Professor Harry Zheng: Existence and Construction of Smooth Solutions to HJB Equations and Applications

Abstract: In this talk we discuss utility maximization in a Black-Scholes world. Utility functions are not required to be differentiable or strictly concave, and may not even be concave. We show that there exists a classical solution to the HJB equation with the dual control method. We suggest two constructive methods to find the optimal control, one is a feedback control in terms of the smooth solution to the HJB equation, and the other is a replicating strategy for the optimal wealth process. There is no need to use ``trial and error’’ method to solve the HJB equation. We provide some nontrivial examples to illustrate these constructive methods. We then apply the results to study a long-term investment problem and show a simple proof to the turnpike property of the optimal policy and give an estimate to the rate of convergence.