\E^\P\left[\int_{t_0}^{t_0+\eta}M_s\mathcal L^{t_0,x_0}\varphi(t_0,\widehat a^{t_0}_s,\nu^{\P,t_0}_s)ds\right].\\ \nonumber&\leq \E^\P\left[\int_{t_0}^{t_0+\eta}M_s\left(\mathcal L^{t_0,x_0}\varphi(s,\widehat a^{t_0}_s,\nu^{\P,t_0}_s)-\mathcal L^{t_0,x_0}\varphi(t_0,\widehat a^{t_0}_s,\nu^{\P,t_0}_s)\right)ds\right]\\ &\hspace{0.0em}+\underset{(a,\nu)\in\mathbb S^{>0}_d\times\mathfrak V^m}{\sup}\mathcal L^{t_0,x_0}\varphi(t_0,a,\nu)\