The hyperbolic model of the corridor fiat banker

The basic idea is to model the borrowers and lenders as a Weiner process, show that the hyperbolic differential equation is a solution in tanh, then show that the tanh function is a suitable equation for principal and interest, then define the relationship between principal balances the generate balance. The result will be both the lending and borrowing rates as well as the target principal amounts.

This basic hyperbolic form has tanh as a solution but when we add in the lending and deposit outstanding balances we can make a solution for this Weiner process:

In which t is replaced by the principal balance ratio. Then we find the appropriate polynomial that makes the function: t*tanh(x) t+tanh,a polynomial, and that defines the solution set in x and t, both quantized. t is the principal balance target and should count sequentially. The banker then adjusts x, the angle between lend and receive, along the finite set of hyperbolic angles and targets t. That makes a no arbitrage solution.

Now its plain to see that sinh is principal returned plus interest received while cosh is principal returns plus interest earned, hence tanh will be a flow from near zero to near one, always positive.   So we can define one the deposit and one the loan.

The solution looks like: tanh(a)^2 = 1-1/t. Some how we will make that a polynomial, likely using Lucas numbers.  Tanh goes to one as t goes to infinity. Does this allow a zero inflation rate? In otherwords, can the banker always retreive excess fiat from the system by making loans attractive and keeping principal balances near target? I am working on that. But I think tanh can go negative of positive when t goes to less than one. That would switch the positions and allow the bank to 'gain' permanent money, as long as the economy has spare entropy.

Use the last form of tanh, where the e^-2x and e^2x can be rates on a unit one principal balance:

The ratio t which scales tanh  will force an eccentricity in the hyperbola, and that should change. The discrete set of x which are selected will be separated by the allowable error in the system. Transaction rates are relative to total transactions and term length a resultant, not an input.  But they should be the shortest terms in the yield curve.

The basic  idea is correct, the hyperbolics can track a Weiner process.