Risk Aversion Asymptotics for Power Utility Maximization

TL;DR

This paper investigates the asymptotic behavior of optimal consumption and investment strategies under power utility as risk aversion approaches extreme values, connecting these limits to exponential and logarithmic utilities.

Contribution

It provides a unified analysis of the convergence of optimal strategies in general semimartingale models and continuous models, using advanced mathematical tools.

Findings

Convergence of optimal consumption for general semimartingale models.

Convergence of optimal trading strategies for continuous models.

Limits relate to exponential and logarithmic utility functions.

Abstract

We consider the economic problem of optimal consumption and investment with power utility. We study the optimal strategy as the relative risk aversion tends to infinity or to one. The convergence of the optimal consumption is obtained for general semimartingale models while the convergence of the optimal trading strategy is obtained for continuous models. The limits are related to exponential and logarithmic utility. To derive these results, we combine approaches from optimal control, convex analysis and backward stochastic differential equations (BSDEs).