Downside risk minimization via a large deviations approach

TL;DR

This paper investigates minimizing the probability of wealth falling below a target in an incomplete market, analyzing its asymptotic behavior through a dual risk-sensitive control framework and solving related ergodic H-J-B equations.

Contribution

It establishes a duality between risk minimization probability asymptotics and ergodic risk-sensitive control, providing explicit formulas via Legendre transforms and H-J-B equations.

Findings

Derived the asymptotic limit of the probability as a Legendre transform.

Connected risk minimization to an ergodic control problem.

Solved the associated H-J-B equation for Markovian models.

Abstract

We consider minimizing the probability of falling below a target growth rate of the wealth process up to a time horizon $T$ in an incomplete market model, and then study the asymptotic behavior of minimizing probability as $T\to\infty$. This problem can be closely related to an ergodic risk-sensitive stochastic control problem in the risk-averse case. Indeed, in our main theorem, we relate the former problem concerning the asymptotics for risk minimization to the latter as its dual. As a result, we obtain an expression of the limit value of the probability as the Legendre transform of the value of the control problem, which is characterized as the solution to an H-J-B equation of ergodic type, in the case of a Markovian incomplete market model.