Before revision:  The application to stochastic volatility models include the small-time and, for certain models, the large-strike asymptotics of the Gyongy-Dupire's local volatility function , the final product being asymptotic formulae that can (i) motivate parameterizations of the local volatility surface and (ii) be used to extrapolate local volatilities in a given model.
After revision:  The application to stochastic volatility models include the small-time and, for certain models, the large-strike asymptotics of the Gyongy-Dupire's local volatility function . The final product are asymptotic formulae that can (i) motivate parameterizations of the local volatility surface and (ii) be used to extrapolate local volatilities in a given model.