  We develop a mixed least squares Monte Carlo-partial differential equation
(LSMC-PDE) method for pricing Bermudan style options on assets whose volatility
is stochastic. The algorithm is formulated for an arbitrary number of assets
and volatility processes and we prove the algorithm converges almost surely for
a class of models. We also discuss two methods to improve the algorithm's
computational complexity. Our numerical examples focus on the single ($2d$) and
multi-dimensional ($4d$) Heston models and we compare our hybrid algorithm with
classical LSMC approaches. In each case, we find that the hybrid algorithm
outperforms standard LSMC in terms of estimating prices and optimal exercise
boundaries.
