The first project is a joint work with Jianfeng Zhang (USC).  We consider a dynamic Principal-Agent problem over time period $[0, T]$. The standard literature considers only the static problem, namely the optimal contract at time $0$. However, when we consider the problem dynamically, the problem is typically time inconsistent in the sense that the optimal contract found at time $0$ does not remain optimal at a later time $t$ (when considering the problem over $[t,T]$). Such time inconsistency is irrelevant if the contract will be enforced once it is signed. However, it will be a serious issue if the principal can fire the agent or if the agent can quit before the expiry date T. In this work we focus on the case that the agent can quit, but with certain cost. There are one principal and a family of agents parameterized by their quality. When an agent (with certain quality) quits, the principal will hire a new agent with possibly different quality (and different individual reservation value). We shall take the equilibrium approach to deal with the time inconsistency issue. The principal's utility at the equilibrium contract will be characterized through a system of HJB equations, where the HJB system is parametrized by the quality of the agents. The solutions could be discontinuous at the boundaries, and thus certain face lifting is needed. We find that the principal's utility may or may not be lower by allowing the agent to quit. Moreover, if the quitting cost has a uniform lower bound, the the principal will only see agents quitting for finitely many times. \\

The second project is a joint work with Weidong Tian (UNCC).  This work  studies a retirement portfolio problem by reformulating into an optimal portfolio choice problem. The model is under the framework of Merton (1971) but with the constraint that  the investment dollar amount in the risky asset is always bound from above by a fixed constant, which is named by risk capacity constraint. The existing literature (e.g.  Zariphopoulou and Vila (1997)) mainly focus on the leverage constraint which does not include our case.  We solve the problem by characterizing the value function as the unique viscosity solution of an HJB equation and shows that the constraint is binding at some free boundary, which is a positive constant $W^*$ in our model. After characterizing the region, we solve the HJB equation as ODEs in both constrained and unconstrained domain. In addition, we get the numerical solution for this HJB equation.