I have some trouble understanding stochastic HJB equations. There are basically two forms of this equation that I have encountered in books, lecture notes etc... (one-dimensional case)
1) $rv(x)=\pi (x)+\mu v'(x)+\frac{1}{2} \sigma^2v''(x)$
where v is the value function, r the discount rate and $\pi$ is the current payoff.
2) $0=sup_{u∈U}[f^{v}(t,x)+\frac{∂v}{∂s}(t,x)+\mu(t,x, u)\frac{∂v}{∂x}+\frac{1}{2}σ²(t,x, u)\frac{∂²v}{∂x²}]$ for all $t,x$
I would guess the two versions are equivalent, but I honestly don't understand how so. I'm especially confused by the term $\frac{∂v}{∂s}(t,x)$ that shows up in the second equation. Is $-\frac{∂v}{∂s}(t,x)$ somehow the same as $rv(x)$?
Thanks a lot!!