An Introduction to Difference Equations by Saber Elaydi and Difference Equations by Paul Cull, Mary Flahive et al have me thinking of second order difference equations.

x(t+2) = a + b  x(t+1) + c  x(t) + u(t) where u ( t ) is a random variable.

The fibonacci equation has b and c = 1 and x (t) = x(t-1 ) + x (t-2) , x(0) = 1 , x( 1) = 2 and the series is 1,2,3,5,8,13,21,… the simple way to solve for the x(t) in such a series is to make it a binomial (x) (x) - x -1 = 0 and then a very beautiful set of roots described in Edspec in a math team problem comes about with a(( 1 + 5 to 1/2)/2)to the 1/2 and a (( 1 -5 to 1/2)/2) to 1/2 term as the multiples to apply in the solution.

Okay, I think the second order difference equations have many applications in many markets. I propose to simulate the best fit for a second order difference equation with say the last 10 sets of 10 observations, and then to predict where it will go. I believe that a simplicity based solution using only the constants 1, 2, and 3 and no higher powers than a square should be found.

I propose that this would be predictive and the divergences from the prediction would be tradable. The exercise violates our rule that the higher the mathematics the less the use in predictions. But it seems to me a useful reflection that is much more relevant than the essays on stochastic difference equations that one sees in the literature such as this and it's a useful diversion from cross road puzzles and pattern recognition I think.


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