Does the probability distribution associated with this pdf have a name? The pdf is $$f(x)=\frac{ab^a}{(x+b)^{a+1}}$$ for $x\geq0$ and some parameters $a,b$. I've come across it a couple of times in study material for the actuarial exams.

This is just a power-law or Pareto distribution.  The Wikipedia page says it is sometimes called a "Lomax distribution" when it is shifted to start at zero like in your case, but I wouldn't call that a significant distinction.
