I am reading a book by Cartea & Jaimungal and on page 145 they claim the following:
We want to solve the following PDE: for $t \in [0,T]$ $$0=(\partial _t+\frac{1}{2}\sigma^2\partial_{ss})H-\phi q^2+\frac{1}{4K}\frac{\Big[(s\partial_x-b\partial_s-\partial_q)H\Big]^2}{\partial_xH}$$ with final condition $$H(T,x,s,q)=x+sq-\alpha q^2.$$
We make the ansatz for the solution to a PDE: $$H(t,x,s,q)=x+sq+h(t,s,q)$$$$ h(T,s,q)=-\alpha q^2.$$
Upon substituting this ansatz into the PDE, we get
$$0=(\partial _t+\frac{1}{2}\sigma^2\partial_{ss})h-\phi q^2+\frac{1}{4K}\Big[b(q+\partial_s h)+\partial_q h\Big]^2.$$
So far so good. But now they claim that since the PDE does not depend on $s$ and that the final condition is independent of $s$, it is enough to conclude that $\partial_sh(t,s,q)=0$ and so we can simplify the ansatz even further to $$H(t,x,s,q)=x+sq+g(t,q),$$ where $h(t,s,q)=g(t,q).$
Is there any intuition behind this? I cannot understand how the above two things mean that $h$ will be independent of the variable $s$.