Converting DSGE to Martingale process, an ongoing project

Since I am part of the complaint about non linearity in economics, I figured I should do a little work on how to improve the DSGE, and what I describe is has likely already been tried. Here is the outline I am following. This will day a day or two, and I am in no hurry. So mathematicians, please jump out here and do this work, and I will give you full credit and recommend you get large quantities in your personal account.

We are dealing with this,
Ce ^(c t) + Xe ^(x t) + G e ^(g t) - Y e^ (y t) = 0


Which are the  Consumer, Investment and Government growth exponentials with initial conditions, in a closed economy. I took it from a lecture note, here.  I want that converted into a Martingale probability distribution.

So these are four variables, and the idea is to construct the four variable martingale process that matches them. And I want to do this by using a finite polynomial approximation of these exponentials.  I want the initial conditions at time zero, and I would use this: (1+xt/N)^N, to obtain the polynomial, and xt here will be a multiplicative expression of all the variables. including their exponents.   Now time is not in these variables, they are tagged with time to separate them out as growth variables, not initial conditions. I think this is the case, the system is built on the arbitrary compounding by exponents.

Next I need to multiply through by all variables raised to negative powers to make the standard polynomial with positive exponents..

These three equations, in one variable tell me how to make the martingale. So I need to assign four sets of Wt, each with their own integer quantizer in the form for Wt to the left.

These are optimum samplers, sync functions, and they will re-introduce time as an index variable, a sequential path over which I can do Ito's calculus.

Anyway, this is how far I have gone, not much but the basic idea is here.  When its finished, then, the result is a probability surface over which I can compute the various probabilities. For example, the probability that interest costs get a bit tight for G. The order I select should correspond to the typical error, say 3%, that is generally left as a price variance in the economy. The Wt function has a zero mean, unit variance. So there is a scaling issue somewhere that needs to be handled. And there are more than one type of Weiner to try out, some have drift, most of them on the Schramm-Loewner evolution indices can be used.


Mathematicians, this is what the economists want, a probability surface derived from their DSGE, basically a Hamiltonian.  So this is important to saving out economic butts and working the complete system pay top dollar in the finance community.