\underset{(a,\nu)\in\mathbb S^{>0}_d\times\mathfrak V^m}{\sup}\mathcal L^{t_0,x_0}\varphi(t_0,a,\nu)\underset{\mathbb P\in\mathcal P^{t_0,\kappa}_h}{\sup}\E^\P\left[\underset{t_0\leq s\leq T}{\sup}M_s\right].\end{aligned}$$ 