Stability of stochastic differential equations with respect to semimartingales.

*(English)*Zbl 0724.60059
Pitman Research Notes in Mathematics Series, 251. Harlow: Longman Scientific & Technical; New York: John Wiley & Sons, Inc. 276 p. £22.00 (1991).

This is a systematic account of the author’s work since 1983 on stability of stochastic differential equations driven by a general semimartingale. The crucial technical tools are the concept of Lyapunov function and Gronwall-Bellman type inequalities for the Lebesgue-Stieltjes case. What one typically gets is almost sure or moment exponential stability. In between there is a lot of stochastic analysis, and it is fun to see it all work.

The book consists of the following eight chapters: 1. Stochastic differential equations; 2. Inequalities; 3. Stochastic stability; 4. Stability of stochastic perturbed systems; 5. Stability of stochastic delay equations; 6. Polynomial stability of stochastic differential equations; 7. Comparison theorems and stability; 8. A transformation formula and stability.

The book consists of the following eight chapters: 1. Stochastic differential equations; 2. Inequalities; 3. Stochastic stability; 4. Stability of stochastic perturbed systems; 5. Stability of stochastic delay equations; 6. Polynomial stability of stochastic differential equations; 7. Comparison theorems and stability; 8. A transformation formula and stability.

Reviewer: L.Arnold

##### MSC:

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

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

93E15 | Stochastic stability in control theory |