2nd Mini-Workshop on
Mathematical Finance at KHU
Mathematical Finance at KHU
- Variational Inequalities arising from Mathematical Finance -
Room 509, College of Electronics Information and Applied Science
Kyung Hee University, Global Campus, Yong-in, Korea
August, 23, 2023
13:30 - 14:00
Title: Various Free Boundary Problems arising from Mathematical Finance
Abstract: TBA
14:20 - 14:50
Title: A Double Obstacle Problem in an Optimal Investment Problem
Abstract: In this paper, we study a double obstacle problem in partial differential equation that arises in an optimization problem in finance. Precisely, we consider the double obstacle problem which is related to the optimal investment problem with proportional transaction costs of an investor with the logarithmic utility in finite time under the constant elasticity of variance (CEV) model. First, we construct a solution of the double obstacle problem and prove the monotonicity of its free boundaries. From the solution to the double obstacle problem, we construct the solution of the optimization problem. Hence, our result regarding monotonicity indicates the optimal strategy for the optimization problem.
15:10 - 15:40
Title: Target-Dated Portfolio Optimization with Sustainable Consumption
Abstract: TBA
16:00 - 16:30
Title: The Finite-Horizon Consumption-Investment and Retirement Problem with Borrowing Constraint
Abstact: In this paper, we study the optimization problem of an economic agent who chooses the best time for retirement as well as consumption and investment in the presence of a mandatory retirement date. Moreover, the agent faces the borrowing constraint which is constrained in the ability to borrow against future income during working. By utilizing the dual-martingale method for the borrowing constraint, we derive a dual two-person zero-sum game between a singular-controller and a stopper over finite-time horizon. The value of the game satisfies a min-max type of parabolic variational inequality involving both obstacle and gradient constraints, which gives rise to two time-varying free boundaries that correspond to the optimal retirement and the wealth binding, respectively. Using partial differential equation (PDE) techniques, including many technical and non-standard arguments, we establish the uniqueness and existence of a strong solution to the variational inequality, as well as the monotonicity and smoothness of the two free boundaries. Furthermore, the value of game is shown to be the solution to the variational inequality, and we establish a duality theorem to characterize the optimal strategy. To the best our knowledge, this paper is the first to study the zero-sum games between a singular-controller and a stopper over finite-time horizon in the mathematical finance literature.