Lifetime Portfolio Selection under Uncertainty: The Continuous-Time Case

TLDR

Portfolio selection models have evolved from one‑period frameworks to discrete‑time, multi‑period, and continuous‑time formulations, with seminal work by Samuelson, Tobin, Phelps, and Mirrlees. This study investigates optimal portfolio selection and consumption rules for an individual in a continuous‑time setting where income derives from stochastic asset returns. The authors derive optimality equations for a multi‑asset continuous‑time model driven by Wiener Brownian motion, analyze a two‑asset constant relative risk‑aversion case, and present a general technique applicable to a broad class of intertemporal economic problems. An explicit solution is.

Abstract

OST models of portfolio selection have M been one-period models. I examine the combined problem of optimal portfolio selection and consumption rules for an individual in a continuous-time model whzere his income is generated by returns on assets and these returns or instantaneous growth rates are stochastic. P. A. Samuelson has developed a similar model in discrete-time for more general probability distributions in a companion paper [8]. I derive the optimality equations for a multiasset problem when the rate of returns are generated by a Wiener Brownian-motion process. A particular case examined in detail is the two-asset model with constant relative riskaversion or iso-elastic marginal utility. An explicit solution is also found for the case of constant absolute risk-aversion. The general technique employed can be used to examine a wide class of intertemporal economic problems under uncertainty. In addition to the Samuelson paper [8], there is the multi-period analysis of Tobin [9]. Phelps [6] has a model used to determine the optimal consumption rule for a multi-period example where income is partly generated by an asset with an uncertain return. Mirrless [5] has developed a continuous-time optimal consumption model of the neoclassical type with technical progress a random variable.