Parameter Estimation in Gaussian Models and Nonlinear Diffusions

Selected Fractional, Tempered, and Mean-Reverting Models
1. Auflage 2027
ISBN: 978-1-041-38925-5
Verlag: Taylor & Francis Ltd

Buch, Englisch, 328 Seiten, Format (B × H): 178 mm x 254 mm

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.

Zielgruppe

Academic, Postgraduate, and Professional Reference

Autoren/Hrsg.

Mishura, Yuliya

Ralchenko, Kostiantyn

Yakovliev, Mykyta

Fachgebiete

• Mathematik | Informatik Mathematik Stochastik Stochastische Prozesse

Inhaltsverzeichnis

Part 1: Drift estimation for Gaussian process: maximum likelihood approach  1. Maximum likelihood estimation of the drift for Gaussian processes with linear drift and stationary increments 2. Maximum likelihood estimation of the drift for Gaussian processes with nonlinear drift and non-stationary increments  Part 2: Parameter estimation in linear fractional models  3. Parameter estimation in the model with Brownian and fractional Brownian motion 4. Parameter estimation in the model with two fractional Brownian motions 5. Drift estimation in Vasicek models driven by tempered fractional Brownian motions Part 3: Parameter estimation in nonlinear non-Gaussian models  6. Drift parameter estimation in the Cox–Ingersoll–Ross process 7. Parameter estimation for the CKLS model under continuous observation

Über Autor(innen)

Yuliya Mishura received her PhD in probability and statistics from Kyiv University in
1978 and her Doctor of Sciences degree in the same field in 1990. She is currently a Professor
in the Department of Probability, Statistics, and Actuarial Mathematics at Taras
Shevchenko National University of Kyiv. She has authored or coauthored more than 350
research papers and more than 20 books. Her research interests include the theory and
statistics of stochastic processes, stochastic differential equations, fractional calculus and
fractional stochastic processes, stochastic analysis, functional limit theorems, entropies of
probability distributions and stochastic systems, financial mathematics, and other applications
of stochastic methods. She has been an invited speaker at numerous international
congresses and conferences and has organized a series of scientific conferences. She is Editorin-
Chief of the journal Theory of Probability and Mathematical Statistics and Co-Editor-in-
Chief of Modern Stochastics: Theory and Applications. She has also served as a team leader
and participated in numerous international research projects.

Kostiantyn Ralchenko received his PhD in probability and statistics from Taras
Shevchenko National University of Kyiv in 2012 and his Doctor of Sciences degree in the
same field in 2019. He is currently a Professor in the Department of Probability, Statistics,
and Actuarial Mathematics at Taras Shevchenko National University of Kyiv. His research
interests include the theory and statistical analysis of stochastic processes, fractional and
multifractional processes, stochastic differential equations and stochastic partial differential
equations, entropy measures of probability distributions and stochastic systems, and financial
mathematics. He has published more than 60 peer-reviewed research papers in these
areas and has coauthored four scientific monographs on parameter estimation in fractional
diffusion models, approximations and projections of fractional Brownian motion, discretetime
approximations and limit theorems with applications to financial markets, and entropy
functionals for fractional processes.

Mykyta Yakovliev received his PhD in probability and statistics from Taras Shevchenko
National University of Kyiv in 2024 with a thesis entitled “Estimation of Parameters in
Linear Models with Errors in Variables and with Mixed Fractional Brownian Motion”. He
is currently a Lecturer in the Department of Mathematical Analysis, Faculty of Mechanics
and Mathematics, at Taras Shevchenko National University of Kyiv. His research focuses on
asymptotic normality of parameter estimators in errors-in-variables models and on parameter
estimation for mixed fractional Brownian motion, both with and without deterministic
trends. He has coauthored research papers on parameter estimation in stochastic models
driven by fractional Brownian motion and has presented his work at national and international
scientific conferences.