umu.sePublications
Change search
CiteExportLink to record
Permanent link

Direct link
Cite
Citation style
  • apa
  • ieee
  • modern-language-association-8th-edition
  • vancouver
  • Other style
More styles
Language
  • de-DE
  • en-GB
  • en-US
  • fi-FI
  • nn-NO
  • nn-NB
  • sv-SE
  • Other locale
More languages
Output format
  • html
  • text
  • asciidoc
  • rtf
The Skorohod problem and weak approximation of stochastic differential equations in time-dependent domains
Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
2009 (English)Doctoral thesis, comprehensive summary (Other academic)
Abstract [en]

This thesis consists of a summary and four scientific articles. All four articles consider various aspects of stochastic differential equations and the purpose of the summary is to provide an introduction to this subject and to supply the notions required in order to fully understand the articles.

In the first article we conduct a thorough study of the multi-dimensional Skorohod problem in time-dependent domains. In particular we prove the existence of cádlág solutions to the Skorohod problem with oblique reflection in time-independent domains with corners. We use this existence result to construct weak solutions to stochastic differential equations with oblique reflection in time-dependent domains. In the process of obtaining these results we also establish convergence results for sequences of solutions to the Skorohod problem and a number of estimates for solutions, with bounded jumps, to the Skorohod problem.

The second article considers the problem of determining the sensitivities of a solution to a second order parabolic partial differential equation with respect to perturbations in the parameters of the equation. We derive an approximate representation of the sensitivities and an estimate of the discretization error arising in the sensitivity approximation. We apply these theoretical results to the problem of determining the sensitivities of the price of European swaptions in a LIBOR market model with respect to perturbations in the volatility structure (the so-called ‘Greeks’).

The third article treats stopped diffusions in time-dependent graph domains with low regularity. We compare, numerically, the performance of one adaptive and three non-adaptive numerical methods with respect to order of convergence, efficiency and stability. In particular we investigate if the performance of the algorithms can be improved by a transformation which increases the regularity of the domain but, at the same time, reduces the regularity of the parameters of the diffusion.

In the fourth article we use the existence results obtained in Article I to construct a projected Euler scheme for weak approximation of stochastic differential equations with oblique reflection in time-dependent domains. We prove theoretically that the order of convergence of the proposed algorithm is 1/2 and conduct numerical simulations which support this claim.

Place, publisher, year, edition, pages
Umeå: Umeå universitet , 2009. , 36 p.
Series
Doctoral thesis / Umeå University, Department of Mathematics, ISSN 1102-8300 ; 42
Keyword [en]
Skorohod problem, weak approximation, time-dependent domain, stochastic differential equations, parabolic partial differential equations, oblique reflection, stopped diffusions, Euler scheme, adaptive methods, sensitivity analysis, financial derivatives, 'Greeks'
National Category
Mathematics
Research subject
Mathematics
Identifiers
URN: urn:nbn:se:umu:diva-25429ISBN: 978-91-7264-823-4 (print)OAI: oai:DiVA.org:umu-25429DiVA: diva2:231754
Distributor:
Institutionen för matematik och matematisk statistik, 90187, Umeå
Public defence
2009-09-18, MA121, MIT-huset, Umeå universitet, Umeå, 13:15 (English)
Opponent
Supervisors
Available from: 2009-08-31 Created: 2009-08-17 Last updated: 2010-11-03Bibliographically approved
List of papers
1. The Skorohod oblique reflection problem in time-dependent domains
Open this publication in new window or tab >>The Skorohod oblique reflection problem in time-dependent domains
2010 (English)In: Annals of Probability, ISSN 0091-1798, E-ISSN 2168-894X, Vol. 38, no 6, 2170-2223 p.Article in journal (Refereed) Published
Abstract [en]

The deterministic Skorohod problem plays an important role in the construction and analysis of diffusion processes with reflection. In the form studied here, the multidimensional Skorohod problem was introduced, in time-independent domains, by H. Tanaka and further investigated by P.-L. Lions and A.-S. Sznitman in their celebrated article. Subsequent results of several researchers have resulted in a large literature on the Skorohod problem in time-independent domains. In this article we conduct a thorough study of the multidimensional Skorohod problem in time-dependent domains. In particular, we prove the existence of càdlàg solutions (x, λ) to the Skorohod problem, with oblique reflection, for (D,,w) assuming, in particular, that D is a time-dependent domain (Theorem 1.2). In addition, we prove that if w is continuous, then x is continuous as well (Theorem 1.3). Subsequently, we use the established existence results to construct solutions to stochastic differential equations with oblique reflection (Theorem 1.9) in time-dependent domains. In the process of proving these results we establish a number of estimates for solutions to the Skorohod problem with bounded jumps and, in addition, several results concerning the convergence of sequences of solutions to Skorohod problems in the setting of time-dependent domains.

Keyword
Skorohod problem, oblique reflection, time-dependent domain, stochastic differential equations
National Category
Mathematics
Research subject
Mathematics
Identifiers
urn:nbn:se:umu:diva-37418 (URN)10.1214/10-AOP538 (DOI)000283010500003 ()
Available from: 2010-11-02 Created: 2010-11-02 Last updated: 2017-12-12Bibliographically approved
2. Pricing and hedging of financial derivatives using a posteriori error estimates and adaptive methods for stochastic differential equations
Open this publication in new window or tab >>Pricing and hedging of financial derivatives using a posteriori error estimates and adaptive methods for stochastic differential equations
2010 (English)In: Journal of Computational and Applied Mathematics, ISSN 0377-0427, E-ISSN 1879-1778, Vol. 235, 563-592 p.Article in journal (Refereed) Published
Abstract [en]

The efficient and accurate calculation of sensitivities of the price of financial derivatives with respect to perturbations of the parameters in the underlying model, the so-called `Greeks', remains a great practical challenge in the derivative industry. This is true regardless of whether methods for partial differential equations or stochastic differential equations (Monte Carlo techniques) are being used. The computation of the `Greeks' is essential to risk management and to the hedging of financial derivatives and typically requires substantially more computing time as compared to simply pricing the derivatives. Any numerical algorithm (Monte Carlo algorithm) for stochastic differential equations produces a time-discretization error and a statistical error in the process of pricing financial derivatives and calculating the associated `Greeks'. In this article we show how a posteriori error estimates and adaptive methods for stochastic differential equations can be used to control both these errors in the context of pricing and hedging of financial derivatives. In particular, we derive expansions, with leading order terms which are computable in a posteriori form, of the time-discretization errors for the price and the associated `Greeks'. These expansions allow the user to simultaneously first control the time-discretization errors in an adaptive fashion, when calculating the price, sensitivities and hedging parameters with respect to a large number of parameters, and then subsequently to ensure that the total errors are, with prescribed probability, within tolerance.

Place, publisher, year, edition, pages
Elsevier, 2010
Keyword
Sensitivity analysis, Parabolic partial differential equations, Stochastic differential equations, Euler scheme, A posteriori error estimate, Adaptive algorithms, hedging, Financial derivatives
National Category
Mathematics
Research subject
Mathematics
Identifiers
urn:nbn:se:umu:diva-37419 (URN)10.1016/j.cam.2010.06.009 (DOI)000282394500004 ()
Available from: 2010-11-03 Created: 2010-11-02 Last updated: 2017-12-12Bibliographically approved
3. On Monte Carlo algorithms applied to Dirichlet problems for parabolic operators in the setting of time-dependent domains
Open this publication in new window or tab >>On Monte Carlo algorithms applied to Dirichlet problems for parabolic operators in the setting of time-dependent domains
2009 (English)In: Monte Carlo Methods and Applications, ISSN 1569-3961, Vol. 15, no 1, 11-47 p.Article in journal (Refereed) Published
Abstract [en]

Dirichlet problems for second order parabolic operators in space-time domains Ω⊂ Rn+1  are of paramount importance in analysis, partial differential equations and applied mathematics. These problems can be approached in many different ways using techniques from partial differential equations, potential theory, stochastic differential equations, stopped diffusions and Monte Carlo methods. The performance of any technique depends on the structural assumptions on the operator, the geometry and smoothness properties of the space-time domain Ω, the smoothness of the Dirichlet data and the smoothness of the coefficients of the operator under consideration. In this paper, which mainly is of numerical nature, we attempt to further understand how Monte Carlo methods based on the numerical integration of stochastic differential equations perform when applied to Dirichlet problems for uniformly elliptic second order parabolic operators and how their performance vary as the smoothness of the boundary, Dirichlet data and coefficients change from smooth to non-smooth. Our analysis is set in the genuinely parabolic setting of time-dependent domains, which in itself adds interesting features previously only modestly discussed in the literature. The methods evaluated and discussed include elaborations on the non-adaptive method proposed by Gobet [4] based on approximation by half spaces and exit probabilities and the adaptive method proposed in [3] for weak approximation of stochastic differential equations.

Place, publisher, year, edition, pages
Berlin New York: de Gruyter, 2009
Keyword
time-dependent domain, non-smooth domain, heat equation, parabolic partial differential equations, Cauchy-Dirichlet problem, stochastic differential equations, stopped diffusion, Euler scheme, adaptive methods
National Category
Mathematical Analysis
Research subject
Mathematics
Identifiers
urn:nbn:se:umu:diva-21728 (URN)10.1515 /MCMA.2009.002 (DOI)
Available from: 2009-08-06 Created: 2009-04-17 Last updated: 2012-08-16Bibliographically approved
4. Weak approximation of obliquely reflected diffusions in time-dependent domains
Open this publication in new window or tab >>Weak approximation of obliquely reflected diffusions in time-dependent domains
2010 (English)In: Journal of Computational Mathematics, ISSN 0254-9409, E-ISSN 1991-7139, Vol. 28, no 5, 579-605 p.Article in journal (Refereed) Published
Abstract [en]

In an earlier paper, we proved the existence of solutions to the Skorohod problem with oblique reflection in time-dependent domains and, subsequently, applied this result to the problem of constructing solutions, in time-dependent domains, to stochastic differential equations with oblique reflection. In this paper we use these results to construct weak approximations of solutions to stochastic differential equations with oblique reflection, in time-dependent domains in R^d, by means of a projected Euler scheme. We prove that the constructed method has, as is the case for normal reflection and time-independent domains, an order of convergence equal to 1/2 and we evaluate the method empirically by means of two numerical examples. Furthermore, using a well-known extension of the Feynman-Kac formula, to stochastic differential equations with reflection, our method gives, in addition, a Monte Carlo method for solving second order parabolic partial differential equations with Robin boundary conditions in time-dependent domains.

Keyword
Stochastic differential equations, Oblique reflection, Robin boundary conditions, Skorohod problem, Time-dependent domain, Weak approximation, Monte Carlo method, Parabolic partial differential equations, Projected Euler scheme
National Category
Mathematics
Research subject
Mathematics
Identifiers
urn:nbn:se:umu:diva-37420 (URN)10.4208/jcm.1003-m2957 (DOI)000281879100002 ()
Available from: 2010-11-03 Created: 2010-11-02 Last updated: 2017-12-12Bibliographically approved

Open Access in DiVA

fulltext(272 kB)1311 downloads
File information
File name FULLTEXT01.pdfFile size 272 kBChecksum SHA-512
9a13bb168aaa070500d2948b37da93bb255e1269202d07e8557138d9305f9dfb342a4ed2aedbac09069c5733b0943f5275b3d31af6ef66a093ebf894946933e3
Type fulltextMimetype application/pdf

Search in DiVA

By author/editor
Önskog, Thomas
By organisation
Department of Mathematics and Mathematical Statistics
Mathematics

Search outside of DiVA

GoogleGoogle Scholar
Total: 1311 downloads
The number of downloads is the sum of all downloads of full texts. It may include eg previous versions that are now no longer available

isbn
urn-nbn

Altmetric score

isbn
urn-nbn
Total: 281 hits
CiteExportLink to record
Permanent link

Direct link
Cite
Citation style
  • apa
  • ieee
  • modern-language-association-8th-edition
  • vancouver
  • Other style
More styles
Language
  • de-DE
  • en-GB
  • en-US
  • fi-FI
  • nn-NO
  • nn-NB
  • sv-SE
  • Other locale
More languages
Output format
  • html
  • text
  • asciidoc
  • rtf