We have done a paper on how to calibrate local vol when both volatility and rates are stochastics.
I'd be interested in any comments.
Lol.I just was a little confused by the 2.1st assumption. The risk neutral asset dynamics is usually has the risk free drift coefficient. Whether [$]\mu ( t , x ) [$] is the risk free rate. For more formal reason it might make sense introduce vector of economic variables on original probability space as it is common in general theory of the stoch processes. Then explain the necessity to make the change original probability measure to arrive at the risk neutral world. The mathematical paper will be looks much better.