Probabilistic interpretation of HJB equations by the representation theorem for generators of BSDEs

The purpose of this note is to propose a new approach for the probabilistic interpretation of Hamilton-Jacobi-Bellman equations associated with stochastic recursive optimal control problems, utilizing the representation theorem for generators of backward stochastic differential equations. The key idea of our approach for proving this interpretation consists of transmitting the signs between the solution and generator via the identity given by representation theorem. Compared with existing methods, our approach seems to be more applicable for general settings. This can also be regarded as a new application of such representation theorem.