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**Stochastic differential equations with Markovian switching.**
*(English)*
Zbl 1126.60002

Hackensack, NJ: World Scientific (ISBN 1-86094-701-8/hbk; 978-1-86094-884-8/ebook). xviii, 409 p. (2006).

The monograph under review deals with stochastic differential equations with Markovian switching. These are stochastic differential equations of ItĂ´ type where the drift and dispersion functions change according to a Markov chain in continuous time. For these kind of equations a general theory is introduced, stability and numerical methods are discussed and some further models and applications are treated. As the authors state the text is mainly based on their own work in this field.

Although in the first part of the book basic concepts in probability theory are introduced. This part serves as a review rather than an introduction. Thus, the reader should have some knowledge on stochastic processes and stochastic analysis. The monograph is very well written and gives a nice overview of a new type of stochastic differential equations with numerous possible applications. The proofs seem to be understandable and the amount of typos is moderate.

A more detailed description is the following: the first chapter reviews some basics on stochastic processes and the second collects some important inequalities from calculus and probability theory. The third chapter introduces stochastic differential equations with Markovian switching and discusses several results on strong existence and uniqueness as well as standard properties of the solution process. The following fourth chapter introduces several methods to approximate the solution. Obviously, the main emphasis is on the Euler-Maruyama method, but also other methods like Caratheodory’s approximation, backward Euler-scheme and stochastic theta method are discussed. The fifth chapter is devoted to boundedness and stability properties of solutions. Here the main technique is Lyapunov’s second method. In the sixth chapter stability properties of numerical methods for asymptotic behavior are considered, i.e. the question how far asymptotic stability of the solution of the original equation is reflected in the numerical solution and vice versa. The following three chapters deal with various types of stochastic differential equations with Markovian switching from functional equations to interval systems. The last chapter introduces, finally, some applications of stochastic differential equations with Markovian switching, e.g., stochastic population dynamics, financial mathematics and neural networks.

Although in the first part of the book basic concepts in probability theory are introduced. This part serves as a review rather than an introduction. Thus, the reader should have some knowledge on stochastic processes and stochastic analysis. The monograph is very well written and gives a nice overview of a new type of stochastic differential equations with numerous possible applications. The proofs seem to be understandable and the amount of typos is moderate.

A more detailed description is the following: the first chapter reviews some basics on stochastic processes and the second collects some important inequalities from calculus and probability theory. The third chapter introduces stochastic differential equations with Markovian switching and discusses several results on strong existence and uniqueness as well as standard properties of the solution process. The following fourth chapter introduces several methods to approximate the solution. Obviously, the main emphasis is on the Euler-Maruyama method, but also other methods like Caratheodory’s approximation, backward Euler-scheme and stochastic theta method are discussed. The fifth chapter is devoted to boundedness and stability properties of solutions. Here the main technique is Lyapunov’s second method. In the sixth chapter stability properties of numerical methods for asymptotic behavior are considered, i.e. the question how far asymptotic stability of the solution of the original equation is reflected in the numerical solution and vice versa. The following three chapters deal with various types of stochastic differential equations with Markovian switching from functional equations to interval systems. The last chapter introduces, finally, some applications of stochastic differential equations with Markovian switching, e.g., stochastic population dynamics, financial mathematics and neural networks.

Reviewer: Markus Riedle (Berlin)

### MSC:

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

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

60H30 | Applications of stochastic analysis (to PDEs, etc.) |