The limiting behaviour of Solow's model with uncertainty when the variance goes to zero

Abstract

It is shown that the solution of stochastic differential equation that describes Solow's model with uncertainty via the population dynamics converges in probability uniformly on bounded intervals to the classical deterministic solution as the variance of the population approaches zero. To achieve this, it was necessary to use methods pertaining to the realm of Ventsel' and Freidlin theory of small random perturbations of dynamical systems. We also show the convergence of the expectations of the steady-state as well as the vague convergence of the steady state distribution to the deterministic equilibrium and the degenerated distribution concentrated on the deterministic equilibrium respectively.