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We consider a classical finite horizon optimal control problem forcontinuous-time pure jump Markov processes described by means of a ratetransition measure depending on a control parameter and controlled by afeedback law. For this class of problems the value function can often bedescribed as the unique solution to the corresponding Hamilton-Jacobi-Bellmanequation. We prove a probabilistic representation for the value function, knownas nonlinear Feynman-Kac formula. It relates the value function with a backwardstochastic differential equation (BSDE) driven by a random measure and with asign constraint on its martingale part. We also prove existence and uniquenessresults for this class of constrained BSDEs. The connection of the controlproblem with the constrained BSDE uses a control randomization method recentlydeveloped in the works of I. Kharroubi and H. Pham and their co-authors. Thisapproach also allows to prove that the value function of the originalnon-dominated control problem coincides with the value function of an auxiliarydominated control problem, expressed in terms of equivalent changes ofprobability measures.

Constrained BSDEs representation of the value function in optimal control of pure jump Markov processes