I have a question concerning the generator of a backwards SDE. Suppose that $f(\omega,t,y,z):\Omega\times\mathbb{R}^+\times\mathbb{R}\times\mathbb{R}^d\rightarrow\mathbb{R}$ is almost surely jointly continuous in $(t,y,z)$. Suppose further that $f$ has suitable growth conditions (e.g. linear in $y$ and quadratic in $z$.
In the Kobylanski 2000 paper, it was asserted that there is a sequence $f^{(n)}$ of $C^\infty$ functions such that:
$$f+\frac{1}{2^{n+1}}\leq f^{(n)} \leq f+\frac{1}{2^n}$$
I'm at a bit of a loss as to how to prove the existence of such a sequence. In the said paper, it was said that the existence of this sequence is given by a standard argument of regularisation, but I don't quite see how. Any help or reference would be greatly appreciated. Thanks!