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Stochastic stability of differential equations. With contributions by G. N. Milstein and M. B. Nevelson. 2nd completely revised and enlarged ed. (English) Zbl 1241.60002
Stochastic Modelling and Applied Probability 66. Berlin: Springer (ISBN 978-3-642-23279-4/hbk; 978-3-642-23280-0/ebook). xvii, 339 p. (2012).
This is the second edition of the English translation of this classic monograph (for reviews of the first edition (1980), see Zbl 0441.60060, and for the Russian original (1960), see Zbl 0214.15903) dedicated to the stability of random/stochastic differential equations. Compared to the first edition, there is a new appendix on moment Lyapunov exponents and on the stability index. This appendix was written jointly with G. N. Milstein. Apart from that there are several smaller changes, and the bibliography is widely extended.
As the author points out in the introduction, the main questions treated in this book are the following:
1.
Under which conditions does the solution to a random/stochastic differential equation not explode to infinity in finite time?
2.
If \(X_t \equiv 0\) is a solution of the system, when is it stable in some stochastic sense?
3.
Under which conditions are the solutions bounded for all times (in a stochastic sense)?
4.
If the stochastic forcing is periodic or stationary, then under which conditions do periodic or stationary solutions exist?
5.
If there exists a stationary or periodic solution, under which conditions will every other solution converge to it?
The first two chapters are devoted to random differential equations, i.e., to ordinary differential equations with random forcing of the form \[ \frac{dx}{dt} = G(x,t,\xi(t,\omega)), \] respectively, most of the time \[ \frac{dx}{dt} = F(x,t) + \sigma(x,t) \xi(t,\omega). \] The conditions on \(\xi\) are kept very general (usually some integrability assumption). In Chapter 1, existence and uniqueness of solutions to such systems is studied, as well as the boundedness of solutions. Stability is studied with the help of Lyapunov functions. In Chapter 2, \(\xi\) is additionally assumed to be periodic/stationary, and it is studied under which conditions this implies the existence of a periodic/stationary solution \(x\), and under which conditions every solution converges to the periodic/stationary solution.
Chapters 3 to 7 are dedicated to stochastic differential equations (SDEs) driven by Brownian motion. The third chapter motivates the study of SDEs (using a process with independent increments leads to solutions that are Markov processes; Brownian motion as forcing gives a good approximation of many “real-life” noises). Global (i.e., for all times) existence and uniqueness of solutions to SDEs under Lipschitz conditions but also under more general conditions are studied, as well as existence of stationary/periodic solutions. The latter is again achieved with the help of Lyapunov functions. Also, the relation between SDEs and partial differential equations is studied. Chapter 4 studies ergodic theorems for solutions of SDEs. Chapters 5–7 are devoted to stability theory for SDEs. In Chapter 8 (written with M. B. Nevelson), the results of Chapters 5–7 are applied to study optimal stabilization of controlled systems.
Even more than 40 years after the publication of the first Russian edition, this monograph is still an excellent source for graduate students and researchers alike. It is a great introduction to the subject of stochastic stability. Important concepts such as Lyapunov functions and moment Lyapunov exponents are carefully explained, and many different types of stochastic stability are studied. The monograph is also a valuable reference for researchers – not only researchers in mathematics, but in fact it should also be useful for engineers and physicists interested in stability of systems subject to random forcing.

MSC:
60-02 Research exposition (monographs, survey articles) pertaining to probability theory
60G52 Stable stochastic processes
60G10 Stationary stochastic processes
60H10 Stochastic ordinary differential equations (aspects of stochastic analysis)
60H25 Random operators and equations (aspects of stochastic analysis)
60F99 Limit theorems in probability theory
60J25 Continuous-time Markov processes on general state spaces
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