Deterministic mean-variance-optimal consumption and investment

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

“…We fill that gap, allowing for a slightly more general model with non-constant Black-Scholes market parameters. By applying a Pontryagin maximum principle, we additionally verify that the sufficient conditions of Christiansen and Steffensen [9] for optimal controls are actually necessary. Furthermore, we present an alternative numerical algorithm for the calculation of optimal controls.…”

“…In Christiansen and Steffensen [9], where a Hamilton-Jacobi-Bellman approach is used, the existence of optimal control functions is related to the existence of solutions for the Hamilton-Jacobi-Bellman partial differential equation. However, only numerical solutions are available, so the general existence of solutions is unclear.…”

“…Similarly to m 1 and m 2 , also n 1 , n 2 , p 1 , p 2 , and k solve a system of ordinary differential equations but with terminal instead of initial conditions, see Christiansen and Steffensen [9].…”

“…Christiansen and Steffensen [9] identify characterizing equations for optimal investment and consumption rate by using a Hamilton-Jacobi-Bellman approach. Here, we show that those characterizing equations are indeed necessary by using a Pontryagin maximum principle (cf.…”

“…The present paper is motivated by Christiansen and Steffensen [9], where meanvariance-optimal deterministic consumption and investment is discussed in a BlackScholes market. Sufficient conditions for optimal strategies are derived from a Hamilton-Jacobi-Bellman approach, but only numerical solutions and no analytical solutions are given.…”