Optimal mean-variance portfolio selection

Assuming that the wealth process $$X^u$$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 $$\mu \in \mathbb {R}$$μ∈R and volatility $$\sigma >0$$σ>0) and its remaining fraction $$1 -u$$1-u in a riskless bond (whose price compounds exponentially with interest rate $$r \in \mathbb {R}$$r∈R), and letting $$\mathsf{P}_{t,x}$$Pt,x denote a probability measure under which $$X^u$$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 $$T>0$$T>0, the supremum is taken over admissible controls u, and $$c>0$$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 $$\begin{aligned} u_*(t,x) = \frac{\delta }{2\; c\; \sigma }\; \frac{1}{x}\, e^{(\delta ^2-r)(T-t)} \end{aligned}$$u∗(t,x)=δ2cσ1xe(δ2-r)(T-t)where $$\delta = (\mu -r)/\sigma $$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.