Pontryagin Maximum Principle for reflected BSDEs

We study a stochastic optimal control problem where the cost is represented by the solution to a reflected backward stochastic differential equation (RBSDE) constrained by a lower obstacle. The presence of the reflection term renders the system inherently nonsmooth, preventing the use of classical variational analysis. To overcome this, we apply a penalization approach, approximating the RBSDE via a family of standard BSDEs with penalization terms. Using spike variation and duality methods, we derive a Pontryagin-type maximum principle for the penalized problems and then pass to the limit. In the limit, we obtain a new adjoint equation involving a singular measure term, supported on the contact set where the reflection constraint is active. We interpret this measure as a stochastic analogue of a Lagrange multiplier, providing a rigorous Hamiltonian formulation of the maximum principle under state inequality constraints. We also present an illustrative example along with its numerical study.