  We put forward a complete theory on moment explosion for fairly general
state-spaces. This includes a characterization of the validity of the affine
transform formula in terms of minimal solutions of a system of generalized
Riccati differential equations.
  Also, we characterize the class of positive semidefinite processes, and
provide existence of weak and strong solutions for Wishart SDEs. As an
application, we answer a conjecture of M.L. Eaton on the maximal parameter
domain of non-central Wishart distributions.
  The last chapter of this thesis comprises three individual works on affine
models, such as a characterization of the martingale property of exponentially
affine processes, an investigation of the jump-behaviour of processes on
positive semidefinite cones, and an existence result for transition densities
of multivariate affine jump-diffusions and their approximation theory in
weighted Hilbert spaces.
