Optimal Investment with Stopping in Finite Horizon

In this paper, we investigate dynamic optimization problems featuring both stochastic control and optimal stopping in a finite time horizon. The paper aims to develop new methodologies, which are significantly different from those of mixed dynamic optimal control and stopping problems in the existing literature, to study a manager's decision. We formulate our model to a free boundary problem of a fully nonlinear equation. Furthermore, by means of a dual transformation for the above problem, we convert the above problem to a new free boundary problem of a linear equation. Finally, we apply the theoretical results to challenging, yet practically relevant and important, risk-sensitive problems in wealth management to obtain the properties of the optimal strategy and the right time to achieve a certain level over a finite time investment horizon

Global classical solution of Muskat free boundary problem

AbstractIn this paper the Muskat problem which describes a two-phase flow of two fluids, for example, oil and water, in porous media is discussed. The problem involves in seeking two time-dependent harmonic functions u1(x,y,t) and u2(x,y,t) in oil and water regions, respectively, and the interface between oil and water, i.e., the free boundary Γ:y=ρ(x,t), such that on the free boundary u1=u2,Vn=−k1∂u1∂n=−k2∂u2∂n, where n the unit normal vector on the free boundary toward oil region, Vn is the normal velocity of the free boundary Γ, k1 and k2 are positive constants satisfying k1>k2. We prove the existence of classical solution globally in time under some reasonable assumptions. The argument developed in this paper can be used in any multidimensional case

Dynkin Game of Convertible Bonds and Their Optimal Strategy

This paper studies the valuation and optimal strategy of convertible bonds as a Dynkin game by using the reflected backward stochastic differential equation method and the variational inequality method. We first reduce such a Dynkin game to an optimal stopping time problem with state constraint, and then in a Markovian setting, we investigate the optimal strategy by analyzing the properties of the corresponding free boundary, including its position, asymptotics, monotonicity and regularity. We identify situations when call precedes conversion, and vice versa. Moreover, we show that the irregular payoff results in the possibly non-monotonic conversion boundary. Surprisingly, the price of the convertible bond is not necessarily monotonic in time: it may even increase when time approaches maturity.Comment: 28 pages, 9 figures in Journal of Mathematical Analysis and Application, 201

American lookback option with fixed strike price—2-D parabolic variational inequality

AbstractIn this paper we study a 2-dimensional parabolic variational inequality with financial background. We define a suitable weak formula and obtain existence and uniqueness of the problem. Moreover we analyze the behaviors of the free boundary surface