Estimation de la densité de la solution de l'équation de Zakaï robuste

We study the behaviour in small time of the density of the robust Zakaï equation under the weak Hörmander's hypothesis. We avoid large deviations phenomena by supposing that the drift at the departure is equal to zero. We get an asymptotic expansion over the diagonal of the density where the observation is involved. In the case where all the terms of that asymptotic expansion are equal to zero, a pathology which is related to the Bismut's condition, we give an estimation of the decay by using a probabilistic analogous of the Gevrey methods.