Hi,Is there analytical approximations to the pricing of vanilla European options on SABR extended with mean reversion speed ? By that I mean SABR where the vol process is exp(OU). Thanks!

If the correlation is zero, there will be a mixing solution. If the vol-of-vol is small, you could try to develop an expansion in powers of that, using the general approach in 'Option Valuation under Stochastic Volatility', Ch. 3. I know some small-T expansion with mean-reversion isdiscussed in Labordere's book, but I don't think it's for your process, but merely tacking on a mean-reverting drift [$]d \sigma = (a - b \sigma) \, dt + .. [$]on the ordinary SABR gbm vol process. But maybe that will suffice for you.

For the log-normal stochastic volatility model with the mean-reversion:d_sigma(t)=kappa*(theta-sigma(t))*dt+volvol*sigma(t)*dW(t)there exists a very accurate approximation for the moment generating function / characteristic function which can be then applied to value options. This method can be even considered as a precise solution based on comparison of option prices obtained using this method to option prices obtained by Monte Carlo simulations.The idea behind this method is that the moment generating function under the log-normal stochastic volatility model (and, in general, under the log-normal volatility model with polynomial drift and volatility functions) can be decomposed into a leading affine term with an explicit closed-form solution and a residual term whose estimate is very small and depends on the higher order moments of the volatility (not just on the volatility of volatility volvol).Details can be found in Log-Normal Stochastic Volatility Model: Affine Decomposition of Moment Generating Function and Pricing of Vanilla Options