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Abstract: We present analytical investigations of a multiplicative stochastic processthat models a simple investor dynamics in a random environment. The dynamics ofthe investor-s budget, $xt$, depends on the stochasticity of the return oninvestment, $rt$, for which different model assumptions are discussed. Thefat-tail distribution of the budget is investigated and compared withtheoretical predictions. Weare mainly interested in the most probable value$x mp$ of the budget that reaches a constant value over time. Based on ananalytical investigation of the dynamics, we are able to predict $x mp^stat$.We find a scaling law that relates the most probable value to thecharacteristic parameters describing the stochastic process. Our analyticalresults are confirmed by stochastic computer simulations that show a very goodagreement with the predictions.

Author: Emeterio Navarro, Ruben Cantero, Joao Rodrigues, Frank Schweitzer



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