“Mathematics is the art of giving the same name to different things.”
If you look carefully at the above proofs they demo straight that the compound interest rate function and the Laplace transform are virtually the same function.
The only difference is the Laplace transform has an integration wrapped around it. The difference is the kernel of the compound interest rate function and the Laplace transform use a different letter for the exponent, that is s instead of r.
Therefore : If s = r and P = 1
Therefore : e^st = e^rt
Therfore : st = rt
Therefore : s = r
They are therefore the same formula and thus the same thing that is to say the laplace transform is an integration of the compound interest rate function. This also indicates that compound interest is by definition a complex number and thus can be translated to the complex plane and spectrographically analyzed on the s plane.
So any impulse (transaction) will create point source interference that will propagate a wave front through the medium which can be de modulated using a bit of code and the correct constant for the market of interest.
Young’s slit experiment.
If many impulses that is to say trades are applied to the medium (market) they will form interference patterns as each wave front decays over time.
Where the interference is generated by the constant impulse responses from transactions that is taking an open position and closing it out at some future time.
This will look very much like a chaotic system but don’t be fooled the reason markets form linear trend lines is because they are linear systems with a chaotic component as there inputs however the way the market reacts is what matters and that is a function of the elasticity of the market in question which remains relatively constant over time.
More the point markets are specifically Moiré patterns. These Moiré patterns appear due to calibration problems inherent in the
To put it simply the market has a resonant frequency that can be measured using the Laplace transform.
The simple fact is that although the Fourier series looks tempting as a market analysis tool it is just the wrong one for the job, as it transforms the price time domains to the price frequency domains which is of little value what I want is a transform that converts from the price time domains to the price profit domain in my accout.
Which is I want to get rid of unwanted oscillations in my equity which is what the Laplace transform does so well and as it turns out like anything in mathematics worth discovering the equations involved are not that complicated.
The above chart is a simple process control function as a single contract track record for the last twenty years the lower panel is the profit per contract traded.
Markets are an elastic medium this is basic economics and are only partially random as such, the inputs maybe a random series of investment decisions. However the overall motion of the market is the result of the superposition of the sum of the impulse responses of all of the trading for a given period when applied to the elastic medium of market.
This presents the basis for an improvement in many of the current investment formulas that are taken as gospel at this time by applying a number of well know engineer Technics that are considered outside of the scope current financial theory.