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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.

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