  We present analytical investigations of a multiplicative stochastic process
that models a simple investor dynamics in a random environment. The dynamics of
the investor's budget, $x(t)$, depends on the stochasticity of the return on
investment, $r(t)$, for which different model assumptions are discussed. The
fat-tail distribution of the budget is investigated and compared with
theoretical predictions. Weare mainly interested in the most probable value
$x_mp$ of the budget that reaches a constant value over time. Based on an
analytical investigation of the dynamics, we are able to predict $x_mp^stat$.
We find a scaling law that relates the most probable value to the
characteristic parameters describing the stochastic process. Our analytical
results are confirmed by stochastic computer simulations that show a very good
agreement with the predictions.
