 $$\begin{aligned} &\int_{t_0}^{t_0+\eta}\left(\rho(s-t_0)+\mathbb E^{\P(a,\nu)}\left[\underset{t_0\leq s\leq T}{\sup}M'_s\rho({\left|B_s^{t_0}\right|})\right]+{\left|s-t_0\right|}+\mathbb E^{\P(a,\nu)}\left[\underset{t_0\leq s\leq T}{\sup}M'_s{\left|B_s^{t_0}\right|}\right]\right)ds\\ &\leq C\left(\eta\rho(\eta)+\eta^0+\eta^{0/0}+\