_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}+\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].\