Lifetime Portfolio Selection under Uncertainty: the Continuous time Case
M OST models of portfolio selection have been one period models. I examine the combined problem of optimal portfolio selection and consumption rules for an individual in a continuous time model where 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 C81. 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 [S] 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 S has developed a continuous time optimal consumption model of the neoclassical type with technical progress a random variable.