  This paper investigates the position (state) distribution of the single step
binomial (multi-nomial) 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 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.

Mathematical Finance