Step in proof of Itô formula

Question

I am reading a book on stochastic processes. The author proved Itô formula for $f(t,w(t))$ where $w(t)$ is brownian motion with filtration $F_t$. Then he wants to prove Itô formula for $x(t)=a(t)+b(t)w(t)$ where $a(t)=a(0)$ and $b(t)=b(0)$ and $a(0)$, $b(0)$ are measurable with respect to $F_0$. The author considered a function $g(t,a_0,b_0,x)=f(t,a_0+b_0x)$, where $a_0$, $b_0$ are constants, then he writes that we can apply Itô formula for function $g$, and then he substitutes $a(t)$, $b(t)$ instead of $a_0$, $b_0$. How can we justify this substitution? Can we use a formal definition of Itô integral, using integrals for step functions, or we can try conditioning on $F_0$? The book is Lectures on the theory of stochastic processes by Skorokhod.

Answer 1

I believe you can condition on $F_0$. Under the regular conditional probability induced by $F_0$, $a_0$ and $b_0$ are deterministic, and of course $w$ is stil a Brownian motion. Then you can apply the usual Ito formula.