  We prove that any Brownian moving average
\[X_t=\int_{-\infty}^t\bigl(f(s-t)-f(s)\bigr) dB_s,\qquad t\ge0,\] satisfies
the conditional full support condition introduced by Guasoni, R\'{a}sonyi and
Schachermayer [Ann. Appl. Probab. 18 (2008) 491--520].
