Solving the optimal control of stochastic differential equations (SDEs) using the dynamic programming method requires writing the problem in terms of the so-called value function. This paper presents conditions to assure that the value function is convex away from the origin, a concept that allows the value function be non-convex in a region close to the origin. In contrast, for regions away from the origin, the value function remains convex under some mild conditions. Stochastic ordering is used to prove this result. A numerical example illustrates the potential benefits of our approach.
