β Scribed by Yuri Kabanov, Sergei Pergamenshchikov (auth.)
No coin nor oath required. For personal study only.
Two-scale systems described by singularly perturbed SDEs have been the subject of ample literature. However, this new monograph develops subjects that were rarely addressed and could be given the collective description "Stochastic Tikhonov-Levinson theory and its applications." The book provides a mathematical apparatus designed to analyze the dynamic behaviour of a randomly perturbed system with fast and slow variables. In contrast to the deterministic Tikhonov-Levinson theory, the basic model is described in a more realistic way by stochastic differential equations. This leads to a number of new theoretical questions but simultaneously allows us to treat in a unified way a surprisingly wide spectrum of applications like fast modulations, approximate filtering, and stochastic approximation.Two-scale systems described by singularly perturbed SDEs have been the subject of ample literature. However, this new monograph develops subjects that were rarely addressed and could be given the collective description "Stochastic Tikhonov-Levinson theory and its applications." The book provides a mathematical apparatus designed to analyze the dynamic behaviour of a randomly perturbed system with fast and slow variables. In contrast to the deterministic Tikhonov-Levinson theory, the basic model is described in a more realistic way by stochastic differential equations. This leads to a number of new theoretical questions but simultaneously allows us to treat in a unified way a surprisingly wide spectrum of applications like fast modulations, approximate filtering, and stochastic approximation.
Front Matter....Pages I-XIV
Warm-up....Pages 1-17
Toolbox: Moment Bounds for Solutions of Stable SDEs....Pages 19-41
The Tikhonov Theory for SDEs....Pages 43-85
Large Deviations....Pages 87-109
Uniform Expansions for Two-Scale Systems....Pages 111-144
Two-Scale Optimal Control Problems....Pages 145-192
Applications....Pages 193-222
Back Matter....Pages 223-266
Probability Theory and Stochastic Processes; Systems Theory, Control
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