Optimum consumption and portfolio rules in a continuous-time model☆
TL;DR: In this paper, the authors considered the continuous-time consumption-portfolio problem for an individual whose income is generated by capital gains on investments in assets with prices assumed to satisfy the geometric Brownian motion hypothesis, which implies that asset prices are stationary and lognormally distributed.
About: This article is published in Journal of Economic Theory.The article was published on 1971-12-01 and is currently open access. It has received 4952 citations till now. The article focuses on the topics: Geometric Brownian motion & Intertemporal portfolio choice.
Citations
More filters
TL;DR: The paper gives a complete treatment of the existence and nonexistence of optimal policies and new theorems for the optimal control of degenerate diffusions are given, as well as explicit formulas for the value function.
Abstract: An agent can distribute his wealth between two investments, one with a fixed rate of return r and the other with a random rate of return (modeled as a diffusion) with mean r. The agent seeks to maximize total discounted utility from consumption over an infinite horizon. Consumption may be constrained from below. Various models for bankruptcy, including welfare, are considered. The agent has a strictly concave utility function for consumption; however, it is shown that the utility function for wealth may have convex portions, thus the agent may be risk seeking. The paper gives a complete treatment of the existence and nonexistence of optimal policies. New theorems for the optimal control of degenerate diffusions are given, as well as explicit formulas for the value function.
55 citations
Cites background or methods from "Optimum consumption and portfolio r..."
...This paper also gives a careful treatment of this decision problem with two important additional features generalizing the original work of Samuelson [14] or Merton [11], [12]....
[...]
...The original work of Samuelson [14] or Merton [11], [12] does not examine the agent's behavior if bankruptcy can occur....
[...]
...It follows that in a bankruptcy context the two fund or three fund mutual fund theorems of Merton [11] will hold only over the concave portion....
[...]
...The previous work of Merton [11], [12] and Richard [13, note 9] has given a derivation of optimal policies for single agent investment-consumption decision problems without settling the question of the existence of an optimal policy....
[...]
...1) (see Merton [11]), one would be forced to set 0 < y < 1 and r = 0....
[...]
01 Jan 2016
TL;DR: In this paper, the authors formulate rigorous axiomatic probability theory based on Lebesgue integration and show how to formulate rigorous probability theory with axiomatization and show that it is possible to obtain axiomatic probability theory without applying the integral.
Abstract: When H. Lebesgue invented the Lebesgue integral, it was regarded as an abstract concept without applications. It was A.N. Kolmogorov [52] who first showed how to formulate rigorous axiomatic probability theory based on Lebesgue integration.
54 citations
TL;DR: In this article, the Integrated Asset Allocation (IAA) problem is addressed and the authors present an integrated asset allocation strategy for the first time, which is based on the following:
Abstract: (1987). Integrated Asset Allocation. Financial Analysts Journal: Vol. 43, No. 5, pp. 25-32.
54 citations
TL;DR: In this paper, the authors studied the intertemporal optimal consumption and investment problem in a continuous-time economy with a divisible durable good, where consumption services are proportional to the stock of the good held and adjustment of the stock is costly.
53 citations
TL;DR: This work proves existence and uniqueness of the solution to a general class of BSDEs, encompassing the solutions to the portfolio choice and valuation problems as special cases, and provides an explicit decomposition of the excess return on an asset into a risk premium and an ambiguity premium.
Abstract: We solve, theoretically and numerically, the problems of optimal portfolio choice and indifference valuation in a general continuous-time setting. The setting features (i) ambiguity and time-consistent ambiguity-averse preferences, (ii) discontinuities in the asset price processes, with a general and possibly infinite activity jump part next to a continuous diffusion part, and (iii) general and possibly nonconvex trading constraints. We characterize our solutions as solutions to backward stochastic differential equations (BSDEs). Generalizing Kobylanski's result for quadratic BSDEs to an infinite activity jump setting, we prove existence and uniqueness of the solution to a general class of BSDEs, encompassing the solutions to our portfolio choice and valuation problems as special cases. We provide an explicit decomposition of the excess return on an asset into a risk premium and an ambiguity premium, and a further decomposition into a piece stemming from the diffusion part and a piece stemming from the ju...
53 citations
References
More filters
TL;DR: In this paper, the combined problem of optimal portfolio selection and consumption rules for an individual in a continuous-time model was examined, where his income is generated by returns on assets and these returns or instantaneous "growth rates" are stochastic.
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.
4,908 citations
Book•
01 Jan 1965
TL;DR: This book should be of interest to undergraduate and postgraduate students of probability theory.
Abstract: This book should be of interest to undergraduate and postgraduate students of probability theory.
3,597 citations
TL;DR: In this paper, the optimal consumption-investment problem for an investor whose utility for consumption over time is a discounted sum of single-period utilities, with the latter being constant over time and exhibiting constant relative risk aversion (power-law functions or logarithmic functions), is discussed.
Abstract: Publisher Summary This chapter reviews the optimal consumption-investment problem for an investor whose utility for consumption over time is a discounted sum of single-period utilities, with the latter being constant over time and exhibiting constant relative risk aversion (power-law functions or logarithmic functions). It presents a generalization of Phelps' model to include portfolio choice and consumption. The explicit form of the optimal solution is derived for the special case of utility functions having constant relative risk aversion. The optimal portfolio decision is independent of time, wealth, and the consumption decision at each stage. Most analyses of portfolio selection, whether they are of the Markowitz–Tobin mean-variance or of more general type, maximize over one period. The chapter only discusses special and easy cases that suffice to illustrate the general principles involved and presents the lifetime model that reveals that investing for many periods does not itself introduce extra tolerance for riskiness at early or any stages of life.
2,369 citations
Book•
17 Jan 2012
TL;DR: In this article, a book on stochastic stability and control dealing with Liapunov function approach to study of Markov processes is presented, which is based on the work of this article.
Abstract: Book on stochastic stability and control dealing with Liapunov function approach to study of Markov processes
1,293 citations
987 citations