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Topics on subelliptic parabolic equations structured on Hörmander vector fields
Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
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 [en]
subelliptic, parabolic, obstacle problem, boundary behaviour, weak approximation
National Category
Mathematical Analysis
Research subject
Mathematics
Identifiers
URN: urn:nbn:se:umu:diva-51604ISBN: 978-91-7459-354-9 (print)OAI: oai:DiVA.org:umu-51604DiVA: diva2:484740
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
List of papers
1. The obstacle problem for parabolic non-divergence form operators of Hörmander type
Open this publication in new window or tab >>The obstacle problem for parabolic non-divergence form operators of Hörmander type
2012 (English)In: Journal of Differential Equations, ISSN 0022-0396, E-ISSN 1090-2732, Vol. 252, no 9, 5002-2041 p.Article in journal (Refereed) Published
Abstract [en]

In this paper we establish the existence and uniqueness of strong solutions to the obstacle problem for a class of parabolic sub-elliptic operators in non-divergence form structured on a set of smooth vector fields in Rn, X={X1,…,Xq}X={X1,…,Xq}, q⩽n, satisfying Hörmanderʼs finite rank condition. We furthermore prove that any strong solution belongs to a suitable class of Hölder continuous functions. As part of our argument, and this is of independent interest, we prove a Sobolev type embedding theorem, as well as certain a priori interior estimates, valid in the context of Sobolev spaces defined in terms of the system of vector fields.

Place, publisher, year, edition, pages
Elsevier, 2012
Keyword
obstacle problem, parabolic equations, Hormander condition, hypo-elliptic, embedding theorem, a priori estimates
National Category
Mathematical Analysis
Research subject
Mathematics
Identifiers
urn:nbn:se:umu:diva-51517 (URN)10.1016/j.jde.2012.01.032 (DOI)000301090200014 ()
Note

Originally published in thesis in manuscript form.

Available from: 2012-01-25 Created: 2012-01-24 Last updated: 2017-12-08Bibliographically approved
2. Regularity in the obstacle problem for parabolic non-divergence operators of Hörmander type
Open this publication in new window or tab >>Regularity in the obstacle problem for parabolic non-divergence operators of Hörmander type
(English)Manuscript (preprint) (Other academic)
Keyword
obstacle problem, subelliptic, regularity
National Category
Mathematical Analysis
Research subject
Mathematics
Identifiers
urn:nbn:se:umu:diva-51518 (URN)
Note
SubmittedAvailable from: 2012-01-25 Created: 2012-01-24 Last updated: 2012-01-27Bibliographically approved
3. Non-divergence form parabolic equations associated with non-commuting vector fields: Boundary behavior of nonnegative solutions
Open this publication in new window or tab >>Non-divergence form parabolic equations associated with non-commuting vector fields: Boundary behavior of nonnegative solutions
Show others...
2012 (English)In: Annali della Scuola Normale Superiore di Pisa (Classe Scienze), Serie V, ISSN 0391-173X, E-ISSN 2036-2145, Vol. 11, no 2, 437-474 p.Article in journal (Refereed) Published
Abstract [en]

In a cylinder Omega(T) = Omega x (0, T) subset of R-+(n+1) we study the boundary behavior of nonnegative solutions of second order parabolic equations of the form

H u = Sigma(m)(i,j=1) a(ij)(x, t)XiX (j)u - partial derivative(t)u = 0, (x, t) is an element of R-+(n+1),

where X = {X-l, . . . , X-m} is a system of C-infinity vector fields inR(n) satisfying Hormander's rank condition (1.2), and Omega is a non-tangentially accessible domain with respect to the Carnot-Caratheodory distance d induced by X. Concerning the matrix-valued function A = {a(ij)}, we assume that it is real, symmetric and uniformly positive definite. Furthermore, we suppose that its entries a(ij) are Holder continuous with respect to the parabolic distance associated with d. Our main results are: I) a backward Harnack inequality for nonnegative solutions vanishing on the lateral boundary (Theorem 1.1); 2) the Holder continuity up to the boundary of the quotient of two nonnegative solutions which vanish continuously on a portion of the lateral boundary (Theorem 1.2); 3) the doubling property for the parabolic measure associated with the operator H (Theorem 1.3). These results generalize to the subelliptic setting of the present paper, those in Lipschitz cylinders by Fabes, Safonov and Yuan in [20, 39]. With one proviso: in those papers the authors assume that the coefficients a(ij) be only bounded and measurable, whereas we assume Holder continuity with respect to the intrinsic parabolic distance.

National Category
Mathematical Analysis
Identifiers
urn:nbn:se:umu:diva-47921 (URN)000309320600009 ()
Available from: 2011-10-07 Created: 2011-10-03 Last updated: 2017-12-08Bibliographically approved
4. Adaptive stochastic weak approximation of degenerate parabolic equations of Kolmogorov type
Open this publication in new window or tab >>Adaptive stochastic weak approximation of degenerate parabolic equations of Kolmogorov type
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
Keyword
Degenerate parabolic, Weak approximation, Adaptivity, Malliavin calculus, Option pricing
National Category
Mathematical Analysis
Research subject
Mathematics
Identifiers
urn:nbn:se:umu:diva-47919 (URN)10.1016/j.cam.2009.12.011 (DOI)000276372100012 ()
Available from: 2011-10-04 Created: 2011-10-03 Last updated: 2017-12-08Bibliographically approved

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