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Optimal portfolio and consumption decisions for a “small investor” on a finite horizon. (English) Zbl 0644.93066

The authors consider a general consumption/investment decision problem for a single agent, whose actions cannot affect the market prices. The objective of the agent is to maximize (with respect to \((\pi,C))\) a linear combination of the total expected discounted utility from consumption over [0,T]: \[ J_ 1(x;\pi,C)=E\int^{T}_{0}\exp (- \int^{t}_{0}\beta (s)ds)U_ 1(C_ t)dt, \] and the expected utility from terminal wealth: \[ J_ 2(x;\pi,C)=E[\exp (-\int^{T}_{0}\beta (s)ds)U_ 2(X_ T)], \] where a portfolio process \(\pi =\{\pi(t)\}\) is an \(\{\) \({\mathcal F}_ t\}\)-adapted \({\mathbb{R}}^ d\)-valued process on a probability space (\(\Omega\),\({\mathcal F},P;{\mathcal F}_ t)\), a consumption process \(C=\{C_ t\}\) is an (\({\mathcal F}_ t\}\)-adapted process with values in [0,\(\infty)\), \(X=\{X_ t\}\) is the agent’s wealth at time t, which depends upon (\(\pi\),C) and the general coefficients of the market model, with an initial one \(X_ 0=x\geq 0\), a discount process \(\{\beta\) (t)\(\}\) is \(\{\) \({\mathcal F}_ t\}\)-adapted and uniformly bounded, and functions \(U_ i\) \((i=1,2)\) are strictly increasing, strictly concave and C 1-functions from \((0,\infty)\) to \({\mathbb{R}}\) with \(\lim_{c\downarrow 0}U_ i(c)\geq -\infty\), and \(\lim_{c\uparrow \infty}U'(c)=0\). Under general conditions the authors consider separately the two elementary problems of maximizing \(J_ 1\) and \(J_ 2\), and then by appropriately composing them they construct optimal consumption and wealth processes for the original problem. In the case of a market model with constant coefficients, they explicitly give the optimal portfolio and consumption rules in feedback form.
Reviewer: Y.Ohtsubo

MSC:

93E20 Optimal stochastic control
60G44 Martingales with continuous parameter
91B62 Economic growth models
49K45 Optimality conditions for problems involving randomness
90B50 Management decision making, including multiple objectives
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