120 research outputs found
A finite difference method for pricing European and American options under a geometric Lévy process
In this paper we develop a numerical approach to a fractional-order differential Linear Complementarity Problem (LCP) arising in pricing European and American options under a geometric Lévy process. The LCP is first approximated by a nonlinear penalty fractional Black-Scholes (fBS) equation. We then propose a finite difference scheme for the penalty fBS equation. We show that both the continuous and the discretized fBS equations are uniquely solvable and establish the convergence of the numerical solution to the viscosity solution of the penalty fBS equation by proving the consistency, stability and monotonicity of the numerical scheme. We also show that the discretization has the 2nd-order truncation error in both the spatial and time mesh sizes. Numerical results are presented to demonstrate the accuracy and usefulness of the numerical method for pricing both European and American options under the geometric Lévy process
Super-diffusive Transport Processes in Porous Media
The basic assumption of models for the transport of contaminants through soil is that the movements of solute particles are characterized by the Brownian motion. However, the complexity of pore space in natural porous media makes the hypothesis of Brownian motion far too restrictive in some situations. Therefore, alternative models have been proposed. One of the models, many times encountered in hydrology, is based in fractional differential equations, which is a one-dimensional fractional advection diffusion equation where the usual second-order derivative gives place to a fractional derivative of order α, with 1 < α ≤ 2. When a fractional derivative replaces the second-order derivative in a diffusion or dispersion model, it leads to anomalous diffusion, also called super-diffusion. We derive analytical solutions for the fractional advection diffusion equation with different initial and boundary conditions. Additionally, we analyze how the fractional parameter α affects the behavior of the solutions
Matrix approach to discrete fractional calculus II: partial fractional differential equations
A new method that enables easy and convenient discretization of partial
differential equations with derivatives of arbitrary real order (so-called
fractional derivatives) and delays is presented and illustrated on numerical
solution of various types of fractional diffusion equation. The suggested
method is the development of Podlubny's matrix approach (Fractional Calculus
and Applied Analysis, vol. 3, no. 4, 2000, 359--386). Four examples of
numerical solution of fractional diffusion equation with various combinations
of time/space fractional derivatives (integer/integer, fractional/integer,
integer/fractional, and fractional/fractional) with respect to time and to the
spatial variable are provided in order to illustrate how simple and general is
the suggested approach. The fifth example illustrates that the method can be
equally simply used for fractional differential equations with delays. A set of
MATLAB routines for the implementation of the method as well as sample code
used to solve the examples have been developed.Comment: 33 pages, 12 figure
Optimal convergence rates for semidiscrete finite element approximations of linear space-fractional partial differential equations under minimal regularity assumptions
We consider the optimal convergence rates of the semidiscrete finite element approximations for solving linear space-fractional partial differential equations by using the regularity results for the fractional elliptic problems obtained recently by Jin et al. \cite{jinlazpasrun} and Ervin et al. \cite{ervheuroo}. The error estimates are proved by using two approaches. One approach is to apply the duality argument in Johnson \cite{joh} for the heat equation to consider the error estimates for the linear space-fractional partial differential equations. This argument allows us to obtain the optimal convergence rates under the minimal regularity assumptions for the solution. Another approach is to use the approximate solution operators of the corresponding fractional elliptic problems. This argument can be extended to consider more general linear space-fractional partial differential equations. Numerical examples are given to show that the numerical results are consistent with the theoretical results
Finite Difference Approximations for Fractional Advection-Dispersion Flow Equations
Fractional advection-dispersion equations are used in groundwater hydrology tomqU- the transport of passive tracers carried by fluid flow in a porous mrousq In this paper we develop practical numtical mumti to solve one dimUEBDqyU fractional advection-dispersion equations with variable coefficients on a finite domeqV The practical application of these results is illustrated by mqUEIB# a radial flow problem Use of the fractional derivative allows the model equations to capture the early arrival of tracer observed at a field site
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