On small-noise equations with degenerate limiting system arising from volatility models

The one-dimensional SDE with non Lipschitz diffusion coefficient dXt = b(Xt)dt + σXγ t dBt, X0 = x, γ <1 (1) 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 Xε:= ε1/(1−γ)X: while allowing to turn a space asymptotic problem into a small-ε problem, the process Xε satisfies a SDE inWentzell–Freidlin form (i.e. with driving noise εdB).We prove a pathwise large deviation principle for the process Xε as ε → 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 ε-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.