Paper Info
Title
On a new family of radial basis functions: Mathematical analysis and applications to option pricing.
Abstract
In this paper, we introduce a new family of infinitely smooth and “nearly” locally supported radial basis functions (RBFs), derived from the general solution of a heat equation arising from the American option pricing problem. These basis functions are expressed in terms of “the repeated integrals of the complementary error function” and provide highly efficient tools to solve the free boundary partial differential equation resulting from the related option pricing model. We introduce an integral operator with a function-dependent lower limit which is employed as a basic tool to prove the radial positive definiteness of the proposed basis functions and could be of independent interest in the RBF theory. We then show that using the introduced functions as expansion bases in the context of an RBF-based meshless collocation scheme, we could exactly impose the transparent boundary condition accompanying the heat equation. We prove that the condition numbers of the resulting collocation matrices are orders of magnitude less than those arising from other popular RBF families used in current literature. Some other properties of these bases such as their Fourier transforms as well as some useful representations in terms of positive Borel measures will also be discussed.
Year
DOI
Venue
2018
10.1016/j.cam.2017.06.012
Journal of Computational and Applied Mathematics
Keywords
Field
DocType
65M99,91G20,91G80
Boundary value problem,Mathematical optimization,Radial basis function network,Mathematical analysis,Basis function,Heat equation,Operator (computer programming),Positive definiteness,Partial differential equation,Mathematics,Collocation
Journal
Volume
Issue
ISSN
328
C
0377-0427
Citations 
PageRank 
References 
0
0.34
13
Authors
3
Name
Order
Citations
PageRank
Seyed-Mohammad-Mahdi Kazemi120.73
Mehdi Dehghan23022324.48
Ali Foroush Bastani3104.85