kind
(English)Manuscript (preprint) (Other academic)
National Category
Probability Theory and Statistics
Research subject
Statistics
Identifiers
urn:nbn:se:su:diva-147433 (URN)
Available from: 2017-09-27 Created: 2017-09-27 Last updated: 2017-09-29Bibliographically approved

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