  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 $\theta=\frac{1-R}{1-S}>1$. In
this sense, this paper is complementary to Herdegen, Hobson and Jerome
[arXiv:2107.06593].
  The main novelty of the case $\theta>1$ (as opposed to $\theta\in(0,1)$) 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.
