Stochastic differential equations

The purpose of this thesis is to compare the effectiveness of several interest
rate models in fitting the true value of interest rates. Up until 1990, the universally
accepted models were the equilibrium models, namely the Rendleman-Bartter model,
the Vasicek model, and the Cox-Ingersoll-Ross (CIR) model. While these models
were probably considered relatively accurate around the time of their discovery, they
do not provide a good fit to the initial term structure of interest rates, making them
substandard for use by traders in pricing interest rate options. The fourth model
we consider is the Hull-White one-factor model, which does provide this fit. After
calibrating, simulating, and comparing these four models, we find that the Hull-White
model gives the best fit to our data sets.

We generalize the theory of stochastic impulse control of jump diffusions introduced by Oksendal and Sulem (2004) with milder assumptions. In particular, we assume that the original process is affected by the interventions. We also generalize the optimal central bank intervention problem including market reaction introduced by Moreno (2007), allowing the exchange rate dynamic to follow a jump diffusion process. We furthermore generalize the approximation theory of stochastic impulse control problems by a sequence of iterated optimal stopping problems which is also introduced in Oksendal and Sulem (2004). We develop new results which allow us to reduce a given impulse control problem to a sequence of iterated optimal stopping problems even though the original process is affected by interventions.