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21	Процедура коррекции области построения для численного метода решения дифференциальных игр быстродействия : магистерская диссертация / Correction procedure for the area of construction of a numerical method solving differential games of optimal time Munts, N., Мунц, Н. В. January 2015 (has links) The paper describes the software implementation of the numerical method proposed by M.Bardi and M.Falcone solving optimal time games. Examples of numerical calculations are given. The question of the applicability of this method for solving differential games with life line, i.e. with a set where the second player escapes and wins unconditionally, is discussed and examined. Currently, the study has not been completed and will be continued in the future. / В работе приведено описание программной реализации численного метода, предложенного М.Барди и М.Фальконе для решения игр быстродействия. Приведены примеры численного счета. Обсуждается и исследуется вопрос о применимости данного метода для решения дифференциальных игр быстродействия с линией жизни, то есть с множеством, при попадании системы на которое второй игрок безусловно выигрывает. В настоящее время это исследование не доведено до конца и будет продолжено в дальнейшем. MASTER'S THESIS DIFFERENTIAL GAMES OF OPTIMAL TIME VALUE FUNCTION NUMERICAL METHOD HAMILTON-JACOBI EQUATION VISCOSITY SOLUTION GAMES WITH LIFE LINE ФУНКЦИЯ ЦЕНЫ ЧИСЛЕННЫЙ МЕТОД ВЯЗКОСТНОЕ РЕШЕНИЕ ИГРЫ С ЛИНИЕЙ ЖИЗНИ
22	Numerical methods for the solution of the HJB equations arising in European and American option pricing with proportional transaction costs Li, Wen January 2010 (has links) This thesis is concerned with the investigation of numerical methods for the solution of the Hamilton-Jacobi-Bellman (HJB) equations arising in European and American option pricing with proportional transaction costs. We first consider the problem of computing reservation purchase and write prices of a European option in the model proposed by Davis, Panas and Zariphopoulou [19]. It has been shown [19] that computing the reservation purchase and write prices of a European option involves solving three different fully nonlinear HJB equations. In this thesis, we propose a penalty approach combined with a finite difference scheme to solve the HJB equations. We first approximate each of the HJB equations by a quasi-linear second order partial differential equation containing two linear penalty terms with penalty parameters. We then develop a numerical scheme based on the finite differencing in both space and time for solving the penalized equation. We prove that there exists a unique viscosity solution to the penalized equation and the viscosity solution to the penalized equation converges to that of the original HJB equation as the penalty parameters tend to infinity. We also prove that the solution of the finite difference scheme converges to the viscosity solution of the penalized equation. Numerical results are given to demonstrate the effectiveness of the proposed method. We extend the penalty approach combined with a finite difference scheme to the HJB equations in the American option pricing model proposed by Davis and Zarphopoulou [20]. Numerical experiments are presented to illustrate the theoretical findings. Hamilton-Jacobi equations Finite differences Viscosity solutions Transaction costs -- Mathematical models Option pricing Transaction costs Penalty approach Finite difference scheme Viscosity solution Partial differential equation Optimal control
23	Fully linear elliptic equations and semilinear fractionnal elliptic equations Chen, Huyuan 10 January 2014 (has links) Cette thèse est divisée en six parties. La première partie est consacrée à l'étude de propriétés de Hadamard et à l'obtention de théorèmes de Liouville pour des solutions de viscosité d'équations aux dérivées partielles elliptiques complètement non-linéaires avec des termes de gradient, ... / This thesis is divided into six parts. The first part is devoted to prove Hadamard properties and Liouville type theorems for viscosity solutions of fully nonlinear elliptic partial differential equations with gradient term ... Propriété d'Hadamard Théorème de typr Liouville Solutions de ciscosité Laplacien fractionnaire Existence Unicité Comportement asymptotique Grandes solutions Mesures de Radon Mesure de Dirac Noyau de Green Capacité de Bessel Singularités isolées Solutions faibles Singularité faible Singularité forte Hadamard property Liouville type theorem Viscosity solution Fully nonlinear elliptic PDE Fractional Laplacian Existence Uniqueness Asymptotic behavior Blow-up solution Radon measure Dirac mass Green kernel Bessel capacities Isolated singularity Weak solution Weak singular solution Strong singular solution Propiedad de Hadamard Teorema del tipo de Liouville Soluciones Viscosas EDP elípticas completamente no lineales Laplaciano fraccionario Existencia, Unicidad Comportamiento asimptótico Soluciones blow-up Medida de Radon Masa de Dirac Núcleo de Green Capacidades de Bessel Singularidad aislada Soluciones débiles Soluciones singulares débiles Soluciones singulares fuertes

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Numerical methods for the solution of the HJB equations arising in European and American option pricing with proportional transaction costs

Fully linear elliptic equations and semilinear fractionnal elliptic equations