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A Stochastic Calculus Model of Continuous Trading: Optimal Portfolios

The problem of choosing a portfolio of securities so as to maximize the expected utility of wealth at a terminal planning horizon is solved via stochastic calculus and convex analysis. This problem is decomposed into two subproblems. With security prices modeled as semimartingales and trading strategies modeled as predictable processes, the set of terminal wealths is identified as a subspace in a space of integrable random variables. The first subproblem is to find the terminal wealth that maximizes expected utility. Convex analysis is used to derive necessary and sufficient conditions for optimality and an existence result. The second subproblem of finding the admissible trading strategy that generates the optimal terminal wealth is a martingale representation problem. The primary advantage of this approach is that explicit formulas can readily be derived for the optimal terminal wealth and the corresponding expected utility, as is shown for the case of an exponential utility function and a risky security modeled as geometric Brownian motion.