STOCHASTIC LOTKA-VOLTERRA SYSTEMS OF COMPETING AUTO-CATALYTIC AGENTS LEAD GENERICALLY TO TRUNCATED PARETO POWER WEALTH DISTRIBUTION, TRUNCATED LEVY DISTRIBUTION OF MARKET RETURNS, CLUSTERED VOLATILITY, BOOMS AND CRACHES *

We give a microscopic representation of the stock-market in which the microscopic agents are the individual traders and their capital. Their basic dynamics consists in the auto-catalysis of the individual capital and in the global competition/cooperation between the agents mediated by the total wealth/index of the market. We show that such systems lead generically to (truncated) Pareto power-law distribution of the individual wealth. This, in turn, leads to intermittent market returns parametrized by a (truncated) Levy-stable distribution. We relate the truncation in the Levy-stable distribution to the fact that at each moment no trader can own more than the total wealth in the market. In the cases where the system is dominated by the largest traders, the dynamics looks similar to noisy low-dimensional chaos. By introducing traders memory and/or feedback between global and individual wealth fluctuations, one obtains auto-correlations in the time evolution of the “volatility” as well as market booms and crashes.

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