An expansion in the model space in the context of utility maximization

TL;DR

This paper derives a second-order expansion formula for the power investor's value function in incomplete markets, enabling accurate first-order approximations of optimal controls with numerical validation.

Contribution

It introduces a novel second-order expansion approach for the value function in incomplete markets, extending previous first-order methods.

Findings

The expansion formula accurately approximates the value function.

Numerical examples demonstrate the method's effectiveness.

First-order controls are effectively approximated using the expansion.

Abstract

In the framework of an incomplete financial market where the stock price dynamics are modeled by a continuous semimartingale (not necessarily Markovian) an explicit second-order expansion formula for the power investor's value function - seen as a function of the underlying market price of risk process - is provided. This allows us to provide first-order approximations of the optimal primal and dual controls. Two specific calibrated numerical examples illustrating the accuracy of the method are also given.