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chaotically damped exponential...ie: the stock market

In a conversation I had with a friend last night, I mentioned that I understood much of the mathematics and physics that I learned as an undergraduate engineering student by approaching the problems from the perspective of my visual sense. I have used this sense to understand areas of mathematical complexity that I probably would understand differently or not at all, had I not had the art related visual ability to create shapes of the abstract ideas that I could then use to relate and form an understanding. During the conversation I mentioned the stock market and described it as a "chaotically damped exponential" system. What follows is a rational for that description that I wrote up to flesh out this idea that I had otherwise not fully explored other than visually.

In mathematics, the term "chaos" is used to describe signals or ensembles (collections of signals) that are varying in a random way across an infinitely long sample or wide enough space. Exponential's how ever are non linear functions that are easy to predict (they form the heart of many theories, radiative decay being a prominent example) but exponential functions can rise with time or they can go down with time depending on the sign associated with the functional exponent. So in my view the stock market works as an exponential function with exponents that are a chaos function or a random function or variable. In statistical dynamics , random variables are a powerful tool used to study variance across samples of some type of data. Mind you I am talking about the average behavior of the market as taken by one of the various indexes, S&P 500 for example. The actual market of stocks is much larger and the indexes are really nothing more than samples of the true total market that are said to reflect gyrations in the whole.

The stock market works like a collection or ensemble of independent agents (companies) whose value rises and falls with the *perception* of the market (investors) who buy or sell based on how they feel the stock is doing, what they want the stock to be doing, what they need in their own personal lives. The reasons for any particular agent viewing a stock vary, and thus the gyrations of the indexes over smaller intervals approximate perfect noise (since on those scales the gyration is simply an average of investor desires, which can be simultaneously divergent) note this is different from what happens with regard to the gyrations of a particular stock only in that the average is accounted for in a different way, using a smaller set of data (ie. only investors in the stock) but even here the agents still employ the same reason for buying or not buying stock in a company within a given market session and those reasons are often divergent. They do however have the ability to converge very rapidly, should some news be released about the company, say indicating that it will be closing a factory or buying another company. Such news could cause immediate effect on a particular company stock but leave little impact on the wider index and market as a whole. The ultimate conclusion seems to be that market gyrations vary with scale but at every scale are still subject to random events that can shift the curves (be they individual stock or a market index) in unpredictable ways. A good example to illustrate the market is the analogy of a group of people in a park, if we took samples of groups of people in the park we would find that the numbers vary with many events, what day of the week it is, what is he nature of the weather, if their is a band playing in the nearby ampitheatre, these events are correlated with various levels of people showing up but they are not predictors of any precise number of people in the park for any particular day. So though it is possible to say "today there will be more people than yesterday" it is impossible to say "today their will be 39 more people than yesterday" the variance in possible people acts as a random variable over all the days that a sample of individuals is taken in the park. I posit, that the stock market works the same way with an inability to predict specific gyration values. Company forms, company exists or survives for a time frame, possibly thrives and then company dies. Unlike living entities though , where collections have relatively known "death" times after they are born, companies can live indefinitely, so long as they perform well by competing in their respective markets and keeping investor interest. They often, during their "lives" purchase other companies or provide markets to them, over an average of companies this obvious statement is true "those that survive last longer than those that die", which is to say, there are more companies maintaining or thriving on average than their are those dying. I have only an intuitive sense of this being true for now but it is the reason for my view that the behavior of the markets is exponential but damped in a random way, which accounts for the fact that over it's history (assume the dow jones average), it has slowly gained but if we add in this latest downturn (the greatest since the great stock crash of 29) we are probably average zero growth or near it, over the history of the markets, which is exactly what we would expect over a chaotically damped system. It is important to realize that the dynamics over all companies extant and extinct define the true nature of the market and none of the averages is a total sample of all companies, the dow jones only tracks 30, it is also important to recognize that apparent trends in growth can be illusions of multi-decade trends in the local and world economy. For example, over the last 145 years, the industrial revolution has been the engine of a flowering of growth of companies to exploit potential in many industries. As the world economy equalizes production over the entire planet, the ability for any local index to exercise a dominant growth results with respect to the other indexes will fade. So over time , I predict that the indexes will reflect more globally random behavior and will lose much of their effectiveness as effective gauges for the viability of a local economy. As companies globalize and nations provide the work force that can satisfy the needs of business at reduced prices, growth will slow even while production may be increasing within specific industries. This prediction is one that we may see play out in the next 50 - 75 years, as I predict by that time that the equalization of production capability should have run its course around the world. More specifically, it will occur once all the potential areas manufacturing, resource and labor are fully mobilized to contribute to the world economy. Currently vast areas in Africa, Central and South America and Central and South Asia remain ripe with potential. Have another view, let us discuss your ideas in the comments.


Links:

http://focus.aps.org/story/v4/st10

http://en.wikipedia.org/wiki/Dow_Jones_Industrial_Average

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