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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)
##### Keywords:
Malliavin calculus; non-smooth drift; density function
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##### References:
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