Before revision:  in this case our approach still works if the drift of the volatility takes a specific functional form and there is no local volatility component, and our results include the SABR model for \beta=1, \rho \le \lm\ne \lm \to 0pt%DIFAUXCMD , and we verify our results numerically for the SABR model using Monte Carlo simulation and the exact closed-form solution given in Antonov\&Spector\mbox{%DIFAUXCMD AS12 }\hspace{0pt}%DIFAUXCMD for the case \rho=} 0.
After revision:  in this case our approach still works if the drift of the volatility takes a specific functional form and there is no local volatility component, and our results include the SABR model for \beta=1, \rho \le 0. We later augment the model with a single jump-to-default with intensity\lm\ne \lm \to 0pt%DIFAUXCMD , and we verify our results numerically for the SABR model using Monte Carlo simulation and the exact closed-form solution given in Antonov\&Spector\mbox{%DIFAUXCMD AS12 }\hspace{0pt}%DIFAUXCMD for the case \rho=} 0.