CREST：セミナー（Dr. Stefano De Marco）

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 (density 'tail' estimates). These results specify and extend some results by Kusuoka and Stroock ('85), but our approach is substantially diffeerent and based on a technique to estimate the Fourier transform inspired from Fournier ('08) and Bally ('07). This study is motivated by existing models for financial securities which rely on SDEs with non-Lipschitz coefficients. We indeed apply our results to a square root-type diffusion (CIR or CEV) with coefficients depending on the state variable, i.e. 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 behaviour at the origin (the singular point).