Parameter estimation in stochastic differential equations.

*(English)*Zbl 1134.62058
Lecture Notes in Mathematics 1923. Berlin: Springer (ISBN 978-3-540-74447-4/pbk). xiii, 264 p. (2008).

This book deals with a variety of statistical inference problems for stochastic differential equations (SDEs) of the type:
\[
dX_t= \mu(\theta, t,X_t)\,dt+ \sigma(\theta, t,X_t)\,dW_t,\quad t\geq 0,\quad X_0= \zeta.
\]
Here the drift coefficient \(\mu(\cdot)\) and/or the diffusion coefficient \(\sigma^2(\cdot)\) depend on unknown parameter(s) \(\theta\). Hence the general problem is to estimate \(\theta\) based on the availble data which are either continuous in time or discrete. This topic has been a long standing challenge for scientists in applied stochastics over the last 4–5 decades. This is not surprising because SDEs of the above type, in the sense of ItĂ´ or Stratonovich, are used for modelling complex dynamical systems under different kinds of random influences, or perturbations.

The material is organized in the following 10 chapters: 1. Parameter stochastic differential equations. 2. Rates of weak convergence of estimators in homogeneous diffusions. 3. Large deviations of estimators in homogeneous diffusions. 4. Local asymptotic mixed normality for nohomogeneous diffusions. 5. Bayes and sequential estimation in stochastic PDEs. 6. Maximum likelihood estimation in fractional diffusions. 7. Approximate MLEs in nonhomogeneous dissusions. 8. Rates of weak convergence of estimators in the Ornstein-Uhlenbeck process. 9. Local asymptotic normality for discretely observed homogeneous diffusions. 10. Estimating functions for discretely observed homogeneous diffusions.

In each chapter the author starts with useful introductory notes clearly describing the specific models and the problems. Then he presents basic results available in the literature. He has found a good balance between his own contributions to the area and a large number of results of other authors. The nature of the models, the problems and the results are such that we need a lot of technicalities. It is clear that even specific cases lead to non-trivial problems. The proofs show that the conditions imposed are essential. The author pays equal atention to the methods of estimation, the specific constructions of the estimators and the derivation of their properties, in particular when showing convergence properties. A series of interesting and well commented examples are provided as an illustration. The book ends with an extended references list of 342 basic works in this area. There is also a list of basic notations and a Subject index.

Among the readers who can benefit from this carefully written book are researchers and postgraduate students in stochastic modelling, especially those working in areas such as physics, engineering, biology and finance.

The material is organized in the following 10 chapters: 1. Parameter stochastic differential equations. 2. Rates of weak convergence of estimators in homogeneous diffusions. 3. Large deviations of estimators in homogeneous diffusions. 4. Local asymptotic mixed normality for nohomogeneous diffusions. 5. Bayes and sequential estimation in stochastic PDEs. 6. Maximum likelihood estimation in fractional diffusions. 7. Approximate MLEs in nonhomogeneous dissusions. 8. Rates of weak convergence of estimators in the Ornstein-Uhlenbeck process. 9. Local asymptotic normality for discretely observed homogeneous diffusions. 10. Estimating functions for discretely observed homogeneous diffusions.

In each chapter the author starts with useful introductory notes clearly describing the specific models and the problems. Then he presents basic results available in the literature. He has found a good balance between his own contributions to the area and a large number of results of other authors. The nature of the models, the problems and the results are such that we need a lot of technicalities. It is clear that even specific cases lead to non-trivial problems. The proofs show that the conditions imposed are essential. The author pays equal atention to the methods of estimation, the specific constructions of the estimators and the derivation of their properties, in particular when showing convergence properties. A series of interesting and well commented examples are provided as an illustration. The book ends with an extended references list of 342 basic works in this area. There is also a list of basic notations and a Subject index.

Among the readers who can benefit from this carefully written book are researchers and postgraduate students in stochastic modelling, especially those working in areas such as physics, engineering, biology and finance.

Reviewer: Jordan M. Stoyanov (Newcastle upon Tyne)

##### MSC:

62M05 | Markov processes: estimation; hidden Markov models |

62-02 | Research exposition (monographs, survey articles) pertaining to statistics |

60H10 | Stochastic ordinary differential equations (aspects of stochastic analysis) |

60H15 | Stochastic partial differential equations (aspects of stochastic analysis) |

62M09 | Non-Markovian processes: estimation |

60J60 | Diffusion processes |

60-02 | Research exposition (monographs, survey articles) pertaining to probability theory |