A transitivity property of Ocone martingales (Q2105373)

For an Ocone martingale \(\theta\), a sufficient condition is given such that \(M_t= \int_{0}^{t} \alpha_s d \theta_s\) is an Ocone martingale, too. A partial converse theorem is obtained for a special class of martingales, for \(\theta_t= \int_{0}^{t}\mathrm{sign}(\alpha_s) d B_s\), where \((B_t)_{t \geq 0}\) is a Brownian motion. Some examples of Ocone martingales are also given.