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Adaptive stochastic weak approximation of degenerate parabolic equations of Kolmogorov type
Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
2010 (English)In: Journal of Computational and Applied Mathematics, ISSN 0377-0427, E-ISSN 1879-1778, Vol. 234, no 1, 146-164 p.Article in journal (Refereed) Published
Abstract [en]

Degenerate parabolic equations of Kolmogorov type occur in many areas of analysis and applied mathematics. In their simplest form these equations were introduced by Kolmogorov in 1934 to describe the probability density of the positions and velocities of particles but the equations are also used as prototypes for evolution equations arising in the kinetic theory of gases. More recently equations of Kolmogorov type have also turned out to be relevant in option pricing in the setting of certain models for stochastic volatility and in the pricing of Asian options. The purpose of this paper is to numerically solve the Cauchy problem, for a general class of second order degenerate parabolic differential operators of Kolmogorov type with variable coefficients, using a posteriori error estimates and an algorithm for adaptive weak approximation of stochastic differential equations. Furthermore, we show how to apply these results in the context of mathematical finance and option pricing. The approach outlined in this paper circumvents many of the problems confronted by any deterministic approach based on, for example, a finite-difference discretization of the partial differential equation in itself. These problems are caused by the fact that the natural setting for degenerate parabolic differential operators of Kolmogorov type is that of a Lie group much more involved than the standard Euclidean Lie group of translations, the latter being relevant in the case of uniformly elliptic parabolic operators.

Place, publisher, year, edition, pages
Elsevier, 2010. Vol. 234, no 1, 146-164 p.
Keyword [en]
Degenerate parabolic, Weak approximation, Adaptivity, Malliavin calculus, Option pricing
National Category
Mathematical Analysis
Research subject
Mathematics
Identifiers
URN: urn:nbn:se:umu:diva-47919DOI: 10.1016/j.cam.2009.12.011ISI: 000276372100012OAI: oai:DiVA.org:umu-47919DiVA: diva2:445335
Available from: 2011-10-04 Created: 2011-10-03 Last updated: 2017-12-08Bibliographically approved
In thesis
1. Topics on subelliptic parabolic equations structured on Hörmander vector fields
Open this publication in new window or tab >>Topics on subelliptic parabolic equations structured on Hörmander vector fields
2012 (English)Doctoral thesis, comprehensive summary (Other academic)
Place, publisher, year, edition, pages
Umeå: Umeå Universitet, 2012. 36 p.
Series
Doctoral thesis / Umeå University, Department of Mathematics, ISSN 1102-8300 ; 51
Keyword
subelliptic, parabolic, obstacle problem, boundary behaviour, weak approximation
National Category
Mathematical Analysis
Research subject
Mathematics
Identifiers
urn:nbn:se:umu:diva-51604 (URN)978-91-7459-354-9 (ISBN)
Public defence
2012-02-24, Mit-huset, MA121, Umeå Universitet, Umeå, 13:00 (English)
Opponent
Supervisors
Available from: 2012-02-03 Created: 2012-01-27 Last updated: 2012-01-27Bibliographically approved

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Frentz, MarieNyström, Kaj

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