## The Annals of Applied Probability

- Ann. Appl. Probab.
- Volume 21, Number 4 (2011), 1282-1321.

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

#### 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).

#### Article information

**Source**

Ann. Appl. Probab., Volume 21, Number 4 (2011), 1282-1321.

**Dates**

First available in Project Euclid: 8 August 2011

**Permanent link to this document**

https://projecteuclid.org/euclid.aoap/1312818837

**Digital Object Identifier**

doi:10.1214/10-AAP717

**Mathematical Reviews number (MathSciNet)**

MR2857449

**Zentralblatt MATH identifier**

1246.60082

**Subjects**

Primary: 60J35: Transition functions, generators and resolvents [See also 47D03, 47D07]

Secondary: 60H10: Stochastic ordinary differential equations [See also 34F05] 60H07: Stochastic calculus of variations and the Malliavin calculus 60J60: Diffusion processes [See also 58J65] 60J70: Applications of Brownian motions and diffusion theory (population genetics, absorption problems, etc.) [See also 92Dxx]

**Keywords**

Smoothness of densities stochastic differential equations locally smooth coefficients tail estimates Malliavin calculus square-root process

#### Citation

De Marco, Stefano. Smoothness and asymptotic estimates of densities for SDEs with locally smooth coefficients and applications to square root-type diffusions. Ann. Appl. Probab. 21 (2011), no. 4, 1282--1321. doi:10.1214/10-AAP717. https://projecteuclid.org/euclid.aoap/1312818837