Smoothness and asymptotic estimates of densities for SDEs with locally smooth coefficients and applications to square root-type diffusions

TL;DR

This paper investigates the smoothness and asymptotic behavior of densities for solutions to SDEs with locally smooth, nondegenerate coefficients, extending existing results and applying to models like CIR and CEV in finance.

Contribution

It introduces a novel Fourier transform-based technique for density estimates of SDEs with locally smooth coefficients, applicable to non-Lipschitz models like CIR and CEV.

Findings

Existence of smooth densities for SDE solutions on open domains.

Upper bounds and tail estimates for densities of square root-type diffusions.

Behavior analysis of densities at the origin for models with singular points.

Abstract

We study smoothness of densities for the solutions of SDEs whose coefficients are smooth and nondegenerate only on an open domain $D$. We prove that a smooth density exists on $D$ and give upper bounds for this density. Under some additional conditions (mainly dealing with the growth of the coefficients and their derivatives), we formulate upper bounds that are suitable to obtain asymptotic estimates of the density for large values of the state variable ("tail" estimates). These results specify and extend some results by Kusuoka and Stroock [J. Fac. Sci. Univ. Tokyo Sect. IA Math. 32 (1985) 1--76], but our approach is substantially different and based on a technique to estimate the Fourier transform inspired from Fournier [Electron. J. Probab. 13 (2008) 135--156] and Bally [Integration by parts formula for locally smooth laws and applications to equations with jumps I (2007) The Royal Swedish Academy of Sciences]. This study is motivated by existing models for financial securities which rely on SDEs with non-Lipschitz coefficients. Indeed, we apply our results to a square root-type diffusion (CIR or CEV) with coefficients depending on the state variable, that is, a situation where standard techniques for density estimation based on Malliavin calculus do not apply. We establish the existence of a smooth density, for which we give exponential estimates and study the behavior at the origin (the singular point).