Parameter Estimation in Fractional Diffusion Models

1. Front Matter

   Pages i-xix

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2. Description and Properties of the Basic Stochastic Models

   • Kęstutis Kubilius, Yuliya Mishura, Kostiantyn Ralchenko

   Pages 1-43

3. The Hurst Index Estimators for a Fractional Brownian Motion

   • Kęstutis Kubilius, Yuliya Mishura, Kostiantyn Ralchenko

   Pages 45-74

4. Estimation of the Hurst Index from the Solution of a Stochastic Differential Equation

   • Kęstutis Kubilius, Yuliya Mishura, Kostiantyn Ralchenko

   Pages 75-123

5. Parameter Estimation in the Mixed Models via Power Variations

   • Kęstutis Kubilius, Yuliya Mishura, Kostiantyn Ralchenko

   Pages 125-160

6. Drift Parameter Estimation in Diffusion and Fractional Diffusion Models

   • Kęstutis Kubilius, Yuliya Mishura, Kostiantyn Ralchenko

   Pages 161-267

7. The Extended Orey Index for Gaussian Processes

   • Kęstutis Kubilius, Yuliya Mishura, Kostiantyn Ralchenko

   Pages 269-320

8. Back Matter

   Pages 321-390

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This book is devoted to parameter estimation in diffusion models involving fractional Brownian motion and related processes. For many years now, standard Brownian motion has been (and still remains) a popular model of randomness used to investigate processes in the natural sciences, financial markets, and the economy. The substantial limitation in the use of stochastic diffusion models with Brownian motion is due to the fact that the motion has independent increments, and, therefore, the random noise it generates is “white,” i.e., uncorrelated. However, many processes in the natural sciences, computer networks and financial markets have long-term or short-term dependences, i.e., the correlations of random noise in these processes are non-zero, and slowly or rapidly decrease with time. In particular, models of financial markets demonstrate various kinds of memory and usually this memory is modeled by fractional Brownian diffusion. Therefore, the book constructs diffusion models with memory and provides simple and suitable parameter estimation methods in these models, making it a valuable resource for all researchers in this field. 

The book is addressed to specialists and researchers in the theory and statistics of stochastic processes, practitioners who apply statistical methods of parameter estimation, graduate and post-graduate students who study mathematical modeling and statistics.

• Fractional Brownian motion
• Diffusion model with memory
• Parameter estimation
• Orey index
• Hurst parameter

“The book presents various diffusion models with memory and provides effective methods for parameter estimation in these models. It may be interesting for researchers and practitioners, graduate and postgraduate students.” (Alex V. Kolnogorov, zbMATH 1388.60006, 2018)

• Institute of Data Science and Digital Technologies, Vilnius University, Vilnius, Lithuania

  Kęstutis Kubilius

• Department of Probability Theory, Statistics and Actuarial Mathematics, Taras Shevchenko National University of Kyiv, Kyiv, Ukraine

  Yuliya Mishura

• Department of Probability Theory, Statistics and Actuarial Mathematics, Taras Shevchenko National University of Kyiv, Kyiv, Lithuania

  Kostiantyn Ralchenko

Prof. Kęstutis Kubilius received his Ph.D. in mathematics at Vilnius University in 1981. Currently he is a professor of mathematics at the same university. His research work mainly focuses on limit theorems for semimartingales, theory of stochastic differential equations, and parameter estimation for fractional diffusion processes. He is the  author of more than 50 published papers.

Prof. Yuliya Mishura received her Ph.D. in probability and statistics at Kyiv University in 1978 and received her postdoctoral degree in probability and statistics (habilitation) in 1990. She is currently a professor at Taras Shevchenko National University of Kyiv. She is the author of more than 250 research papers and 6 books. Her  research interests include theory and statistics of stochastic processes, fractional processes, stochastic analysis and financial mathematics.

Dr. Kostiantyn Ralchenko is a postdoctoral researcher at the Department ofProbability Theory, Statistics and Actuarial Mathematics, Taras Shevchenko National University of Kyiv, where he also completed his Ph.D. in mathematics in 2012. His research interests include stochastic differential equations, fractional and multifractional processes, and statistics of stochastic processes. He is the author of 24 papers.

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