Geodesic
Densities for rough differential equations under Hörmander’s condition
Thomas Cass1 ; Peter Friz2
1Mathematical Institute<br/>24-29 St. Giles’<br/>Oxford OX1 3LB<br/>England
2Weierstrasse Institute for Applied Analysis and Stochastics<br/>Morhenstrasse 39<br/>10117 Berlin<br/>Germany<br/>and <br/>TU Berlin<br/>Fakultät II<br/>Institut für Mathematik, MA 7-2<br/>Strasse des 17. Juni 136<br/>D-10623 Berlin<br/>Germany
Annals of mathematics, Tome 171 (2010) no. 3, pp. 2115-2141
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We consider stochastic differential equations $dY=V\left( Y\right) dX$ 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 $t\in (0,T]$ provided (i) the vector fields $V=\left( V_{1},\dots,V_{d}\right) $ 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.

DOI : 10.4007/annals.2010.171.2115

Thomas Cass  1   ; Peter Friz  2

1 Mathematical Institute<br/>24-29 St. Giles’<br/>Oxford OX1 3LB<br/>England
2 Weierstrasse Institute for Applied Analysis and Stochastics<br/>Morhenstrasse 39<br/>10117 Berlin<br/>Germany<br/>and <br/>TU Berlin<br/>Fakultät II<br/>Institut für Mathematik, MA 7-2<br/>Strasse des 17. Juni 136<br/>D-10623 Berlin<br/>Germany 
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     author = {Thomas Cass and Peter Friz},
     title = {Densities for rough differential equations  under {H\"ormander{\textquoteright}s} condition},
     journal = {Annals of mathematics},
     pages = {2115--2141},
     year = {2010},
     volume = {171},
     number = {3},
     doi = {10.4007/annals.2010.171.2115},
     mrnumber = {2680405},
     zbl = {1205.60105},
     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.4007/annals.2010.171.2115/}
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Thomas Cass; Peter Friz. Densities for rough differential equations  under Hörmander’s condition. Annals of mathematics, Tome 171 (2010) no. 3, pp. 2115-2141. doi: 10.4007/annals.2010.171.2115

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