Optimality conditions for utility maximization in an incomplete market

The problem of maximizing expected utility of final wealth in an incomplete market is investigated. The incomplete market is modelled by a bond and a finite number of stocks, the latter being driven by a d-dimensional Brownian motion. The coefficients of the bond and stock price processes are adapted to this Brownian motion, and the number of stocks is less than or equal to the dimension of the driving Brownian motion. It is shown that there is a way to “fictitiously” complete this market so that the optimal protfolio for the resulting completed market coincides with the optimal portfolio for the original incomplete market. A number of equivalent characterizations of the fictitious completion are given, and examples are provided.

[1]  John C. Cox,et al.  A variational problem arising in financial economics , 1991 .

[2]  R. C. Merton,et al.  Optimum consumption and portfolio rules in a continuous - time model Journal of Economic Theory 3 , 1971 .

[3]  J. Bismut.  Conjugate convex functions in optimal stochastic control , 1973 .

[4]  J. Harrison,et al.  A stochastic calculus model of continuous trading: Complete markets , 1983 .

[5]  Ioannis Karatzas,et al.  Brownian Motion and Stochastic Calculus , 1987 .

[6]  S. Shreve,et al.  Optimal portfolio and consumption decisions for a “small investor” on a finite horizon , 1987 .

[7]  J. Cox,et al.  Optimal consumption and portfolio policies when asset prices follow a diffusion process , 1989 .

[8]  J. Doob.  Stochastic processes , 1953 .

[9]  David M. Kreps,et al.  Martingales and arbitrage in multiperiod securities markets , 1979 .

[10]  R. C. Merton,et al.  Lifetime Portfolio Selection under Uncertainty: The Continuous-Time Case , 1969 .

[11]  Steven E. Shreve,et al.  A Duality Method for Optimal Consumption and Investment Under Short- Selling Prohibition. I. General Market Coefficients , 1992 .

[12]  J. Harrison,et al.  Martingales and stochastic integrals in the theory of continuous trading , 1981 .

[13]  P. Samuelson.  Lifetime Portfolio Selection by Dynamic Stochastic Programming , 1969 .

[14]  I. Karatzas.  Optimization problems in the theory of continuous trading , 1989 .

[15]  S. Shreve,et al.  Martingale and duality methods for utility maximization in a incomplete market , 1991 .

[16]  Neil D. Pearson,et al.  Consumption and portfolio policies with incomplete markets and short-sale constraints: The infinite dimensional case , 1991 .

[17]  Stanley R. Pliska,et al.  A Stochastic Calculus Model of Continuous Trading: Optimal Portfolios , 1986, Math. Oper. Res..