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

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

2011-08-23

arxiv.org/abs/1108.4558v1

Annals of Applied Probability 2011, Vol. 21, No. 4, 1282-1321

Mathematics

Probability

Published in at http://dx.doi.org/10.1214/10-AAP717 the Annals of Applied Probability (http://www.imstat.org/aap/) by the Inst

Scientific paper

10.1214/10-AAP717

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).

Affiliated with

Also associated with

No associations

Rating

Smoothness and asymptotic estimates of densities for SDEs with locally smooth coefficients and applications to square root-type diffusions does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.

If you have personal experience with Smoothness and asymptotic estimates of densities for SDEs with locally smooth coefficients and applications to square root-type diffusions, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Smoothness and asymptotic estimates of densities for SDEs with locally smooth coefficients and applications to square root-type diffusions will most certainly appreciate the feedback.

Rate now

Comments { 0 }

Profile ID: LFWR-SCP-O-67214