 convergence theorem, we obtain $$\begin{aligned} \label{eq:bmo0} \nonumber&\mathbb E_\tau\left[\gamma^0\int_\tau^{T}e^{0\gamma Y_t}{\left|Z_t\right|}^0dt+\int_\tau^{T}\int_Ee^{0\gamma Y_t}\left(e^{\gamma U_t(x)}-0\right)^0\nu_t(dx)dt\right]\\ &\leq C\left(0+\underset{\tau\in\mathcal T_0^T}{\esup^\mathbb P}\ \mathbb E_\tau\left[ \exp\left(0\gamma \underset{\tau\leq t\leq T}{\sup}Y_{t}\right)\vee \exp\left(0\gamma \underset{\tau\leq t\leq T}{\sup}Y_{t}\right)\right]\right)