Asymptotics of the probability minimizing a "down-side" risk

TL;DR

This paper investigates the asymptotic behavior of the probability of an investor's portfolio falling below a target growth rate, linking large deviation control with risk-sensitive stochastic control, and provides explicit solutions in Gaussian models.

Contribution

It establishes a duality between large deviation and risk-sensitive control problems and derives explicit solutions for multidimensional Gaussian models.

Findings

Duality between large deviation and risk-sensitive control.

Explicit solutions in Gaussian models.

Asymptotic analysis of down-side risk probability.

Abstract

We consider a long-term optimal investment problem where an investor tries to minimize the probability of falling below a target growth rate. From a mathematical viewpoint, this is a large deviation control problem. This problem will be shown to relate to a risk-sensitive stochastic control problem for a sufficiently large time horizon. Indeed, in our theorem we state a duality in the relation between the above two problems. Furthermore, under a multidimensional linear Gaussian model we obtain explicit solutions for the primal problem.