  0.  \[ass:gen\_0\] The following finiteness conditions are satisfied $$\operatorname*{ess\,sup}_{s\in [0,T]} \left\|\frac{\partial f}{\partial x}(s,x_s)\right\|_{\mathbb{D}^{0,0}} <\infty, \quad \operatorname*{ess\,sup}_{s\in [0,T]} \left\|\frac{\partial^0 f}{\partial x^0}(s,x_s)\right\|_{\mathbb{D}^{0,0}} <\infty, \quad \operatorname*{ess\,sup}_{s\in [0,T]} \left\|\frac{\partial^0 f}{\partial x^0}(s,x_s)\right\|_{