Investments in Random Environments

TL;DR

This paper analyzes a stochastic model of investor wealth in random environments, deriving the distribution and most probable wealth, and confirming findings with simulations.

Contribution

It provides an analytical framework for understanding investor wealth dynamics in random environments, including predictions of the most probable wealth and its scaling laws.

Findings

Distribution of investor wealth exhibits fat tails.

Most probable wealth reaches a steady state predicted analytically.

Simulation results agree well with theoretical predictions.

Abstract

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.