6 problems

Stochastic control problems can be decomposed into an average of deterministic problems indexed by the realization of the underlying randomness. Lions' conjecture. These determinis…

In a Markovian time-inconsistent stochastic control problem, let $\nu^\star$ be an equilibrium and let $v$ be its value function. The associated system of PDEs is denoted by … , an…

Assume that $V$, $f$, and $g$ solve the extended HJB system and that the supremum in the $V$-equation is attained for every $(t,x)$. Let $\hat{\mathbf u}$ be the equilibrium law gi…

Assume that $V$, $f$, and $g$ solve the extended HJB system and that the supremum in the $V$-equation is attained for every $(t,x)$. Existence conjecture. There exists an equilibri…

Verification conjecture. $V$ satisfies the extended HJB system, and $\hat{\mathbf u}$ realizes the supremum in the equation.

Let $V$ denote the value function associated with conditional value-at-risk and its HJB equation, as in the preceding representation. Separation of variables refers to solving this…