I am a fresh PhD student in numerical analysis. Recently I am considering finite difference methods and their error analysis for solving HJB equation of the following form: $$v_t=g(a(x)v_x),\quad x\in \mathbb{R},$$ where $a$ is a given function on $\mathbb{R}$ (with possible regularity assumptions) and $g:\mathbb{R}\to \mathbb{R}$ is Lipschitz and monotone.
I will also appreciate if you could suggest me some books which give a systematic introduction to FDM for solving HJB equation? I just want to learn the common techniques to analyze the stability of such numerical methods under different assumptions for $g$. I know there are a lot of papers on this subject, but I hope to start with some books.