Stochastic processes, estimators, and control theory are introduced in this book, which undergraduate students should find useful. After a brief introduction to probability theory in chapter 1, chapter 2 introduces the concept of a random variable and describes its probabilistic characterization. The concept is then extended by an index, such as time, that eventually leads to the development of stochastic processes.

The application of a conditional expectation to dynamic filtering is shown in chapter 3. The discrete-time conditional mean estimator, the famous Kalman filter, is derived and illustrated. An alternate derivation of the Kalman estimator is the least-squares approach. The orthogonal projection lemma, associated with the stochastic gradient of the least-square costs, is an important tool for determining an optimal estimator; this is presented in chapter 4.

The stochastic processes, introduced in chapter 2, are characterized by their own calculus in chapter 5. This is needed for developing the model for estimation problems, found in chapter 6, and for the development of the dynamic programming algorithm needed for the solution of continuous-time optimal control problems, found in chapter 9.

Chapter 7 presents approximate filters because state estimates, such as the conditional mean estimates, cannot be implemented for real-time applications.

Special but important extensions to the basic Kalman filter are developed in chapter 8. An especially important extension is that of the linear exponential Gaussian estimator, derived in chapter 10.

All chapters end with a set of exercises. Solutions are not provided. The index spans only three pages and is rather poor.

Most often, the figures used in the book are very helpful to understand the challenging subject matter. However, some of the pictures are apparently just for decoration, such as the sketch of a car in Figure 3.5, or the colorful plotting of six sine waves in Figure 1.1, which is accompanied by a curious statement: “For no apparent reason, we plot this collection” (page 2). The book would have benefited from more thorough editing, to eliminate various spelling errors, repetitions, “unfettered prose” (page 9)--as the authors themselves call it--and to improve the quality of some figures.

Overall, the level of explanation is excellent in this very interesting book.