Merton's optimal investment problem with jump signals

TL;DR

This paper introduces a novel framework for Merton's investment problem incorporating jump signals using Meyer σ-fields, enabling investors to react to impending jumps and optimize strategies accordingly.

Contribution

It develops a new approach using Meyer σ-fields for signals about jumps, providing explicit solutions for power utilities and analyzing the impact of signal quality and quantity.

Findings

Investors may disinvest completely after positive signals in jump scenarios.

The framework allows comparison between improving signal quality versus quantity.

Explicit solutions are derived for Gaussian jump cases.

Abstract

This paper presents a new framework for Merton's optimal investment problem which uses the theory of Meyer $\sigma$-fields to allow for signals that possibly warn the investor about impending jumps. With strategies no longer predictable, some care has to be taken to properly define wealth dynamics through stochastic integration. By means of dynamic programming, we solve the problem explicitly for power utilities. In a case study with Gaussian jumps, we find, for instance, that an investor may prefer to disinvest completely even after a mildly positive signal. Our setting also allows us to investigate whether, given the chance, it is better to improve signal quality or quantity and how much extra value can be generated from either choice.