} \nonumber&\mathbb E^{\P(a,\nu)}\Big[\underset{t_0\leq s\leq T}{\sup}M'_s\int_{t_0}^{t_0+\eta}\Big|f(s,B_s^{t_0,x_0},\varphi(s,B_s^{t_0,x_0}),D\varphi(s,B_s^{t_0,x_0}),\Kc\varphi(s,B_s^{t_0,x_0},\cdot),a,\nu)\\ \nonumber &\hspace{0.0em}-f(t_0,x_0,\varphi(t_0,x_0),D\varphi(t_0,x_0),\Kc\varphi(t_0,x_0,\cdot),a,\nu)\Big|ds\Big]\\ 