Constrained optimal stopping games

In this thesis, we consider four optimal stopping problems with stopping constraints. Chapter 2 introduces a new class of Dynkin games, where the two players are allowed to make their stopping decisions at a sequence of exogenous Poisson arrival times. The value function and the associated optimal stopping strategy are characterized by the solution of a backward stochastic differential equation. Furthermore, the chapter applies the model to study the optimal conversion and calling strategies of convertible bonds, and their asymptotics when the Poisson intensity goes to infinity. Chapter 3 generalizes the work in Chapter 2 from the risk-neutral criteria and common signal times for both players to the risk-sensitive criteria and two heterogeneous signal times. Chapter 4 considers a two-player zero-sum optimal switching games with stopping constraints. We prove the chain of inequalities involving the four values of the game, and the values of both the static and dynamic games exist in the case when the running and terminal rewards are separated. Chapter 5 studies a mixed stochastic control and constrained optimal stopping problem which models rollover debt decisions in an incomplete market. In addition to the rollover decisions, the creditor can also choose a control strategy to trade in risky assets correlated with the fundamental assets. In the case of exponential utility, we prove the complete characterization and obtain the exponential indifference bond price and its associated optimal mixed strategy.