2021/12/13 by Herdegen, Martin, Hobson, David, Jerome, Joseph · 3 citations
#60H20 #91B16 #93E20 #FOS: Economics and business #Mathematical Finance (q-fin.MF)
paper · doi:10.48550/arxiv.2112.06708
In this article, we consider the optimal investment-consumption problem for an agent with preferences governed by Epstein--Zin stochastic differential utility (EZ-SDU) who invests in a constant-parameter Black-Scholes-Merton market over the infinite horizon. The parameter combinations that we consider in this paper are such that the risk aversion parameter R and the elasticity of intertemporal complementarity S satisfy θ=(1-R)/(1-S)>1. In this sense, this paper is complementary to Herdegen, Hobson and Jerome [arXiv:2107.06593]. The main novelty of the case θ>1 (as opposed to θ∈(0,1)) is that there is an infinite family of utility processes associated to every nonzero consumption stream. To deal with this issue, we introduce the economically motivated notion of a proper utility process, where, roughly speaking, a utility process is proper if it is nonzero whenever future consumption is nonzero. We then proceed to show that for a very wide class of consumption streams C, there exists a proper utility process V associated to C. Furthermore, for a wide class of consumption streams C, the proper utility process V is unique. Finally, we solve the optimal investment-consumption problem in a constant parameter financial market, where we optimise over the right-continuous attainable consumption streams that have a unique proper utility process associated to them.