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Frentz, Marie
Publications (6 of 6) Show all publications
Frentz, M., Garofalo, N., Götmark, E., Munive, I. & Nyström, K. (2012). Non-divergence form parabolic equations associated with non-commuting vector fields: Boundary behavior of nonnegative solutions. Annali della Scuola Normale Superiore di Pisa (Classe Scienze), Serie V, 11(2), 437-474
Open this publication in new window or tab >>Non-divergence form parabolic equations associated with non-commuting vector fields: Boundary behavior of nonnegative solutions
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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, p. 437-474Article 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: 2018-06-08Bibliographically approved
Frentz, M., Götmark, E. & Nyström, K. (2012). The obstacle problem for parabolic non-divergence form operators of Hörmander type. Journal of Differential Equations, 252(9), 5002-2041
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, p. 5002-2041Article 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
Keywords
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: 2018-06-08Bibliographically approved
Frentz, M. (2012). Topics on subelliptic parabolic equations structured on Hörmander vector fields. (Doctoral dissertation). Umeå: Umeå Universitet
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. p. 36
Series
Doctoral thesis / Umeå University, Department of Mathematics, ISSN 1102-8300 ; 51
Keywords
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: 2018-06-08Bibliographically approved
Frentz, M. & Nyström, K. (2010). Adaptive stochastic weak approximation of degenerate parabolic equations of Kolmogorov type. Journal of Computational and Applied Mathematics, 234(1), 146-164
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, p. 146-164Article 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
Keywords
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: 2018-06-08Bibliographically approved
Frentz, M., Nyström, K., Pascucci, A. & Polidoro, S. (2010). Optimal regularity in the obstacle problem for Kolmogorov operators related to American Asian options. Mathematische Annalen, 347(4), 805-838
Open this publication in new window or tab >>Optimal regularity in the obstacle problem for Kolmogorov operators related to American Asian options
2010 (English)In: Mathematische Annalen, ISSN 0025-5831, E-ISSN 1432-1807, Vol. 347, no 4, p. 805-838Article in journal (Refereed) Published
Abstract [en]

In this paper we prove optimal interior regularity for solutions to the obstacle problem for a class of second order differential operators of Kolmogorov type. We treat smooth obstacles as well as non-smooth obstacles. All our proofs follow the same line of thought and are based on blow-ups, compactness, barriers and arguments by contradiction. The problem considered arises in financial mathematics, when considering path-dependent derivative contracts with early exercise feature.

Place, publisher, year, edition, pages
Springer, 2010
National Category
Mathematical Analysis
Identifiers
urn:nbn:se:umu:diva-47920 (URN)10.1007/s00208-009-0456-z (DOI)000277957100003 ()
Available from: 2011-10-04 Created: 2011-10-03 Last updated: 2018-06-08Bibliographically approved
Frentz, M.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)
Keywords
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: 2018-06-08Bibliographically approved
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