Duality in optimal consumption–investment problems with alternative data

Abstract

This study investigates an optimal consumption–investment problem in which the unobserved stock trend is modulated by a hidden Markov chain that represents different economic regimes. In the classic approach, the hidden state is estimated using historical asset prices, but recent technological advances now enable investors to consider alternative data in their decision-making. These data, such as social media commentary, expert opinions, COVID-19 pandemic data and GPS data, come from sources other than standard market data sources but are useful for predicting stock trends. We develop a novel duality theory for this problem and consider a jump-diffusion process for alternative data series. This theory helps investors identify “useful” alternative data for dynamic decision-making by providing conditions for the filter equation that enable the use of a control approach based on the dynamic programming principle. We apply our theory to provide a unique smooth solution for an agent with constant relative risk aversion once the distributions of the signals generated from alternative data satisfy a bounded likelihood ratio condition. In doing so, we obtain an explicit consumption–investment strategy that takes advantage of different types of alternative data that have not been addressed in the literature.