McKean–Vlasov Limit in Portfolio Optimization

Abstract

This article considers a sector-wise allocation in a portfolio consisting of a very large number of stocks. Their interdependence is captured by the dependence of the drift coefficient of each stock on an averaged effect of the sectors. This leads to a decoupled dynamics in the limit of large numbers, akin to the “mean field” limit leading to the McKean–Vlasov equation in statistical physics. This gives a more compact description using a time-varying drift characterized in terms of a measure-valued process that satisfies a nonlinear parabolic equation. The classical portfolio optimization problem is then addressed in this framework.