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November 2006 Lower bounds for the density of locally elliptic Itô processes
Vlad Bally
Ann. Probab. 34(6): 2406-2440 (November 2006). DOI: 10.1214/009117906000000458


We give lower bounds for the density pT(x, y) of the law of Xt, the solution of dXt=σ(Xt) dBt+b(Xt) dt, X0=x, under the following local ellipticity hypothesis: there exists a deterministic differentiable curve xt, 0≤tT, such that x0=x, xT=y and σσ*(xt)>0, for all t∈[0, T]. The lower bound is expressed in terms of a distance related to the skeleton of the diffusion process. This distance appears when we optimize over all the curves which verify the above ellipticity assumption.

The arguments which lead to the above result work in a general context which includes a large class of Wiener functionals, for example, Itô processes. Our starting point is work of Kohatsu-Higa which presents a general framework including stochastic PDE’s.


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Vlad Bally. "Lower bounds for the density of locally elliptic Itô processes." Ann. Probab. 34 (6) 2406 - 2440, November 2006.


Published: November 2006
First available in Project Euclid: 13 February 2007

zbMATH: 1123.60037
MathSciNet: MR2294988
Digital Object Identifier: 10.1214/009117906000000458

Primary: 60J35
Secondary: 60H07 , 60H30 , 60J60

Keywords: Density of the low , Itô processes , lower bounds , Malliavin calculus

Rights: Copyright © 2006 Institute of Mathematical Statistics

Vol.34 • No. 6 • November 2006
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