  We use commutator techniques and calculations in solvable Lie groups to
investigate certain evolution Partial Differential Equations (PDEs for short)
that arise in the study of stochastic volatility models for pricing contingent
claims on risky assets. In particular, by restricting to domains of bounded
volatility, we establish the existence of the semi-groups generated by the
spatial part of the operators in these models, concentrating on those arising
in the so-called "SABR stochastic volatility model with mean reversion." The
main goal of this work is to approximate the solutions of the Cauchy problem
for the SABR PDE with mean reversion, a parabolic problem the generator of
which is denoted by $L$. The fundamental solution for this problem is not known
in closed form. We obtain an approximate solution by performing an expansion in
the so-called volvol or volatility of the volatility, which leads us to study a
degenerate elliptic operator $L_0$, corresponding the the zero-volvol case of
the SABR model with mean reversion, to which the classical results do not
apply. However, using Lie algebra techniques we are able to derive an exact
formula for the solution operator of the PDE $\partial_t u - L_0 u = 0$. We
then compare the semi-group generated by $L$--the existence of which does
follows from standard arguments--to that generated by $L_0$, thus establishing
a perturbation result that is useful for numerical methods for the SABR PDE
with mean reversion. In the process, we are led to study semigroups arising
from both a strongly parabolic and a hyperbolic problem.
