The book constitutes an introduction to stochastic calculus, stochastic
differential equations and related topics such as Malliavin calculus. On
the other hand it focuses on the techniques of stochastic integration
and calculus via regularization initiated by the authors. The
definitions relies on a smoothing procedure of the integrator process,
they generalize the usual Itô and Stratonovich integrals for Brownian
motion but the integrator could also not be a semimartingale and the
integrand is allowed to be anticipating. The resulting calculus requires
a simple formalism: nevertheless it entails pathwise techniques even
though it takes into account randomness. It allows connecting different
types of pathwise and non pathwise integrals such as Young, fractional,
Skorohod integrals, enlargement of filtration and rough paths. The
covariation, but also high order variations, play a fundamental role in
the calculus via regularization, which can also be applied for irregular
integrators. A large class of Gaussian processes, various
generalizations of semimartingales such that Dirichlet and weak
Dirichlet processes are revisited. Stochastic calculus via
regularization has been successfully used in applications, for instance
in robust finance and on modeling vortex filaments in turbulence. The
book is addressed to PhD students and researchers in stochastic analysis
and applications to various fields.
