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  • 1
    Publication Date: 2011-06-28
    Description:    We consider stochastic control problems with jump-diffusion processes and formulate an algorithm which produces, starting from a given admissible control π , a new control with a better value. If no improvement is possible, then π is optimal. Such an algorithm is well-known for discrete-time Markov Decision Problems under the name Howard’s policy improvement algorithm . The idea can be traced back to Bellman. Here we show with the help of martingale techniques that such an algorithm can also be formulated for stochastic control problems with jump-diffusion processes. As an application we derive some interesting results in financial portfolio optimization. Content Type Journal Article Pages 1-14 DOI 10.1007/s00245-011-9141-1 Authors Nicole Bäuerle, Institute for Stochastics, Karlsruhe Institute of Technology, 76128 Karlsruhe, Germany Ulrich Rieder, Department of Optimization and Operations Research, University of Ulm, 89069 Ulm, Germany Journal Applied Mathematics & Optimization Online ISSN 1432-0606 Print ISSN 0095-4616
    Print ISSN: 0095-4616
    Electronic ISSN: 1432-0606
    Topics: Mathematics
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  • 2
    Publication Date: 2011-06-11
    Description:    We study finite convergence of the modified cyclic subgradient projections (MCSP) algorithm for the convex feasibility problem (CFP) in the Euclidean space. Expanding control sequences allow the indices of the sets of the CFP to re-appear and be used again by the algorithm within windows of iteration indices whose lengths are not constant but may increase without bound. Motivated by another development in finitely convergent sequential algorithms that has a significant real-world application in the field of radiation therapy treatment planning, we show that the MCSP algorithm retains its finite convergence when used with an expanding control that is repetitive and fulfills an additional condition. Content Type Journal Article Pages 1-13 DOI 10.1007/s00245-011-9139-8 Authors Yair Censor, Department of Mathematics, University of Haifa, Mt. Carmel, Haifa, 31905 Israel Wei Chen, Department of Computer Science, The Graduate Center, City University of New York, 365 Fifth Avenue, New York, NY 10016, USA Homeira Pajoohesh, Department of Mathematics, Medgar Evers College, City University of New York, Brooklyn, New York, NY 11225, USA Journal Applied Mathematics & Optimization Online ISSN 1432-0606 Print ISSN 0095-4616
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    Topics: Mathematics
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  • 3
    Publication Date: 2011-06-28
    Description:    The paper deals with optimal control of heterogeneous systems, that is, families of controlled ODEs parameterized by a parameter running over a domain called domain of heterogeneity . The main novelty in the paper is that the domain of heterogeneity is endogenous: it may depend on the control and on the state of the system. This extension is crucial for several economic applications and turns out to rise interesting mathematical problems. A necessary optimality condition is derived, where one of the adjoint variables satisfies a differential inclusion (instead of equation) and the maximization of the Hamiltonian takes the form of “min-max”. As a consequence, a Pontryagin-type maximum principle is obtained under certain regularity conditions for the optimal control. A formula for the derivative of the objective function with respect to the control from L ∞ is presented together with a sufficient condition for its existence. A stylized economic example is investigated analytically and numerically. Content Type Journal Article Pages 1-25 DOI 10.1007/s00245-011-9140-2 Authors Anton O. Belyakov, ORCOS, Institute of Mathematical Methods in Economics, Vienna University of Technology, Argentinierstrasse 8/105-4, 1040 Vienna, Austria Tsvetomir Tsachev, Institute of Mathematics and Informatics, Bulgarian Academy of Sciences, Acad. G. Bonchev str., Block 8, 1113 Sofia, Bulgaria Vladimir M. Veliov, ORCOS, Institute of Mathematical Methods in Economics, Vienna University of Technology, Argentinierstrasse 8/105-4, 1040 Vienna, Austria Journal Applied Mathematics & Optimization Online ISSN 1432-0606 Print ISSN 0095-4616
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    Topics: Mathematics
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  • 4
    Publication Date: 2011-06-06
    Description:    We investigate an initial-boundary value problem for the quasilinear Westervelt equation which models the propagation of sound in fluidic media. We prove that, if the initial data are sufficiently small and regular, then there exists a unique global solution with optimal L p -regularity. We show furthermore that the solution converges to zero at an exponential rate as time tends to infinity. Our techniques are based on maximal L p -regularity for abstract quasilinear parabolic equations. Content Type Journal Article Pages 1-15 DOI 10.1007/s00245-011-9138-9 Authors Stefan Meyer, Naturwissenschaftliche Fakultät II, Institut für Mathematik, Martin-Luther-Universität Halle-Wittenberg, 06099 Halle (Saale), Germany Mathias Wilke, Naturwissenschaftliche Fakultät II, Institut für Mathematik, Martin-Luther-Universität Halle-Wittenberg, 06099 Halle (Saale), Germany Journal Applied Mathematics & Optimization Online ISSN 1432-0606 Print ISSN 0095-4616
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  • 5
    Publication Date: 2011-12-06
    Description:    This paper focuses on the study of a linear eigenvalue problem with indefinite weight and Robin type boundary conditions. We investigate the minimization of the positive principal eigenvalue under the constraint that the absolute value of the weight is bounded and the total weight is a fixed negative constant. Biologically, this minimization problem is motivated by the question of determining the optimal spatial arrangement of favorable and unfavorable regions for a species to survive. For rectangular domains with Neumann boundary condition, it is known that there exists a threshold value such that if the total weight is below this threshold value then the optimal favorable region is like a section of a disk at one of the four corners; otherwise, the optimal favorable region is a strip attached to the shorter side of the rectangle. Here, we investigate the same problem with mixed Robin-Neumann type boundary conditions and study how this boundary condition affects the optimal spatial arrangement. Content Type Journal Article Pages 1-36 DOI 10.1007/s00245-011-9153-x Authors M. Hintermüller, Humboldt-University of Berlin, Unter den Linden 6, 10099 Berlin, Germany C.-Y. Kao, Department of Mathematics and Computer Science, Claremont McKenna College, Claremont, CA 91711, USA A. Laurain, Department of Mathematics, Johann-von-Neumann-Haus, Rudower Chaussee 25, 12489 Berlin-Adlershof, Germany Journal Applied Mathematics & Optimization Online ISSN 1432-0606 Print ISSN 0095-4616
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  • 6
    Publication Date: 2012-03-10
    Description:    The goal of this work is to study a model of the strongly damped wave equation with dynamic boundary conditions and nonlinear boundary/interior sources and nonlinear boundary/interior damping. First, applying the nonlinear semigroup theory, we show the existence and uniqueness of local in time solutions. In addition, we show that in the strongly damped case solutions gain additional regularity for positive times t 〉0. Second, we show that under some restrictions on the initial data and if the interior source dominates the interior damping term and if the boundary source dominates the boundary damping, then the solution grows as an exponential function. Moreover, in the absence of the strong damping term, we prove that the solution ceases to exists and blows up in finite time. Content Type Journal Article Pages 1-42 DOI 10.1007/s00245-012-9165-1 Authors Philip Jameson Graber, Department of Mathematics, University of Virginia, Charlottesville, VA 22904, USA Belkacem Said-Houari, Division of Mathematical and Computer Sciences and Engineering, King Abdullah University of Science and Technology (KAUST), Thuwal, Kingdom of Saudi Arabia Journal Applied Mathematics & Optimization Online ISSN 1432-0606 Print ISSN 0095-4616
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    Topics: Mathematics
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  • 7
    Publication Date: 2012-11-12
    Description:    In this paper we derive a Cameron-Storvick theorem for the analytic Feynman integral of functionals on product abstract Wiener space B 2 . We then apply our result to obtain an evaluation formula for the analytic Feynman integral of unbounded functionals on B 2 . We also present meaningful examples involving functionals which arise naturally in quantum mechanics. Content Type Journal Article Pages 1-18 DOI 10.1007/s00245-012-9186-9 Authors Jae Gil Choi, Department of Mathematics, Dankook University, Cheonan, 330-714 Korea Seung Jun Chang, Department of Mathematics, Dankook University, Cheonan, 330-714 Korea Journal Applied Mathematics & Optimization Online ISSN 1432-0606 Print ISSN 0095-4616
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    Topics: Mathematics
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  • 8
    Publication Date: 2012-09-24
    Description:    The paper is concerned with the analysis of a new variational model to restore point-like and curve-like singularities in biological images. To this aim we investigate the variational properties of a suitable energy which governs these pathologies. Finally in order to realize numerical experiments we minimize, in the discrete setting, a regularized version of this functional by fast descent gradient scheme. Content Type Journal Article Pages 1-24 DOI 10.1007/s00245-012-9181-1 Authors Gilles Aubert, Laboratoire J.A. Dieudonné, Université de Nice Sophia Antipolis, Parc Valrose, 06108 Nice Cedex 2, France Laure Blanc-Féraud, Morpheme CNRS/INRIA/UNSA Sophia Antipolis, Inria, 2004 route des Lucioles, BP 93, 06902 Sophia Antipolis Cedex, France Daniele Graziani, Morpheme CNRS/INRIA/UNSA Sophia Antipolis, Inria, 2004 route des Lucioles, BP 93, 06902 Sophia Antipolis Cedex, France Journal Applied Mathematics & Optimization Online ISSN 1432-0606 Print ISSN 0095-4616
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  • 9
    Publication Date: 2012-09-24
    Description:    We consider the problem of maximization of expected terminal power utility (risk sensitive criterion). The underlying market model is a regime-switching diffusion model where the regime is determined by an unobservable factor process forming a finite state Markov process. The main novelty is due to the fact that prices are observed and the portfolio is rebalanced only at random times corresponding to a Cox process where the intensity is driven by the unobserved Markovian factor process as well. This leads to a more realistic modeling for many practical situations, like in markets with liquidity restrictions; on the other hand it considerably complicates the problem to the point that traditional methodologies cannot be directly applied. The approach presented here is specific to the power-utility. For log-utilities a different approach is presented in Fujimoto et al. (Preprint, 2012 ). Content Type Journal Article Pages 1-40 DOI 10.1007/s00245-012-9180-2 Authors Kazufumi Fujimoto, Corporate Risk Management Division, The Bank of Tokyo-Mitsubishi UFJ, Ltd., Marunouchi 2-7-1, 100-8388 Chiyoda-ku, Tokyo, Japan Hideo Nagai, Division of Mathematical Science for Social Systems, Graduate School of Engineering Science, Osaka University, Machikaneyama 1-3, 560-8531 Toyonaka, Osaka, Japan Wolfgang J. Runggaldier, Dipartimento di Matematica Pura ed Applicata, Università di Padova, Via Trieste 63, 35121 Padova, Italy Journal Applied Mathematics & Optimization Online ISSN 1432-0606 Print ISSN 0095-4616
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    Topics: Mathematics
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  • 10
    Publication Date: 2012-10-07
    Description:    In this paper we develop the large deviations principle and a rigorous mathematical framework for asymptotically efficient importance sampling schemes for general, fully dependent systems of stochastic differential equations of slow and fast motion with small noise in the slow component. We assume periodicity with respect to the fast component. Depending on the interaction of the fast scale with the smallness of the noise, we get different behavior. We examine how one range of interaction differs from the other one both for the large deviations and for the importance sampling. We use the large deviations results to identify asymptotically optimal importance sampling schemes in each case. Standard Monte Carlo schemes perform poorly in the small noise limit. In the presence of multiscale aspects one faces additional difficulties and straightforward adaptation of importance sampling schemes for standard small noise diffusions will not produce efficient schemes. It turns out that one has to consider the so called cell problem from the homogenization theory for Hamilton-Jacobi-Bellman equations in order to guarantee asymptotic optimality. We use stochastic control arguments. Content Type Journal Article Pages 1-39 DOI 10.1007/s00245-012-9183-z Authors Konstantinos Spiliopoulos, Division of Applied Mathematics, Brown University, Providence, RI 02912, USA Journal Applied Mathematics & Optimization Online ISSN 1432-0606 Print ISSN 0095-4616
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    Topics: Mathematics
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