This thesis deals with the infinite-horizon optimal investment and consumption problem for a utility maximising agent with Epstein–Zin stochastic differential utility (EZ-SDU) preferences. In particular, it achieves four main goals.

The first achievement is to provide a detailed introduction to the problem of continuous-time optimal investment and consumption under constant relative risk aversion (CRRA) utility preferences—a restriction of EZ-SDU preferences—in a Black–Scholes–Merton financial market. This is significantly simpler than the optimal investment-consumption problem under EZ-SDU, but even in this case, features of the problem take it outside the standard settings of classical primal stochastic control. This means that existing primal verification proofs rely on parameter restrictions, restrictions on the admissible strategies, or intricate approximation arguments. We show in Chapter II that these complications can be overcome using a simple and elegant argument involving a stochastic perturbation of the utility function.

The second achievement is to provide a detailed introduction to infinite-horizon EZ-SDU, including a discussion of which parameter combinations lead to a wellposed investment-consumption problem. To do this, we introduce a slightly different formulation of EZ-SDU to that which is traditionally used in the literature. This highlights the necessity and appropriateness of certain restrictions on the parameters governing EZ-SDU.We provide a thorough comparison of our formulation of EZ-SDU to the classical formulation.
Thirdly, and most importantly, we tackle the existence and uniqueness of infinite horizon EZ-SDU—a result currently lacking from the literature. To do this, it is necessary to make case distinctions depending on the parameters governing the agent’s temporal and risk preferences. Specifically, if R is the agent’s risk aversion and S is the agent’s elasticity of complementarity (which are both defined in Chapter I), we must distinguish between the case when # = 1