Optimal consumption and investment with bounded downside risk measures for logarithmic utility functions

TL;DR

This paper derives explicit dynamic strategies for optimal consumption and investment in a Black-Scholes market, constrained by risk measures like VaR and Expected Shortfall, specifically for logarithmic utility functions.

Contribution

It extends previous work on power utilities to include logarithmic utility, providing explicit solutions under downside risk constraints.

Findings

Explicit dynamic strategies derived for constrained utility maximization.

Comparison and interpretation of solutions under risk restrictions.

Extension of prior power utility results to logarithmic utility.

Abstract

We investigate optimal consumption problems for a Black-Scholes market under uniform restrictions on Value-at-Risk and Expected Shortfall for logarithmic utility functions. We find the solutions in terms of a dynamic strategy in explicit form, which can be compared and interpreted. This paper continues our previous work, where we solved similar problems for power utility functions.