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Numerical Methods for Stochastic Control Problems with Applications in Financial MathematicsBlechschmidt, Jan 25 May 2022 (has links)
This thesis considers classical methods to solve stochastic control problems and valuation problems from financial mathematics numerically.
To this end, (linear) partial differential equations (PDEs) in non-divergence form or the optimality conditions known as the (nonlinear) Hamilton-Jacobi-Bellman (HJB) equations are solved by means of finite differences, volumes and elements.
We consider all of these three approaches in detail after a thorough introduction to stochastic control problems and discuss various solution terms including classical solutions, strong solutions, weak solutions and viscosity solutions.
A particular role in this thesis play degenerate problems.
Here, a new model for the optimal control of an energy storage facility is developed which extends the model introduced in [Chen, Forsyth (2007)].
This four-dimensional HJB equation is solved by the classical finite difference Kushner-Dupuis scheme [Kushner, Dupuis (2001)] and a semi-Lagrangian variant which are both discussed in detail.
Additionally, a convergence proof of the standard scheme in the setting of parabolic HJB equations is given.
Finite volume schemes are another classical method to solve partial differential equations numerically.
Sharing similarities to both finite difference and finite element schemes we develop a vertex-centered dual finite volume scheme.
We discuss convergence properties and apply the scheme to the solution of HJB equations, which has not been done in such a broad context, to the best of our knowledge.
Astonishingly, this is one of the first times the finite volume approach is systematically discussed for the solution of HJB equations.
Furthermore, we give many examples which show advantages and disadvantages of the approach.
Finally, we investigate novel tailored non-conforming finite element approximations of second-order PDEs in non-divergence form, utilizing finite-element Hessian recovery strategies to approximate second derivatives in the equation.
We study approximations with both continuous and discontinuous trial functions.
Of particular interest are a-priori and a-posteriori error estimates as well as adaptive finite element methods.
In numerical experiments our method is compared with other approaches known from the literature.
We discuss implementations of all three approaches in MATLAB (finite differences and volumes) and FEniCS (finite elements) publicly available in GitHub repositories under https://github.com/janblechschmidt.
Many numerical experiments show convergence properties as well as pros and cons of the respective approach.
Additionally, a new postprocessing procedure for policies obtained from numerical solutions of HJB equations is developed which improves the accuracy of control laws and their incurred values.
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Mathematics as one of the basic Pillars of physical Theory: a historical and epistemological SurveyŠebesta, Juraj 13 September 2017 (has links)
No description available.
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The Interrelation between Mathematics and Physics at the Universities Jena, Halle-Wittenberg and Leipzig – a ComparisonSchlote, Karl-Heinz, Schneider, Martina 13 September 2017 (has links)
No description available.
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764 |
Der erste Professor für Theoretische Physik an der Universität Hamburg: Wilhelm LenzReich, Karin 13 September 2017 (has links)
No description available.
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765 |
Geometry as Physics: Oswald Veblen and the Princeton SchoolRitter, Jim 13 September 2017 (has links)
No description available.
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766 |
Mathematische Physik bei Hermann Weyl – zwischen „Hegelscher Physik“ und „symbolischer Konstruktion der Wirklichkeit“Scholz, Erhard 13 September 2017 (has links)
No description available.
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Henri Poincaré, theoretical Physics, and Relativity Theory in ParisWalter, Scott 13 September 2017 (has links)
No description available.
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Indeterminismus vor der Quantenmechanik: Richard von Mises’ wahrscheinlichkeitstheoretischer Purismus in der Theorie physikalischer ProzesseSiegmund-Schultze, Reinhard 13 September 2017 (has links)
No description available.
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769 |
Mathematical Foundations and physical Visions: Pascual Jordan and the Field Theory ProgramLehner, Christoph 13 September 2017 (has links)
No description available.
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From Matrices to Hilbert Spaces: The Interplay of Physics and Mathematics in the Rise of Quantum MechanicsLacki, Jan 13 September 2017 (has links)
No description available.
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