CONTINUOUS-TIME MARKOWITZ'S PROBLEMS IN AN INCOMPLETE MARKET, WITH CONSTRAINED PORTFOLIOS

Xun Yu Zhou

Abstract

Continuous-time Markowitz's mean-variance portfolio selection problems with finite-time horizons are investigated in an arbitrage-free yet incomplete market. Models with various constraints on portfolios, including those unconstrained, shorting prohibited, bankruptcy prohibited, and both shorting and bankruptcy pro hibited, are respectively tackled. The sets of the terminal wealths that can be replicated by admissible portfolios are characterized, in explicit terms, for all the models under consideration.

This enables one to transfer the original dynamic portfolio selection problems into ones of static, albeit constrained, optimization problems in terms of the terminal wealth. Solutions to the latter are obtained via certain dual (static) optimization problems.

When all the market coefficients are deterministic processes, mean-variance efficient portfolios and frontiers are obtained explicitly for all the models.