Continuous-Time Mean-Variance Portfolio Selection: A Stochastic LQ Framework

Abstract

This paper is concerned with a continuous-time mean-variance portfolio selection model that is formulated as a bicriteria optimization problem. The objective is to maximize the expected terminal return and minimize the variance of the terminal wealth. By putting weights on the two criteria one obtains a single objective stochastic control problem which is however not in the standard form due to the variance term involved. It is shown that this nonstandard problem can be "embedded" into a class of auxiliary stochastic linear-quadratic (LQ) problems. The stochastic LQ control model proves to be an appropriate and effective framework to study the mean-variance problem in light of the recent development on general stochastic LQ problems with indefinite control weighting matrices. This gives rise to the efficient frontier in a closed form for the original portfolio selection problem.

Optimal mean-variance portfolio selection

“…The methodology in these papers relies upon the results on multi-index optimisation problems from the paper by Reid and Citron [17] and is more involved (in comparison with the simple conditioning combined with a double application of Lagrange multipliers as done in the present paper). In particular, the results of [10] and [24] do not establish the existence of statically optimal controls in the problems (2.4)-(2.6) although they do derive their closed form expressions in discrete and continuous time respectively. In this context it may be useful to recall that the first to point out that nonlinear dynamic programming problems may be tackled using the ideas of Lagrange multipliers was White in his paper [23].…”

“…Returning to the stream of papers in the finance literature, the paper by Li and Ng [10, Theorems 1&2] in discrete time and the paper by Zhou and Li [24,Theorem 3.1] in continuous time show that if there is statically optimal control in the unconstrained problem (2.4) then this control can be found by solving a linear-quadratic optimal control problem (which in turn also yields statically optimal controls in the constrained problems (2.5) and (2.6)). The methodology in these papers relies upon the results on multi-index optimisation problems from the paper by Reid and Citron [17] and is more involved (in comparison with the simple conditioning combined with a double application of Lagrange multipliers as done in the present paper).…”

Maximum principle for mean-field jump–diffusion stochastic delay differential equations and its application to finance

“…Using the necessary maximum principle, Andersson and Djehiche (2011) obtained a candidate of the optimal control in a simplified control system without jump and delay. Although this candidate of the optimal control coincided with the optimal portfolio strategy found in Zhou and Li (2000)'s pioneering work, only the sufficient condition for optimality can verify that the candidate is indeed an optimal control of the problem. This motivates us to investigate again whether there exists a version of the sufficient maximum principle, which can be used to solve the bicriteria mean-variance problem.…”

“…However, since the weighted average performance functional involves a nonlinear (quadratic) function of the expected term, even this equivalent single-objective optimization problem is time-inconsistent, where both Bellman's optimality principle and Pontryagin's maximum principle do not work. So the bicriteria mean-variance problem has long been investigated by the stochastic LQ theory with an ingenious embedding technique (see Zhou and Li, 2000). Only until recently, the stochastic maximum principle of mean-field type was found useful to solve the bicriteria mean-variance problem (see Anderson and Djehiche, 2011).…”

A Variational Approach for Mean-Variance-Optimal Deterministic Consumption and Investment

“…Since then various authors have improved and extended the results, see for example Korn and Trautmann [12], Korn [13], Zhou and Li [18], Basak and Chabakauri [3], Kryger and Steffensen [15], Kronborg and Steffensen [14], Alp and Korn [1], Björk, Murgoci and Zhou [5] and others.…”