Elaboration of a Problem
By Steven J. Grisafi, PhD.
I thought it would be helpful to elaborate upon what I perceive to be a difficulty of the attempt of economists to develop a continuous time diffusion theory for asset price motions. I believe this is a fundamental difficulty not wholly recognized within the entire academic economics community. I refer to the generalized method of moments, which I suspect may have become a cause for some confusion. Our concern is with systems presumed to be at non-equilibrium conditions.
The method of moments has been my primary tool of inquiry in theoretical analysis since completion of my graduate studies in engineering. It has always been my understanding that one applies the method of moments when one cannot determine the statistical probability distribution that governs the particular process of one’s interest. When one can find the probability distribution, one knows all that one can know about a stochastic process. Oftentimes one cannot solve the governing differential equations for the probability distribution function itself, so one seeks to solve a set of simpler equations taken as moments of the governing equation for the probability distribution function. In principle, if one could evaluate a denumerable infinity of moments, one would acquire all information available from the probability distribution function itself. More often than one would hope, one is confronted with the closure problem of the method of moments, in which lower order moments possess a functional dependence upon higher moments. When this occurs, one is forced to apply closure to the series of moments in the form of either a truncation technique or an equilibrium approximation. A truncation technique would approximate the highest moment required for the analysis with a product of lower moments. An equilibrium approximation would evaluate the highest moment required for the analysis assuming the process has achieved equilibrium. All lower moments would have their non-equilibrium distribution.
The difficulty I see in the approach of economists to the continuous time diffusion of asset price motions is their lack of any governing principle upon which to build the probability distribution for asset price motions. Destitute of any principle, they assume that their diffusion process is a Brownian Motion and adopt all of its principles. However, doing so means that they assume the only knowledge they require, or can ever acquire, for their analysis since Brownian Motion is by definition a stationary stochastic process governed by a Gaussian probability distribution. Asset price motions cannot be a Brownian Motion process because Brownian Motion fluctuations must always be Gaussian and the Gaussian distribution cannot be time varying. The Gaussian distribution can only describe an equilibrium state from which the system must depart in order to reach a different equilibrium state. One cannot simply alter the mean and variance of a Gaussian distribution continuously with time variation and retain conformity with the governing equations of any process. There is no time varying Gaussian solution to the differential equations. Yet, hope springs eternal, and those who model diffusion processes often approximate their analysis with a “Quasi-Static” approximation. That is, they assume that their process transforms so slowly, relative to the time scales upon which they need to take measurements, such that they can assume that their system transforms slowly and continuously from one equilibrium state to another. Hence, a Quasi-Static approximation would imply a slow variation of the mean and variance for the Gaussian distribution of a Brownian Motion process. However, the fly in the ointment, the Ito Calculus, upon which economists build their asset price diffusion model, provides no such mechanism through which the parameters of a Gaussian distribution would be caused to vary. The mathematical theorems all assume stationarity of the process. Consequently, economists ought seek the methods of Finance Rheology, which provides a principle of continuity, for non-equilibrium analysis of their economic systems.