On Differentiability of Controlled Stochastic Differential Equations with Reflection

We prove the differentiability of solutions to one-dimensional reflected stochastic differential equations with respect to controls in drift coefficients using probabilistic method. Based on excursion theory, the derivative of the solutions is first identified on every excursion interval, then the right-continuous modification of the derivative process is characterized by a class of stochastic differential equations with jumps. This result is also verified by penalty method. A formula of Bismut type and a necessary condition for a class of optimal control problems are presented as well.