While the scattering phase for several one-dimensional potentials can be exactly derived, less is known in multi-dimensional quantum systems. This work provides a method to extend the one-dimensional phase knowledge t...While the scattering phase for several one-dimensional potentials can be exactly derived, less is known in multi-dimensional quantum systems. This work provides a method to extend the one-dimensional phase knowledge to multi-dimensional quantization rules. The extension is illustrated in the example of Bogomolny＇s transfer operator method applied in two quantum wells bounded by step potentials of different heights. This generalized semiclassical method accurately determines the energy spectrum of the systems, which indicates the substantial role of the proposed phase correction. Theoretically, the result can be extended to other semiclassical methods, such as Gutzwiller trace formula, dynamical zeta functions, and semielassical Landauer Buttiker formula. In practice, this recipe enhances the applicability of semiclassical methods to multi-dimensional quantum systems bounded by general soft potentials.展开更多
In this paper we present how nonlinear stochastic Itˆo differential equations arising in the modelling of perturbed rigid bodies can be solved numerically in such a way that the solution evolves on the correct manifol...In this paper we present how nonlinear stochastic Itˆo differential equations arising in the modelling of perturbed rigid bodies can be solved numerically in such a way that the solution evolves on the correct manifold.To this end,we formulate an approach based on Runge-Kutta–Munthe-Kaas(RKMK)schemes for ordinary differ-ential equations on manifolds.Moreover,we provide a proof of the mean-square convergence of this stochastic version of the RKMK schemes applied to the rigid body problem and illustrate the effectiveness of our proposed schemes by demonstrating the structure preservation of the stochastic RKMK schemes in contrast to the stochastic Runge-Kutta methods.展开更多
One goal of financial research is to determine fair prices on the financial market.As financial models and the data sets on which they are based are becoming ever larger and thus more complex,financial instruments mus...One goal of financial research is to determine fair prices on the financial market.As financial models and the data sets on which they are based are becoming ever larger and thus more complex,financial instruments must be further developed to adapt to the new complexity,with short runtimes and efficient use of memory space.Here we show the effects of combining known strategies and incorporating new ideas to further improve numerical techniques in computational finance.In this paper we combine an ADI(alternating direction implicit)scheme for the temporal discretization with a sparse grid approach and the combination technique.The later approach considerably reduces the number of“spatial”grid points.The presented standard financial problem for the valuation of American options using the Heston model is chosen to illustrate the advantages of our approach,since it can easily be adapted to other more complex models.展开更多
基金Supported by the National Science Council at Taiwan through Grants No. NSC 97-2112-M-009-008-MY3
文摘While the scattering phase for several one-dimensional potentials can be exactly derived, less is known in multi-dimensional quantum systems. This work provides a method to extend the one-dimensional phase knowledge to multi-dimensional quantization rules. The extension is illustrated in the example of Bogomolny＇s transfer operator method applied in two quantum wells bounded by step potentials of different heights. This generalized semiclassical method accurately determines the energy spectrum of the systems, which indicates the substantial role of the proposed phase correction. Theoretically, the result can be extended to other semiclassical methods, such as Gutzwiller trace formula, dynamical zeta functions, and semielassical Landauer Buttiker formula. In practice, this recipe enhances the applicability of semiclassical methods to multi-dimensional quantum systems bounded by general soft potentials.
基金supported by the bilateral German-Slovakian Project MATTHIAS–Modelling and Approximation Tools and Techniques for Hamilton-Jacobi-Bellman equations in finance and Innovative Approach to their Solution,financed by DAAD and the Slovakian Ministry of EducationFurther the authors acknowledge partial support from the bilateral German-Portuguese Project FRACTAL–FRActional models and CompuTationAL Finance financed by DAAD and the CRUP–Conselho de Reitores das Universidades Portuguesas.
文摘In this paper we present how nonlinear stochastic Itˆo differential equations arising in the modelling of perturbed rigid bodies can be solved numerically in such a way that the solution evolves on the correct manifold.To this end,we formulate an approach based on Runge-Kutta–Munthe-Kaas(RKMK)schemes for ordinary differ-ential equations on manifolds.Moreover,we provide a proof of the mean-square convergence of this stochastic version of the RKMK schemes applied to the rigid body problem and illustrate the effectiveness of our proposed schemes by demonstrating the structure preservation of the stochastic RKMK schemes in contrast to the stochastic Runge-Kutta methods.
基金supported by the bilateral German-Slovakian Project MATTHIAS–Modelling and Approximation Tools and Techniques for Hamilton-Jacobi-Bellman equations in finance and Innovative Approach to their Solution,financed by DAAD and the Slovakian Ministry of Education.Further the authors acknowledge partial support from the bilateral German-Portuguese Project FRACTAL–FRActional models and CompuTationAL Finance financed by DAAD and the CRUP–Conselho de Reitores das Universidades Portuguesas.
文摘One goal of financial research is to determine fair prices on the financial market.As financial models and the data sets on which they are based are becoming ever larger and thus more complex,financial instruments must be further developed to adapt to the new complexity,with short runtimes and efficient use of memory space.Here we show the effects of combining known strategies and incorporating new ideas to further improve numerical techniques in computational finance.In this paper we combine an ADI(alternating direction implicit)scheme for the temporal discretization with a sparse grid approach and the combination technique.The later approach considerably reduces the number of“spatial”grid points.The presented standard financial problem for the valuation of American options using the Heston model is chosen to illustrate the advantages of our approach,since it can easily be adapted to other more complex models.
