The one-dimensional SDE with non Lipschitz diffusion coefficient dX_{t} = b(X_{t})dt + \sigma X_{t}^{\gamma} dB_{t}, \ X_{0}=x, \ \gamma<1 is widely studied in mathematical finance. Several works have proposed asymptotic analysis of densities and implied volatilities in models involving specific instances of this equation, based on a careful implementation of saddle-point methods and (essentially) the explicit knowledge of Fourier transforms. Recent research on tail asymptotics for heat kernels [J-D. Deuschel, P.~Friz, A.~Jacquier, and S.~Violante. Marginal density expansions for diffusions and stochastic volatility, part II: Applications. 2013, arxiv:1305.6765 suggests to work with the rescaled variable X^{\varepsilon}:=\varepsilon^{1/(1-\gamma)} X: while allowing to turn a spatial asymptotic problem into a  fixed terminal point, small-\varepsilon problem, the process X^{\varepsilon} satisfies a SDE in Wentzell--Freidlin form (i.e. with driving noise \varepsilon dB). We prove a pathwise large deviation principle for the process X^{\varepsilon} as \varepsilon \to 0. As it will become clear, the limiting ODE governing the large deviations admits infinitely many solutions, a non-standard situation in the Wentzell--Freidlin theory. The \varepsilon-scaling allows to derive exact log-asymptotics for path functionals of the process ; while on the one hand the resulting formulae are confirmed by the CIR-CEV benchmarks, on the other hand the large deviation approach (i) applies to equations with a more general drift term  (ii) potentially opens the way to heat kernel analysis for higher-dimensional diffusions involving such an SDE as a component.
The one-dimensional SDE with non Lipschitz diffusion coefficient dX_{t} = b(X_{t})dt + \sigma X_{t}^{\gamma} dB_{t}, \ X_{0}=x, \ \gamma<1 is widely studied in mathematical finance. Several works have proposed asymptotic analysis of densities and implied volatilities in models involving instances of this equation, based on a careful implementation of saddle-point methods and (essentially) the explicit knowledge of Fourier transforms. Recent research on tail asymptotics for heat kernels [J-D. Deuschel, P.~Friz, A.~Jacquier, and S.~Violante. Marginal density expansions for diffusions and stochastic volatility, part II: Applications. 2013, arxiv:1305.6765 suggests to work with the rescaled variable X^{\varepsilon}:=\varepsilon^{1/(1-\gamma)} X: while allowing to turn a space asymptotic problem into a small-\varepsilon problem with fixed terminal point, the process X^{\varepsilon} satisfies a SDE in Wentzell--Freidlin form (i.e. with driving noise \varepsilon dB). We prove a pathwise large deviation principle for the process X^{\varepsilon} as \varepsilon \to 0. As it will become clear, the limiting ODE governing the large deviations admits infinitely many solutions, a non-standard situation in the Wentzell--Freidlin theory. As for applications, the \varepsilon-scaling allows to derive exact log-asymptotics for path functionals of the process : while on the one hand the resulting formulae are confirmed by the CIR-CEV benchmarks, on the other hand the large deviation approach (i) applies to equations with a more general drift term and (ii) potentially opens the way to heat kernel analysis for higher-dimensional diffusions involving such an SDE as a component.
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