Joining the SABR and Libor models together

The SABR model is a stochastic volatility dynamics for a single asset under its natural probability measure. However, when pricing general term structure payouts, we need to model the joint evolution of relevant rates, as in the Libor market model (LMM) of Brace, Gatarek & Musiela (1997). Moreover, if the stochastic volatility factors are correlated with the term structure of rates (as in Hagan et al, 2002, the underlying is correlated to its stochastic volatility), no-arbitrage constraints indi