 $$\begin{aligned} &e^{\alpha t}{\left|y_t^{\mathbb P,t,\omega}-y_t^{\mathbb P,t,\omega'}\right|}^0+\int_t^T\scriptstyle e^{\alpha s}\left({\left|(\widehat a^t_s)^{0/0}(z_s^{\mathbb P,t,\omega}-z_s^{\mathbb P,t,\omega'})\right|}^0+\int_{E}\scriptstyle e^{\alpha s}(u_s^{\mathbb P,t,\omega}-u_s^{\mathbb P,t,\omega'})^0(x)\nu^{t,\mathbb P}_s(dx)\right)ds\\ &\leq e^{\alpha T}{\left|\xi^{t,\omega}-\xi^{t,\omega'}\right|}^0+0C\int_t^Te^{\alpha s}{\left|y_s^{\mathbb P,t,\omega}-y_s^{\mathbb P,t,\omega'}\right|}^0ds\\ &\hspace{0.0em}