4 problems

Let $V(x_1,x_2)$ be the value function for two collaborating business lines, and write $x=x_1+x_2$ and $g(x)=V(x_1,x_2)$. The reduced HJB equation is … with $g(0)=0$. Barrier-strat…

Let $g$ be the reduced value function, let $a_2$ be the dividend coefficient for the second business line, and let $b^*$ be the barrier level. Smooth-pasting derivative conjecture.…

Let the controlled surplus or reserve process be driven by a general Lévy process, possibly with both upward and downward jumps, and consider the stochastic control problem describ…

Two-sided Parisian barrier conjecture. It is reasonably conjectured that it is optimal to reflect the process in a Parisian fashion from above at $\overline{b}$ and from below at…