When I was learning the definition of viscosity solution, I had the following confusion：
The definition of viscosity solution in <Continuous-time stochastic control and optimization with financial applications> is as follows:
My question:
How could we know that the maximium of $w^*-\varphi$ is attainable? We know that $w^*$ is a upper-semicontinuous function, $\varphi\in C^2$, so that $w^*-\varphi$ can attain its maximum on any compact set. But $\mathcal{O}$ is just a open set but not compact set.
My guess:
Is it that "the maximium of $w^*-\varphi$" actually means "the local maximium of $w^*-\varphi$"? If it is the case, then we should have this requirement to $\mathcal{O}$:
"For any $x\in \mathcal{O}$, there exists a compact neighborhood $V_x$ of $x$", which means that, we need $\mathcal{O}$ is Locally Compact. Am I right?
It would be very grateful if you can provide some answers or comments. Thank you!