Optimal mean-variance portfolio selection

Assuming that the wealth process (X^u) is generated self-financially from the given initial wealth by holding its fraction u in a risky stock (whose price follows a geometric Brownian motion with drift (mu in mathbb {R}) and volatility (sigma >0)) and its remaining fraction (1 -u) in a riskless bond (whose price compounds exponentially with interest rate (r in mathbb {R})), and letting (mathsf{P}_{t,x}) denote a probability measure under which (X^u) takes value x at time t, we study the dynamic version of the nonlinear mean-variance optimal control problem

Open image in new windowt runs from 0 to the given terminal time (T>0), the supremum is taken over admissible controls u, and (c>0) is a given constant. By employing the method of Lagrange multipliers we show that the nonlinear problem can be reduced to a family of linear problems. Solving the latter using a classic Hamilton-Jacobi-Bellman approach we find that the optimal dynamic control is given by

$$begin{aligned} u_*(t,x) = frac{delta }{2; c; sigma }; frac{1}{x}, e^{(delta ^2-r)(T-t)} end{aligned}$$

where (delta = (mu -r)/sigma ). The dynamic formulation of the problem and the method of solution are applied to the constrained problems of maximising/minimising the mean/variance subject to the upper/lower bound on the variance/mean from which the nonlinear problem above is obtained by optimising the Lagrangian itself.