Continuous interest via Renormalization Group
introduction
Saturday 23th I had just finished a visit to the collection of cuneiform tablets of the British Museum when I found a colleague "ensimismado" at a corner of the Greek rooms. We took the opportunity to walk around the Partenon frise and then down for some coffee in Oxford Circus, putting ourselves in knowledge of each other’ recent work and, as usual between Spaniards, "repairing the world". This latter, in our case, accounted both to trace the paralells between the fallen world we have just contemplated and our own century, and to execute a pledge about the status of Spanish science.
So when I took the train back to Chatham I felt myself in a "revenue" mathematical mood, and my initial thoughts on getting some mortgaged study near Covent Garten soon evolved to the problem of going from a periodic payment to a continuum one, and to wonder if the classical formulae of continous interest could be shown with the scaling scheme of Renormalization Group.
I am writing down the sketch of this small quest in the feeling it could be useful inside some introduction course to RG for economists. The main power of scaling methods, namely to take account of fluctuations, is not shown in this play, but it could serve as starting point to study the more formal, serious and rigorous, approach.
First we will grasp how a scale transformation works in the lattice of continuous parameters, and how must a interest formula be formulated and parametrized. Then we will take a look to the space of parameters and look for the renormalization flow lines and specially for the one coming from a fixed point; obviously such line shall be the asociated to exponential interest formulae. With this knowledge, one can retake the simple interest formula and perturbatibely compound it showing how we approach to the continuum limit, and how this appoaching can be traced in the parameted space. Everything is standard lore, and it is no more that the procedure stated in this short parragraph.
As an additional exercice, we will try to schedule a periodical "amortizacion, incremento", and drive again the route down to the continuum limit, to get the exponential formula of continuous increases.
lattice and scaling
Elementary financial calculus is usually discrete, with periodical payments at intervals ΔT. We call this a one dimensional lattice with spacing a = ΔT.
A compound interest rule in the lattice can be given simply by C(tn + 1) = C(tn)(1+rΔT)
Lets reduce the lattice spacing, for instance a factor two: $\Delta'T=\frac{\Delta T}2$. We all know that the previous formula in the new scale will give different results when iterated along the same period of time.
Renormalization flow in the procedure going back from the small lattice to the greater one, but preserving the dinamics of the finer, smaller one.
The core of renormalization is to ask for the new formulae to have the same physical results than the original ones. This will imply alteration of the parameters (coupling constants, we say) involved in formulae.
So, the formula C(tn + 2) = C(tn)(1+r′Δ′T)2 becomes $$C(t_{n+1})=C(t_n)(1+r {\Delta T over 2})^2= =C(t_n)(1+r' \Delta T+ \frac14 {r'}^2 {\Delta T}^2$$
parameter flow
fixed points
and renormalized limit lines
perturbative theory
formal series
Near the critical point, we can expand on the relevant parameter.
RG triangle
mortgage
integration
Our previous section could had given us the impression of a relationship between one (zero?) dimensional theory of renormalization and the theory of integration.
Integration, if we are to believe the scratchs in folio .. of codex ..., as read by xxx in 1909, was discovered by Aρξιμδϵσ by combining exhaustiond and equilibrium. In this sense it fits nicely with the exposition of this paper.
The more sophisticater RG theory is currently being fitted with integration theory by A. Connes and Dirk Kreimer. We suggest hept-th/9901xxx for indications to recent development.
conclussions
Well, after putting numbers to the above formulae, I find myself not in a position to decide whether to put my scarce money in the previously named flat, or to invite Miss [name deleted] to travel to Asia trying to find some prakritt or sanskritt documentation on the origin of the zero as operational cipher.
Either to take fluctuations into account or to get an increase in my actual salary income, must be needed in order to take a sounder decision.