Speaker: Shuaijie Qian (Harvard CMSA)
Title: Some non-concave dynamic optimization problems in finance
Abstract: Non-concave dynamic optimization problems appear in many areas of finance and economics. Most of existing literature solves these problems using the concavification principle, and derives equivalent, concave optimization problems whose value functions are still concave. In this talk, I will present our recent work on some non-concave dynamic optimization problems, where the concavification principle may not hold and the resulting value function is indeed non-concave.
The first work is about the portfolio selection model with capital gains tax and a bitcoin mining model with exit options and technology innovation, where the average tax basis and the average mining cost serves as an approximation, respectively. This approximation results in a non-concave value function, and the associated HJB equation problem turns out to admit infinitely many solutions. We show that the value function is the minimal (viscosity) solution of the HJB equation problem. Moreover, the penalty method still works and converges to the value function.
The second work is about a non-concave utility maximization problem with portfolio constraints. We find that adding bounded portfolio constraints, which makes the concavification principle invalid, can significantly affect economic insights in the existing literature. As the resulting value function is likely discontinuous, we introduce a new definition of viscosity solution, prove the corresponding comparison principle, and show that a monotone, stable, and consistent finite difference scheme converges to the solution of the utility maximization problem.