Modelling and Estimation of L´evy driven Ornstein Uhlenbeck processes

  • Author / Creator
    Sharma, Neha
  • 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.

  • Subjects / Keywords
  • Graduation date
    Spring 2024
  • Type of Item
  • Degree
    Doctor of Philosophy
  • DOI
  • License
    This thesis is made available by the University of Alberta Libraries with permission of the copyright owner solely for non-commercial purposes. This thesis, or any portion thereof, may not otherwise be copied or reproduced without the written consent of the copyright owner, except to the extent permitted by Canadian copyright law.