A Appendix: Introduction to stochastic differential equations

In this appendix we bring some basic results concerning stochastic differential equations of the Ito type which systems with state-multiplicative noise constitute a special case of. Stochastic differential equations received a comprehensive treatment in [63], mainly aimed at providing a rigorous framework for optimal state estimation of nonlinear stochastic processes. In the present Appendix, we provide only the main facts that are required to assimilate the main concepts and results which are useful in deriving optimal estimators and controllers for linear systems with state-multiplicative white noise. While the expert reader may skip this appendix, graduate students or practicing engineers may find it to be a useful summary of basic facts and concepts, before they read the text of [63]. Comprehensive treatment of stochastic differential equations in a form accessible to graduate students and practicing engineers is given also in [97] where the close connections between Ito type stochastic differential equations and statistical physics are explored and where a few additional topics such as stochastic stability are covered. Also in [97] many communications oriented examples of nonlinear estimation theory can be found.