This book encompasses the study of hybrid switching di usion processes
and their applications. The word \hybrid" signi es the coexistence of c-
tinuous dynamics and discrete events, which is one of the distinct
features of the processes under consideration. Much of the book is
concerned with the interactions of the continuous dynamics and the
discrete events. Our motivations for studying such processes originate
from emerging and - isting applications in wireless communications,
signal processing, queueing networks, production planning, biological
systems, ecosystems, nancial engineering, and modeling, analysis, and
control and optimization of lar- scale systems, under the in uence of
random environments. Displaying mixture distributions, switching di
usions may be described by the associated operators or by systems of
stochastic di erential eq- tions together with the probability
transition laws of the switching actions. We either have
Markov-modulated switching di usions or processes with continuous
state-dependent switching. The latter turns out to be much more
challenging to deal with. Viewing the hybrid di usions as a number of di
usions joined together by the switching process, they may be se- ingly
not much di erent from their di usion counterpart. Nevertheless, the
underlying problems become more di cult to handle, especially when the
switching processes depend on continuous states. The di culty is due to
the interaction of the discrete and continuous processes and the tangled
and hybrid information pattern.
