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Local Hölder continuity property of the densities of solutions of SDEs with singular coefficients. (English) Zbl 1294.60081
The authors consider a homogeneous Itō process \[ X_t= x+ \int^t_0 \sigma(X_s)\,dB_s+ \int^t_0 b(X_s)\,ds,\quad 0\leq t\leq T, \] assuming that:
– \(B\) is a real Brownian motion;
– \(\sigma\) and \(b\) are Borelian bounded on \(I:= ]y-r,y+r[\);
– \(\sigma\in C^\infty_b(I)\) and \(\sigma\geq \sigma_0> 0\) on \(I\); and
– \(b/\sigma\) is an \(\alpha\)-Hölderian on \(I\), for some \(0<\alpha<1\).
Then the authors prove that for any \(0<\gamma<\alpha\) and \(0\leq t\leq T\), \(X_t\) admits a \(\gamma\)-Hölderian density on \(]y-{r\over 6},y+{r\over 6}[\). They proceed by a Girsanov transform supporting the drift term, localization to \(I\), smoothing of the Girsanov exponential martingale, and Malliavin calculus.

MSC:
60H07 Stochastic calculus of variations and the Malliavin calculus
60H10 Stochastic ordinary differential equations (aspects of stochastic analysis)
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