Parameter Estimation in Fractional Stochastic Differential Equations

This book discusses long memory and long range dependence for continuous time financial models. While traditional models are Markovian, which have short memory, models with long memory have not been focused on and only studied in the discrete time series modeling context. The development of increasingly complex financial models products requires the use of advanced mathematical and statistical methods. Though the mathematics behind these models are more complicated, these models are more practical from the perspectives of finance, biology, and physics. The author presents models driven by non-Gaussian fractional Levy processes, which are more useful models in these fields. In addition, the author incorporates long memory into the model by using noise driven by fractional Brownian motion, which is neither a semi martingale nor a Markov process, except the one half Hurst parameter case, where it is Brownian motion. Fractional stochastic differential equations are state-of-the art in continuous time asset pricing and interest rate models. Though pricing has been studied, parameter estimation has not been well studied. Readers will learn advanced mathematical and statistical methods in finance, and special attention is paid to stylized facts such as high dimensional models and data, models with jumps, and models with long-memory.

Berry-Esseen Bound for the Least Squares Estimator in the Fractional Ornstein-Uhlenbeck Process.- Large Deviations in Testing Fractional Ornstein-Uhlenbeck Models.- Minimum Contrast Estimation in Fractional Ornstein-Uhlenbeck Process.- Hypotheses Testing in Nonergodic Fractional Ornstein-Uhlenbeck Models.- Nonparametric Estimation in Heath-Jarrow-Morton Term Structure Models Driven by Fractional Levy Processes.- Bootstrap Confidence Interval for Fractional Diffusions and American Options.

Jaya P.N. Bishwal, Ph.D., is an Associate Professor in the Department of Mathematics and Statistics at the University of North Carolina at Charlotte.  His research interests include inference for stochastic differential equations, mathematical finance and financial econometrics, stochastic analysis and probability, and mathematical biology. Dr. Bishwal has published two Springer books, Parameter Estimation in Stochastic Differential Equations (2007) and Parameter Estimation in Stochastic Volatility Models (2022), and more than 70 research papers. He has taught several graduate courses including Stochastic Calculus for Finance, Advanced Stochastic calculus for Finance, Monte Carlo Methods in Finance, and Statistics and Data Analysis for Finance.