The first book in inference for stochastic processes from a statistical, rather than a probabilistic, perspective. It provides a systematic exposition of theoretical results from over ten years of mathematical literature and presents, for the first time in book form, many new techniques and approaches.

Decision making in all spheres of activity involves uncertainty. If rational decisions have to be made, they have to be based on the past observations of the phenomenon in question. Data collection, model building and inference from the data collected, validation of the model and refinement of the model are the key steps or building blocks involved in any rational decision making process. Stochastic processes are widely used for model building in the social, physical, engineering, and life sciences as well as in financial economics. Statistical inference for stochastic processes is of great importance from the theoretical as well as from applications point of view in model building. During the past twenty years, there has been a large amount of progress in the study of inferential aspects for continuous as well as discrete time stochastic processes. Diffusion type processes are a large class of continuous time processes which are widely used for stochastic modelling. the book aims to bring together several methods of estimation of parameters involved in such processes when the process is observed continuously over a period of time or when sampled data is available as generally feasible.

This book develops alternative methods to estimate the unknown parameters in stochastic volatility models, offering a new approach to test model accuracy. While there is ample research to document stochastic differential equation models driven by Brownian motion based on discrete observations of the underlying diffusion process, these traditional methods often fail to estimate the unknown parameters in the unobserved volatility processes. This text studies the second order rate of weak convergence to normality to obtain refined inference results like confidence interval, as well as nontraditional continuous time stochastic volatility models driven by fractional Levy processes. By incorporating jumps and long memory into the volatility process, these new methods will help better predict option pricing and stock market crash risk. Some simulation algorithms for numerical experiments are provided.

Focussing on stochastic models for the spread of infectious diseases in a human population, this book is the outcome of a two-week ICPAM/CIMPA school on "Stochastic models of epidemics" which took place in Ziguinchor, Senegal, December 5–16, 2015. The text is divided into four parts, each based on one of the courses given at the school: homogeneous models (Tom Britton and Etienne Pardoux), two-level mixing models (David Sirl and Frank Ball), epidemics on graphs (Viet Chi Tran), and statistics for epidemic models (Catherine Larédo). The CIMPA school was aimed at PhD students and Post Docs in the mathematical sciences. Parts (or all) of this book can be used as the basis for traditional or individual reading courses on the topic. For this reason, examples and exercises (some with solutions) are provided throughout.

The seventh volume in the SemStat series, Statistical Methods for Stochastic Differential Equations presents current research trends and recent developments in statistical methods for stochastic differential equations. Written to be accessible to both new students and seasoned researchers, each self-contained chapter starts with introductions to th

The latest cutting-edge research on market microstructure Based on the December 2010 conference on market microstructure, organized with the help of the Institut Louis Bachelier, this guide brings together the leading thinkers to discuss this important field of modern finance. It provides readers with vital insight on the origin of the well-known anomalous "stylized facts" in financial prices series, namely heavy tails, volatility, and clustering, and illustrates their impact on the organization of markets, execution costs, price impact, organization liquidity in electronic markets, and other issues raised by high-frequency trading. World-class contributors cover topics including analysis of high-frequency data, statistics of high-frequency data, market impact, and optimal trading. This is a must-have guide for practitioners and academics in quantitative finance.

This book deals with the mathematical aspects of survival analysis and reliability as well as other topics, reflecting recent developments in the following areas: applications in epidemiology; probabilistic and statistical models and methods in reliability; models and methods in survival analysis, longevity, aging, and degradation; accelerated life models; quality of life; new statistical challenges in genomics. The work will be useful to a broad interdisciplinary readership of researchers and practitioners in applied probability and statistics, industrial statistics, biomedicine, biostatistics, and engineering.

Martingale Methods in Statistics provides a unique introduction to statistics of stochastic processes written with the author’s strong desire to present what is not available in other textbooks. While the author chooses to omit the well-known proofs of some of fundamental theorems in martingale theory by making clear citations instead, the author does his best to describe some intuitive interpretations or concrete usages of such theorems. On the other hand, the exposition of relatively new theorems in asymptotic statistics is presented in a completely self-contained way. Some simple, easy-to-understand proofs of martingale central limit theorems are included. The potential readers include those who hope to build up mathematical bases to deal with high-frequency data in mathematical finance and those who hope to learn the theoretical background for Cox’s regression model in survival analysis. A highlight of the monograph is Chapters 8-10 dealing with Z-estimators and related topics, such as the asymptotic representation of Z-estimators, the theory of asymptotically optimal inference based on the LAN concept and the unified approach to the change point problems via "Z-process method". Some new inequalities for maxima of finitely many martingales are presented in the Appendix. Readers will find many tips for solving concrete problems in modern statistics of stochastic processes as well as in more fundamental models such as i.i.d. and Markov chain models.

This book covers an extensive class of models involving inhomogeneous Poisson processes and deals with their identification, i.e. the solution of certain estimation or hypothesis testing problems based on the given dataset. These processes are mathematically easy-to-handle and appear in numerous disciplines, including astronomy, biology, ecology, geology, seismology, medicine, physics, statistical mechanics, economics, image processing, forestry, telecommunications, insurance and finance, reliability, queuing theory, wireless networks, and localisation of sources. Beginning with the definitions and properties of some fundamental notions (stochastic integral, likelihood ratio, limit theorems, etc.), the book goes on to analyse a wide class of estimators for regular and singular statistical models. Special attention is paid to problems of change-point type, and in particular cusp-type change-point models, then the focus turns to the asymptotically efficient nonparametric estimation of the mean function, the intensity function, and of some functionals. Traditional hypothesis testing, including some goodness-of-fit tests, is also discussed. The theory is then applied to three classes of problems: misspecification in regularity (MiR),corresponding to situations where the chosen change-point model and that of the real data have different regularity; optical communication with phase and frequency modulation of periodic intensity functions; and localization of a radioactive (Poisson) source on the plane using K detectors. Each chapter concludes with a series of problems, and state-of-the-art references are provided, making the book invaluable to researchers and students working in areas which actively use inhomogeneous Poisson processes.

The YUIMA package is the first comprehensive R framework based on S4 classes and methods which allows for the simulation of stochastic differential equations driven by Wiener process, Lévy processes or fractional Brownian motion, as well as CARMA, COGARCH, and Point processes. The package performs various central statistical analyses such as quasi maximum likelihood estimation, adaptive Bayes estimation, structural change point analysis, hypotheses testing, asynchronous covariance estimation, lead-lag estimation, LASSO model selection, and so on. YUIMA also supports stochastic numerical analysis by fast computation of the expected value of functionals of stochastic processes through automatic asymptotic expansion by means of the Malliavin calculus. All models can be multidimensional, multiparametric or non parametric.The book explains briefly the underlying theory for simulation and inference of several classes of stochastic processes and then presents both simulation experiments and applications to real data. Although these processes have been originally proposed in physics and more recently in finance, they are becoming popular also in biology due to the fact the time course experimental data are now available. The YUIMA package, available on CRAN, can be freely downloaded and this companion book will make the user able to start his or her analysis from the first page.