Parameter Estimation in Gaussian Models and Nonlinear Diffusions : Selected Fractional, Tempered, and Mean-Reverting Models

Omtale

Statistical models with Gaussian noise and nonlinear mean-reverting dynamics arise naturally in quantitative finance, actuarial science, energy modeling, and many other applications. Their practical use requires reliable procedures for parameter estimation, while modifications to the noise structure or the diffusion coefficient can substantially affect both the form and the asymptotic properties of estimators. This monograph develops statistical methods for a selected class of Gaussian, fractional, tempered, and nonlinear diffusion models. It covers both continuous and discrete observation schemes and combines rigorous asymptotic analysis with numerical methods and simulation studies. Particular attention is given to models involving fractional and mixed Gaussian noise and to mean-reverting processes such as the Vasicek, Cox-Ingersoll-Ross, and CKLS models. In the book, we achieve the following goals:Develop maximum likelihood estimation for Gaussian regression models with stationary and nonstationary increments under continuous and discrete observations. Study joint parameter estimation in models involving mixed fractional Brownian motion and sums of fractional Brownian motions, using likelihood, quadratic-variation, and ergodic methods. Treat non-ergodic Vasicek models driven by tempered fractional Gaussian processes, including trajectory-growth results and consistent drift estimation. Provide detailed inference results for CIR and CKLS diffusions, including continuous- and high-frequency discrete-observation methods and estimators applicable beyond classical likelihood restrictions. Combine theoretical results on consistency and limiting distributions with numerical algorithms and Monte Carlo studies. The book is intended for researchers, lecturers, doctoral students, and advanced graduate students in probability, mathematical statistics, stochastic processes, and related fields. It will also be useful to quantitatively oriented practitioners and applied scientists who work with stochastic models and require mathematically justified parameter estimation procedures.