A Stochastic Calculus Model of Continuous Trading: Optimal Portfolios

Abstract

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.

Risk- and ambiguity-averse portfolio optimization with quasiconcave utility functionals

“…for some function U : R → R. This result forms the axiomatic basis for the problem of maximization of (subjective) expected utility, 2 which since the seminal contributions of [60] has played a prominent role within mathematical finance; see, among others, [16,45,46,67]. The question of model uncertainty was incorporated in the axiomatic approaches starting with the work of Gilboa and Schmeidler [35], later followed by, among others, Maccheroni et al [57] and Cerreia-Vioglio et al [11]; see also [10,13].…”

Optimal mean-variance portfolio selection

“…The stream of papers in the finance literature starting with the paper by Richardson [19] deals with optimal portfolio selection problems under mean-variance criteria similar/analogous to (2.4)-(2.6) above. Richardson's paper [19] derives a statically optimal control in the constrained problem (2.6) using the martingale method suggested by Pliska [15] who makes use of the Legendre transform (convex analysis) rather than the Lagrange multipliers. For an overview of the martingale method based on Lagrange multipliers see e.g.…”

Does the Failure of the Expectations Hypothesis Matter for Long-Term Investors?

“…To solve this problem, it is convenient to use the martingale technique of Cox and Huang (1989), Karatzas, Lehoczky, and Shreve (1987) and Pliska (1986) generalized to the case of incomplete markets by He and Pearson (1991). He and Pearson show that, for some endogenous pricing kernel, the dynamic budget constraint (21) can be replaced by a static budget constraint.…”