| 1. |
- Jain, P., et al.
(författare)
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Inequalities and properties of some generalized Orlicz classes and spaces
- 2007
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Ingår i: Acta Mathematica Hungarica. - : Springer Science and Business Media LLC. - 0236-5294 .- 1588-2632. ; 117:1-2, s. 161-174
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Tidskriftsartikel (refereegranskat)abstract
- We discuss and complement the knowledge about generalized Orlicz classes and Orlicz spaces X Φ obtained by replacing the space L 1 in the classical construction by an arbitrary Banach function space X. Our main aim is to focus on the task to study inequalities in such spaces. We prove a number of new inequalities and also natural generalizations of some classical ones (e.g., Minkowski’s, Hölder’s and Young’s inequalities). Moreover, a number of other basic facts for further study of inequalities and function spaces are included.
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| 2. |
- Kuliev, K., et al.
(författare)
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An extension of Rothe´s method to noncylindrical domains
- 2007
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Ingår i: Applications of Mathematics. - : Institute of Mathematics, Czech Academy of Sciences. - 0862-7940 .- 1572-9109. ; 52:5, s. 365-389
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Tidskriftsartikel (refereegranskat)abstract
- In this paper Rothe's classical method is extended so that it can be used to solve some linear parabolic boundary value problems in non-cylindrical domains. The corresponding existence and uniqueness theorems are proved and some further results and generalizations are discussed and applied.
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| 3. |
- Okpoti, Christopher, et al.
(författare)
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An equivalence theorem for some integral conditions with general measures related to Hardy´s inequality
- 2007
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Ingår i: Journal of Mathematical Analysis and Applications. - : Elsevier BV. - 0022-247X .- 1096-0813. ; 326:1, s. 398-413
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Tidskriftsartikel (refereegranskat)abstract
- It is proved that, besides the usual Muckenhoupt condition, there exist four different scales of conditions for characterizing the Hardy type inequality with general measures for the case 1<pq<∞. In fact, an even more general equivalence theorem of independent interest is proved and discussed.
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| 4. |
- Persson, Lars-Erik, et al.
(författare)
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Some scales of equivalent weight characterizations of Hardy´s inequality: the case q < p
- 2007
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Ingår i: Mathematical Inequalities & Applications. - : Element d.o.o.. - 1331-4343 .- 1848-9966. ; 10:2, s. 267-279
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Tidskriftsartikel (refereegranskat)abstract
- We consider the weighted Hardy inequality 1/q 1/p q ∞ ∞ x f p (x)v(x)dx f (t)dt u(x)dx C 0 0 0for the case 0 < q < p < ∞, p > 1 . The weights u(x) and v(x) for which this inequalityholds for all f (x) 0 may be characterized by the Mazya-Rosin or by the Persson-Stepanovconditions. In this paper, we show that these conditions are not unique and can be supplementedby some continuous scales of conditions and we prove their equivalence. The results for the dualoperator which do not follow by duality when 0 < q < 1 are also given.
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| 5. |
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| 6. |
- Persson, Lars-Erik, et al.
(författare)
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Weighted inequalities of Hardy type for matrix operators: The case q
- 2007
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Ingår i: Mathematical Inequalities & Applications. - : Element d.o.o.. - 1331-4343 .- 1848-9966. ; 10:4, s. 843-861
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Tidskriftsartikel (refereegranskat)abstract
- A non-negative triangular matrix operator is considered in weighted Lebesgue spaces ofsequences. Under some additional conditions on the matrix, some new weight characterizationsfor discrete Hardy type inequalities with matrix operator are proved for the case 1 < q < p < ∞.Some further results are pointed out.
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| 7. |
- Önskog, Thomas, 1979-, et al.
(författare)
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Pricing and hedging of financial derivatives using a posteriori error estimates and adaptive methods for stochastic differential equations
- 2010
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Ingår i: Journal of Computational and Applied Mathematics. - : Elsevier. - 0377-0427 .- 1879-1778. ; 235, s. 563-592
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Tidskriftsartikel (refereegranskat)abstract
- 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.
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| 8. |
- Önskog, Thomas, 1979-, et al.
(författare)
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Weak approximation of obliquely reflected diffusions in time-dependent domains
- 2010
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Ingår i: Journal of Computational Mathematics. - : Global Science Press. - 0254-9409 .- 1991-7139. ; 28:5, s. 579-605
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Tidskriftsartikel (refereegranskat)abstract
- 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.
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| 9. |
- Önskog, Thomas, 1979-, et al.
(författare)
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The Skorohod oblique reflection problem in time-dependent domains
- 2010
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Ingår i: Annals of Probability. - : Project Euclid. - 0091-1798 .- 2168-894X. ; 38:6, s. 2170-2223
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Tidskriftsartikel (refereegranskat)abstract
- 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.
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| 10. |
- Gaidashev, Denis, 1973-, et al.
(författare)
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A numerical study of infinitely renormalizable area-preserving maps
- 2012
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Ingår i: Dynamical systems. - : Informa UK Limited. - 1468-9367 .- 1468-9375. ; 27:3, s. 283-301
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Tidskriftsartikel (refereegranskat)abstract
- It has been shown in Gaidashev and Johnson [D. Gaidashev and T. Johnson, Dynamics of the universal area-preserving map associated with period doubling: stable sets, J. Mod. Dyn. 3(4) (2009), pp. 555–587.] and Gaidashev et al. [D. Gaidashev, T. Johnson, and M. Martens, Rigidity for infinitely renormalizable area-preserving maps, in preparation.] that infinitely renormalizable area-preserving maps admit invariant Cantor sets with a maximal Lyapunov exponent equal to zero. Furthermore, the dynamics on these Cantor sets for any two infinitely renormalizable maps is conjugated by a transformation that extends to a differentiable function whose derivative is Hölder continuous of exponent α > 0. In this article we investigate numerically the specific value of α. We also present numerical evidence that the normalized derivative cocycle with the base dynamics in the Cantor set is ergodic. Finally, we compute renormalization eigenvalues to a high accuracy to support a conjecture that the renormalization spectrum is real.
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