$\sup_{\alpha}E_P[X_{\alpha} (T)]=Kf(\alpha^*) ?$

Question

Let $X_{\alpha} (t)$ be a Ito process and let $Q_{\alpha}$ and $P$ be equivalent probability measures on $\mathcal{F}_T$ and say we have that $$E_P[X_{\alpha} (T)]=KE_{Q_{\alpha}}[f(\alpha)]$$ for some smooth, positive function $f$ and $\alpha=\alpha (t)$ is a stochastic process. Suppose there exists a non-random $\alpha^*$ such that $f(\alpha)\leq f(\alpha^*)$, for all $\alpha$. Can we conclude that $$\sup_{\alpha}E_P[X_{\alpha} (T)]=Kf(\alpha^*) ?$$