Let [d S_{t}=S_{t}left(b d t+sigma_{t} d B_{t} ight)] where (left(sigma_{t}, t geq 0 ight)) is an adapted

Let

\[d S_{t}=S_{t}\left(b d t+\sigma_{t} d B_{t}\right)\]

where \(\left(\sigma_{t}, t \geq 0\right)\) is an adapted process such that for any \(t, 0

\[\forall t, \mathcal{B S}\left(S_{t}, a, t\right) \leq \mathbb{E}_{\mathbb{Q}}\left(e^{-r(T-t)}\left(S_{T}-K\right)^{+} \mid \mathcal{F}_{t}\right) \leq \mathcal{B S}\left(S_{t}, b, t\right)\]