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We use commutator techniques and calculations in solvable Lie groups toinvestigate certain evolution Partial Differential Equations (PDEs for short)that arise in the study of stochastic volatility models for pricing contingentclaims on risky assets. In particular, by restricting to domains of boundedvolatility, we establish the existence of the semi-groups generated by thespatial part of the operators in these models, concentrating on those arisingin the so-called "SABR stochastic volatility model with mean reversion." Themain goal of this work is to approximate the solutions of the Cauchy problemfor the SABR PDE with mean reversion, a parabolic problem the generator ofwhich is denoted by $L$. The fundamental solution for this problem is not knownin closed form. We obtain an approximate solution by performing an expansion inthe so-called volvol or volatility of the volatility, which leads us to study adegenerate elliptic operator $L_0$, corresponding the the zero-volvol case ofthe SABR model with mean reversion, to which the classical results do notapply. However, using Lie algebra techniques we are able to derive an exactformula for the solution operator of the PDE $\partial_t u - L_0 u = 0$. Wethen compare the semi-group generated by $L$--the existence of which doesfollows from standard arguments--to that generated by $L_0$, thus establishinga perturbation result that is useful for numerical methods for the SABR PDEwith mean reversion. In the process, we are led to study semigroups arisingfrom both a strongly parabolic and a hyperbolic problem.

Heat Kernels, Solvable Lie Groups, and the Mean Reverting SABR Stochastic Volatility Model