Debt Management Problems and Topics in Stackelberg Equilibrium

Open Access
Author:
Jiang, Yilun
Graduate Program:
Mathematics
Degree:
Doctor of Philosophy
Document Type:
Dissertation
Date of Defense:
February 19, 2019
Committee Members:
  • Alberto Bressan, Dissertation Advisor
  • Alberto Bressan, Committee Chair
  • Alexei Novikov, Committee Member
  • Iouri M Soukhov, Committee Member
  • John C Liechty, Outside Member
Keywords:
  • optimal control
  • debt management
  • bankruptcy risk
  • non-cooperative game
  • Stackelberg equilibrium
  • generic property
Abstract:
The dissertation contains two parts. In the first part of the dissertation, we study optimal strategies for a borrower who needs to repay his debt, in an infinite time horizon. An instantaneous bankruptcy risk is present and the borrower refinances the debt by selling bonds to a pool of risk-neutral lenders. We consider both open-loop and feedback strategies. For open-loop strategies, we interpreted them as Stackelberg equilibria, where the borrower announces his repayment strategy at all future times, and lenders adjust the interest rate accordingly. Our analysis shows the existence of optimal open-loop controls, deriving necessary conditions for optimality and characterizing possible asymptotic limits as $t\to +\infty$. For feedback strategies, we study the solution of a Hamilton-Jacobi equations and construct it as the limit of viscous solutions. Under suitable assumptions, this (possibly discontinuous) limit can be interpreted as an equilibrium solution to a non-cooperative differential game with deterministic dynamics. In the second part of the dissertation, we study the structure of the best reply map for the follower and the optimal strategy for the leader in a non-cooperative Stackelberg game. The two players choose their strategies within domains $X\subseteq\R^m$ and $Y\subseteq\R^n$. Two main cases are considered: either $X=Y=[0,1]$, or $X=\R$, $Y=\R^n$ with $n\geq 1$. Using techniques from differential geometry, we prove that for an open dense set of cost functions the Stackelberg equilibrium is unique and is stable w.r.t.~small perturbations of the two cost functions. Then we introduce a concept of ''self consistent'' Stackelberg equilibria for stochastic games in infinite time horizon, where the two players adopt feedback strategies and have exponentially discounted costs. We focus on games in continuous time, described by a controlled Markov process with finite state space. Under generic assumptions, we prove that a unique self-consistent Stackelberg equilibrium exists, provided that either (i) the leader is far-sighted, i.e.~his exponential discount factor is sufficiently small, or (ii) the follower is narrow-sighted, i.e.~his discount factor is large enough.