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Media type:
E-Article
Title:
Densities for rough differential equations under Hörmander's condition
Contributor:
Cass, Thomas;
Friz, Peter
imprint:
Dept. of Mathematics, Princeton University, 2010
Published in:Annals of Mathematics
Language:
English
ISSN:
0003-486X
Origination:
Footnote:
Description:
<p>We consider stochastic differential equations dY = V (Y) dX driven by a multidimensional Gaussian process X in the rough path sense [T. Lyons, Rev. Mat. Iberoamericana 14, (1998), 215—310]. Using Malliavin Calculus we show that Y t admits a density for t ∈ (0,T] provided (i) the vector fields V = (V₁,…,V d ) satisfy Hörmander's condition and (ii) the Gaussian driving signal X satisfies certain conditions. Examples of driving signals include fractional Brownian motion with Hurst parameter H > 1/4, the Brownian bridge returning to zero after time T and the Ornstein-Uhlenbeck process.</p>