Doob-Meyer decomposition for inequalities

Question

Consider a family of probability measures $\mathcal{P}$ such that the processes \begin{equation} X_t:=ess\inf_{\mathbb{P}\in\mathcal{P}}\mathbb{E}^{\mathbb{P}}[ B|\mathcal{F}_t], \end{equation} is a $\mathbb{P}$-submartingales for any $\mathcal{P}$, nonnegative r.v. $B$, and the Doob-Meyer decomposition \begin{equation} X_t=M_t(B)+A_s(B):=\inf_{\mathbb{P}\in\mathcal{P}}\mathbb{E}^{\mathbb{P}}[B]+\int_0^t\psi_s(B)dW_s^{\nu}+A_s(B): \end{equation} where the increasing process $A$ does NOT depend of $\nu$. My question is if the next reasoning is correct: Define $C=B+1$, then \begin{align} Y_t:&=ess\inf_{\mathbb{P}\in\mathcal{P}}\mathbb{E}^{\mathbb{P}}[ C|\mathcal{F}_t]\\ &= ess\inf_{\mathbb{P}\in\mathcal{P}}\mathbb{E}^{\mathbb{P}}[ B|\mathcal{F}_t]+1\\ &=M_t(B)+A_(B)+1 \end{align} In the other hand $Y_t=M_t(C)+A(C)_t$. By the uniqueness of the Doob-Meyer decomposition I conclude that $M(C)=M(B)+1$ and $A(B)=A(C)$. Is this true? Or there is some trouble that I am not taking into account?. Any help or advicce is welcome. Thank you!