On Small-Noise Equations with Degenerate Limiting System Arising from Volatility Models

The one-dimensional SDE with non Lipschitz diffusion coefficient   is widely studied in mathematical finance. Several works have proposed asymptotic analysis of densities and implied volatilities in models involving instances of (1), 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 (Deuschel et al. Comm. in Pure and Applied Math., 67(1):40–82, 2014, [11]) 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, the process \(X^{\varepsilon }\) satisfies a SDE in Wentzell–Freidlin form (i.e. with driving noise \(\varepsilon \textit{dB}\)). We prove a pathwise large deviation principle for the process \(X^{\varepsilon }\) as \(\varepsilon \rightarrow 0\). As it will be seen, 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 leading order asymptotics for path functionals: 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 (1) as a component.