High Dimensional Transformation Algorithms

Abstract

Multi-dimensional transformation algorithms can be an excellent tool for solving multi-variate stochastic models, such as a financial model with features including underlying modeled by a stochastic process, stochastic volatility, stochastic interest rates. Our algorithm builds on the recently developed fast and stable one-dimensional transformation algorithm of Den Iseger (2007). Furthermore, the d-dimensional algorithms exploit the matrix-Kronecker product structure of the key characteristics, which greatly contributes to the speed and transparency of the algorithms. The d-dimensional algorithm inherits one of the two distinctive features of the one{dimensional algorithm: 1, it works both ways (one can obtain the original function given some numerical values of the Laplace transform, or the other way around, obtain the Laplace transform by feeding some numerical values of the original function in the input); 2, one can obtain the function or Laplace transform in arbitrary points (not only predefined points). These features facilitate the iterative use of the algorithms (e.g. dynamic programming-like problems).

Keywords: Fourier transform inversion, Laplace transform inversion, Gaussian quadrature

JEL Classification: G13, C63