  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.
