A Necessary Characteristic Equation of Diffusion Processes Having Gaussian Marginals

Syeda Rabab Mudakkar

doi:10.1155/2012/598590

Abstract

The aim of this work is to characterize one‐dimensional homogeneous diffusion process, under the assumption that marginal density of the process is Gaussian. The method considers the forward Kolmogorov equation and Fourier transform operator approach. The result establishes the necessary characteristic equation between drift and diffusion coefficients for homogeneous and nonhomogeneous diffusion processes. The equation for homogeneous diffusion process leads to characterize the possible diffusion processes that can exist. Two well‐known examples using the necessary characteristic equation are also given.

Highlights

Introduction and PreliminariesThe characterization of drift and diffusion coefficients by considering the marginal density of stochastic process is one of the known methods of estimating the actual underlying process and to construct new diffusion processes 1–6
We first prove the characteristic equation for homogeneous diffusion process and extend our results when the starting position Xt0 is arbitrary
We prove the characteristic equation in case of non-homogeneous diffusion

Summary

Introduction

The characterization of drift and diffusion coefficients by considering the marginal density of stochastic process is one of the known methods of estimating the actual underlying process and to construct new diffusion processes 1–6. We will consider the forward Kolmogorov equation and apply the Fourier transform operator to prove necessary characteristic equation between drift and diffusion coefficients for a homogeneous one-dimensional diffusion process when the marginal density is Gaussian. This equation characterizes the possible diffusion processes. The idea is motivated by Hamza and Klebaner 6 , who constructed an entire family of non-Gaussian martingales given that the marginals are Gaussian.

Objectives

Results

Conclusion