A Visual Criterion for Identifying Ito Diffusions as Martingales or Strict Local Martingales
Quantitative Finance Research Centre, University of Technology, Sydney, Research Paper Number 263
12 Pages Posted: 3 Nov 2012
Date Written: November 1, 2009
Abstract
It is often important, in applications of stochastic calculus to financial modelling, to know whether a given local martingale is a martingale or a strict local martingale. We address this problem in the context of a time-homogenous diffusion process with a finite lower boundary, presented as the solution of a driftless stochastic differential equation. Our main theorem demonstrates that the question of whether or not this process is a martingale may be decided simply by examining the slope of a certain increasing function. Further results establish the connection between our theorem and other results in the literature, while a number of examples are provided to illustrate the use of our criterion.
Keywords: diffusions, first-passage times, Laplace transforms, local martingales, ordinary differential equations
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