Optimal mean-variance portfolio selection

Assuming that the wealth process Xᵘ 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 μ ∈ R and volatility σ >0 ) and its remaining fraction $$1 -u$$ in a riskless bond (whose price compounds exponentially with interest rate r ∈ R ), and letting Pt,x denote a probability measure under which Xᵘ takes value x at time t, we study the dynamic version of the nonlinear mean-variance optimal control problem where t 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 aligned u_*(t,x) = δ /2\; c\; σ \; 1/x\, e(δ ²-r)(T-t) aligned where δ = (μ -r)/σ . 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.