Modelling and Estimation of L´evy driven Ornstein Uhlenbeck processes

Abstract

This dissertation is concerned with the parameter estimation problem for Ornstein-Uhlenbeck processes and Vasicek models and the product formula for multiple Itˆo integrals of L´evy processes. In the first part of the thesis, we study the parameter estimation for Ornstein- Uhlenbeck processes driven by the double exponential compound Poisson process. In chapter 2 a method of moments using ergodic theory is proposed to construct ergodic estimators for the double exponential Ornstein- Uhlenbeck process, where the process is observed at discrete time instants with time step size h. We further also show the existence and uniqueness of the function equations to determine the estimators for fixed time step size h. Also, we show the strong consistency and the asymptotic normality of the estimators. Furthermore, we propose a simulation method of the double exponential Ornstein-Uhlenbeck process and perform some numerical simulations to demonstrate the effectiveness of the proposed estimators. In the next chapter, we consider the parameter estimation problem for Vasicek model driven by the compound Poisson process with double exponential jumps as discussed in Chapter 3. Here we discuss the construction of least square estimators for drift parameters based on continuous time observations. In Chapter 4 of the dissertation, we show the derivation of the product formula for finitely many multiple stochastic integrals of L´evy process, expressed in terms of the associated Poisson random measure. A short proof is found that uses properties of exponential vectors and polarization techniques.