10,971 research outputs found
A Hybrid Radial Basis Function - Pseudospectral Method for Thermal Convection in a 3-D Spherical Shell
A novel hybrid spectral method that combines radial basis function (RBF) and Chebyshev pseudospectral (PS) methods in a “2+1” approach is presented for numerically simulating thermal convection in a 3-D spherical shell. This is the first study to apply RBFs to a full 3D physical model in spherical geometry. In addition to being spectrally accurate, RBFs are not defined in terms of any surface based coordinate system such as spherical coordinates. As a result, when used in the lateral directions, as in this study, they completely circumvent the pole issue with the further advantage that nodes can be “scattered” over the surface of a sphere. In the radial direction, Chebyshev polynomials are used, which are also spectrally accurate and provide the necessary clustering near the boundaries to resolve boundary layers. Applications of this new hybrid methodology are given to the problem of convection in the Earth’s mantle,which is modeled by a Boussinesq fluid at infinite Prandtl number. To see whether this numerical technique warrants further investigation, the study limits itself to an isoviscous mantle.Benchmark comparisons are presented with other currently used mantle convection codes for Rayleigh number 7 · 103 and 105. The algorithmic simplicity of the code (mostly due to RBFs)allows it to be written in less than 400 lines of Matlab and run on a single workstation. We find that our method is very competitive with those currently used in the literature
Pricing and Hedging GLWB in the Heston and in the Black-Scholes with Stochastic Interest Rate Models
Valuing Guaranteed Lifelong Withdrawal Benefit (GLWB) has attracted
significant attention from both the academic field and real world financial
markets. As remarked by Forsyth and Vetzal the Black and Scholes framework
seems to be inappropriate for such long maturity products. They propose to use
a regime switching model. Alternatively, we propose here to use a stochastic
volatility model (Heston model) and a Black Scholes model with stochastic
interest rate (Hull White model). For this purpose we present four numerical
methods for pricing GLWB variables annuities: a hybrid tree-finite difference
method and a hybrid Monte Carlo method, an ADI finite difference scheme, and a
standard Monte Carlo method. These methods are used to determine the
no-arbitrage fee for the most popular versions of the GLWB contract, and to
calculate the Greeks used in hedging. Both constant withdrawal and optimal
withdrawal (including lapsation) strategies are considered. Numerical results
are presented which demonstrate the sensitivity of the no-arbitrage fee to
economic, contractual and longevity assumptions
Introduction to discrete functional analysis techniques for the numerical study of diffusion equations with irregular data
We give an introduction to discrete functional analysis techniques for
stationary and transient diffusion equations. We show how these techniques are
used to establish the convergence of various numerical schemes without assuming
non-physical regularity on the data. For simplicity of exposure, we mostly
consider linear elliptic equations, and we briefly explain how these techniques
can be adapted and extended to non-linear time-dependent meaningful models
(Navier--Stokes equations, flows in porous media, etc.). These convergence
techniques rely on discrete Sobolev norms and the translation to the discrete
setting of functional analysis results
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