Portfolio Dominance and Optimality in Infinite Security Markets

The most natural way of ordering portfolios is by comparing their payoffs. If a portfolio has a payoff higher than the payoff of another portfolio, then it is greater than the other portfolio. This order is called the portfolio dominance order. An important property that a portfolio dominance order may have is the lattice property. It requires that the supremum and the infimum of any two portfolios are well-defined. The lattice property implies that such portfolio investment strategies as portfolio insurance or hedging an option's payoff are well-defined. The lattice property of the portfolio dominance order plays an important role in the optimality and equilibrium analysis of markets with infinitely many securities with simple (i.e., arbitrary finite) portfolio holdings. If the portfolio dominance order is a lattice order and has a Yudin basis, then optimal portfolio allocations and equilibria in securities markets do exist. Yudin basis constitutes a system of mutual funds of securities such that trading mutual funds provides spanning opportunities, and that the restriction of no short sales of mutual funds is equivalent to the restriction of non-negative wealth.