Post by Sapphire Capital on Mar 21, 2009 3:44:59 GMT 4
Laplace Transform Inversion on the Entire Line
Peter Den Iseger
Cardano Risk Management
March 8, 2009
Abstract:
Numerical inversion of Laplace transforms is a powerful tool in computational probability. It greatly enhances the applicability of stochastic models in many fields. The present paper introduces a numerical inversion method which employs a Gaussian quadrature rule in order to perform a numerical inversion on the entire line. The two key improvements of this inversion method w.r.t. its forebears can be summarized as follows. First, the original function f (obtained after the numerical inversion) can be evaluated on an arbitrary grid instead of only on a uniform grid, or let alone one single point. Secondly, the new method works both ways: if some numerical values of the original function f are given, then the algorithm generates its Laplace transform Lf (thus no closed-form analytical expression for f needed!); and vice versa, given some numerical values of the Laplace transform, the original function f can be obtained in arbitrary points. These enhancements open the way for tackling problems as iterative algorithms (for instance recursion, widely employed for dynamic programming style problems), Levy processes based modeling, or problems where as well the inverse transform as the transform itself needs to be computed.
papers.ssrn.com/sol3/Delivery.cfm/SSRN_ID1356510_code863969.pdf?abstractid=1355451&mirid=3
Peter Den Iseger
Cardano Risk Management
March 8, 2009
Abstract:
Numerical inversion of Laplace transforms is a powerful tool in computational probability. It greatly enhances the applicability of stochastic models in many fields. The present paper introduces a numerical inversion method which employs a Gaussian quadrature rule in order to perform a numerical inversion on the entire line. The two key improvements of this inversion method w.r.t. its forebears can be summarized as follows. First, the original function f (obtained after the numerical inversion) can be evaluated on an arbitrary grid instead of only on a uniform grid, or let alone one single point. Secondly, the new method works both ways: if some numerical values of the original function f are given, then the algorithm generates its Laplace transform Lf (thus no closed-form analytical expression for f needed!); and vice versa, given some numerical values of the Laplace transform, the original function f can be obtained in arbitrary points. These enhancements open the way for tackling problems as iterative algorithms (for instance recursion, widely employed for dynamic programming style problems), Levy processes based modeling, or problems where as well the inverse transform as the transform itself needs to be computed.
papers.ssrn.com/sol3/Delivery.cfm/SSRN_ID1356510_code863969.pdf?abstractid=1355451&mirid=3