Path Diffusion, Part I

Abstract

This paper investigates the position (state) distribution of the single step binomial (multinomial) process on a discrete state/time grid under the assumption that the velocity process rather than the state process is Markovian. In this model the particle follows a simple multi-step process in velocity space which also preserves the proper state equation of motion. Many numerical examples of this process are provided. For a smaller grid the probability construction converges into a correlated set of probabilities of hyperbolic functions for each velocity at each state point. It is shown that the two dimensional process can be transformed into a Telegraph equation and via transformation into a Klein-Gordon equation if the transition rates are constant. In the last Section there is an example of multi-dimensional hyperbolic partial differential equation whose numerical average satisfies Newton's equation. There is also a momentum measure provided both for the two-dimensional case as for the multi-dimensional rate matrix.

Keywords: Markov Process, Velocit Convexity Equation, Telegraph Equation, Klein-Gordon Equation, Newston's Equation

JEL Classification: C0, C6