Convex Duality in Constrained Portfolio Optimization

The theory of portfolio selection of Markowitz (1952) applies mean and variance to characterize return and risk for a combination of more than two financial assets traded in a frictionless economy. Unlimited short selling is allowed and the rates of return on these assets are assumed to have finite variances. Rothschild and Stiglitz (1970, 1971) introduced the concept of second degree stochastic dominance. They showed that when there are more than two risky assets, if there exists a portfolio of assets that the second degree stochastically dominates all the portfolio with the same expected rate of return, then this dominant portfolio must have the minimum variance among all the portfolios. This observation is one of the motivations for characterizing portfolios that have the minimum variance for various levels of expected rate of return. The study of the mean-variance efficiency by Gonzalez-Gaverra (1973), Merton (1972) and Roll (1977) expanded Markowitz model to discuss various issues in portfolio management. To understand the formulation of mean-variance as return-risk, Chamberlain (1983) made an effort to characterize the complete family of probability distributions that are necessary and sufficient for the expected utility of terminal wealth to be a function only of the mean and variance of terminal wealth or for mean-variance utility functions. Epstein (1985) showed that the mean-variance utility functions were implied by a set of decreasing absolute risk aversion postulates.

Criteria, Models and Strategies in Portfolio Selection.- A Model for Portfolio Selection with Order of Expected Returns.- A Compromise Solution to Mutual Funds Portfolio Selection with Transaction Costs.- Optimal Portfolio Selection of Assets with Transaction Costs and No Short Sales.- Portfolio Frontier with Different Interest Rates for Borrowing and Lending.- Multi-period Investment.- Mean-Variance-Skewness Model for Portfolio Selection with Transaction Costs.- Capital Asset Pricing: Theory and Methodologies.- Empirical Tests of CAPM for China's Stock Markets.

Arbitrage Pricing Simplified

We consider the problem of optimal investment in a market with borrowing and random coefficients. We assume that the bond interest rate, the borrowing interest rate, the appreciation rate and the volatility of stock, are random and possibly unbounded. Due to the possibility of borrowing, the formulated optimal investment problem is an optimal stochastic control problem with nonlinear system dynamics and possibly unbounded coefficients. Explicit closed-form solutions in terms of a linear backward stochastic differential equation are obtained for the power and logarithmic utility from terminal wealth. The optimal controls turn out to be of a linear state-feedback form.

Problems in nonlinear optimal control can be reformulated as convex optimization problems over a vector space of linear functionals. In this way, methods of convex analysis can be brought to bear on the task of characterizing solutions to such problems. The result is a necessary and sufficient condition of optimality that generalizes well-known sufficient conditions, referred to as verification theorems, in dynamic programming; as a byproduct, we obtain a representation of the minimum cost in terms of the upper envelope of subsolutions to the Hamilton–Jacobi equation. It is a striking illustration of the wide range of problems to which convex analysis, and, in particular, convex duality, is applicable. The approach, applied to parametric problems in the calculus of variations, was pioneered by L. C. Young [Lectures on the Calculus of Variations and Optimal Control Theory, W. B. Saunders, Philadelphia, PA, 1969]. As recent work has shown, however, it is equally fruitful when applied in optimal control. This paper, which is expository, offers a self-contained treatment of the application of methods of convex duality to general nonlinear problems in deterministic optimal control. At the same time, it provides extensions of previously published results in several directions. A simple proof is given of the main “convex closure” theorem relating generalized flows and relaxed arcs; this is based on mollification techniques recently developed by Fleming and Vermes [SIAM J. Control Optim., 27 (1989), pp. 1136–1155] for constructing smooth subsolutions to the Hamilton–Jacobi equation.

Tools from convex analysis are used to show how stability properties and Lyapunov inequalities translate when passing from a linear differential inclusion (LDI) to its dual. In particular, it is proved that a convex, positive definite function is a Lyapunov function for an LDI if and only if its convex conjugate is a Lyapunov function for the LDIs dual. Examples show how such duality effectively doubles the number of tools available for assessing stability of LDIs.

We discuss the problem of pricing contingent claims, such as European call options, based on the fundamental principle of absence of arbitrage and in the presence of constraints on portfolio choice, for example, incomplete markets and markets with short-selling constraints. Under such constraints, we show that there exists an arbitrage-free interval which contains the celebrated Black-Scholes price (corresponding to the unconstrained case); no price in the interior of this interval permits arbitrage, but every price outside the interval does. In the case of convex constraints, the endpoints of this interval are characterized in terms of auxiliary stochastic control problems, in the manner of Cvitanic and Karatzas. These characterizations lead to explicit computations, or bounds, in several interesting cases. Furthermore, a unique fair price $\hat{p}$ is selected inside this interval, based on utility maximization and marginal rate of substitution principles. Again, characterizations are provided for $\hat{p}$, and these lead to very explicit computations. All these results are also extended to treat the problem of pricing contingent claims in the presence of a higher interest rate for borrowing. In the special case of a European call option in a market with constant coefficients, the endpoints of the arbitrage-free interval are the Black-Scholes prices corresponding to the two different interest rates, and the fair price coincides with that of Barron and Jensen.

Portfolio Optimization Under Convex Incentive Schemes

In this paper, we study stability and performance properties of linear fractional transformation (LFT) parameter-dependent systems using duality theory and tools from convex analysis. A pair of conjugate functions, the convex hull and the maximum of a family of quadratic functions, are used for analysis and synthesis of LFT systems. Sufficient synthesis conditions for both robust state feedback and gain-scheduling output feedback control problems are formulated as a set of linear matrix inequalities (LMIs) with linear search over scalar variables. Finally, a numerical example is used to demonstrate the advantages of the proposed approaches.

A posteriori error analysis for elliptic variational inequalities of the second kind

A variety of Bayesian methods have recently been introduced for finding sparse representations from overcomplete dictionaries of candidate features. These methods often capitalize on latent structure inherent in sparse distributions to perform standard MAP estimation, variational Bayes, approximation using convex duality, or evidence maximization. Despite their reliance on sparsity-inducing priors however, these approaches may or may not actually lead to sparse representations in practice, and so it is a challenging task to determine which algorithm and sparse prior is appropriate. Rather than justifying prior selections and modelling assumptions based on the credibility of the full Bayesian model as is commonly done, this paper bases evaluations on the actual cost functions that emerge from each method. Two minimal conditions are postulated that ideally any sparse learning objective should satisfy. Out of all possible cost functions that can be obtained from the methods described above using (virtually) any sparse prior, a unique function is derived that satisfies these conditions. Both sparse Bayesian learning (SBL) and basis pursuit (BP) are special cases. Later, all methods are shown to be performing MAP estimation using potentially non-factorable implicit priors, which suggests new sparse learning cost functions.

We study solvability of convolution equations for functions with discrete support in , a special case being functions with support in the integer points. The more general case is of interest for several grids in Euclidean space, like the body-centred and face-centred tessellations of 3-space, as well as for the non-periodic grids that appear in the study of quasicrystals. The theorem of existence of fundamental solutions by de Boor et al is generalized to general discrete supports, using only elementary methods. We also study the asymptotic growth of sequences and arrays using the Fenchel transformation.

Convex duality in nonlinear optimal transport

Exponential utility maximization for an insurer with time-inconsistent preferences

We consider the problem of optimal investment in a market with random interest rate for borrowing. This represents an optimal stochastic control problem with a nonlinear system dynamics. We solve this problem under the assumption of coupled Hull-White type model for the interest rates. An explicit closed-form solution is derived for the power and logarithmic utility functions from terminal wealth.