A Stochastic Model of Economic Growth in Time-Space

TL;DR

This paper introduces a stochastic model for economic growth in time-space, solving the associated control problem explicitly and analyzing the long-term behavior of optimal growth strategies.

Contribution

It extends deterministic economic growth models to include stochastic disturbances, providing explicit solutions and optimal feedback strategies in a complex infinite-dimensional setting.

Findings

Explicit solution to the stochastic HJB equation

Verification that the solution is the value function

Analysis of asymptotic behavior of optimal trajectories

Abstract

We deal with an infinite horizon, infinite dimensional stochastic optimal control problem arising in the study of economic growth in time-space. Such problem has been the object of various papers in deterministic cases when the possible presence of stochastic disturbances is ignored. Here we propose and solve a stochastic generalization of such models where the stochastic term, in line with the standard stochastic economic growth models, is a multiplicative one, driven by a cylindrical Wiener process. The problem is studied using the Dynamic Programming approach. We find an explicit solution of the associated HJB equation and, using a verification type result, we prove that such solution is the value function and we find the optimal feedback strategies. Finally we use this result to study the asymptotic behavior of the optimal trajectories.