We provide a new proof for regularity of affine processes on general state
spaces by methods from the theory of Markovian semimartingales. On the way to
this result we also show that the definition of an affine process, namely as
stochastically continuous time-homogeneous Markov process with exponential
affine Fourier-Laplace transform, already implies the existence of a c\`adl\`ag
version. This was one of the last open issues in the fundaments of affine
processes.

We provide a general and flexible approach to LIBOR modeling based on the
class of affine factor processes. Our approach respects the basic economic
requirement that LIBOR rates are non-negative, and the basic requirement from
mathematical finance that LIBOR rates are analytically tractable martingales
with respect to their own forward measure. Additionally, and most importantly,
our approach also leads to analytically tractable expressions of multi-LIBOR
payoffs. This approach unifies therefore the advantages of well-known forward
price models with those of classical LIBOR rate models.

We review Fujiwara's scheme, a sixth order weak approximation scheme for the
numerical approximation of SDEs, and embed it into a general method to
construct weak approximation schemes of order $ 2m $ for $ m \in \mathbf{N} $.
Those schemes cannot be seen as cubature schemes, but rather as universal ways
how to extrapolate from a lower order weak approximation scheme, namely the
Ninomiya-Victoir scheme, for higher orders.

We construct default-free interest rate models in the spirit of the
well-known Markov funcional models: our focus is analytic tractability of the
models and generality of the approach. We work in the setting of state price
densities and construct models by means of the so called propagation property.
The propagation property can be found implicitly in all of the popular state
price density approaches, in particular heat kernels share the propagation
property (wherefrom we deduced the name of the approach). As a related matter,
an interesting property of heat kernels is presented, too.

This paper provides the mathematical foundation for stochastically continuous
affine processes on the cone of positive semidefinite symmetric matrices. These
matrix-valued affine processes have arisen from a large and growing range of
useful applications in finance, including multi-asset option pricing with
stochastic volatility and correlation structures, and fixed-income models with
stochastically correlated risk factors and default intensities.