Stochastic integrals and conditional full support

We present conditions that imply the conditional full support (CFS) property, introduced by Guasoni, R\'asonyi, and Schachermayer [Ann. Appl. Probab., 18 (2008), pp. 491--520], for processes Z := H + K \cdot W, where W is a Brownian motion, H is a continuous process, and processes H and K are either progressive or independent of W. Moreover, in the latter case under an additional assumption that K is of finite variation, we present conditions under which Z has CFS also when W is replaced with a general continuous process with CFS. As applications of these results, we show that several stochastic volatility models and the solutions of certain stochastic differential equations have CFS.