On the optimal control of random walks with jumps and barrier

The modeling and optimal control of a class of random walks (RWs) is investigated in the framework of the Chapman-Kolmogorov (CK) and Fokker-Planck (FP) equations. This class of RWs includes jumps driven by a compound Poisson process and are subject to different barriers. A control mechanism is investigated that is included in the CK stochastic transition matrix and the purpose of the control is to track a desired discrete probability density function and attain a desired terminal density configuration. Existence and characterization of optimal controls are discussed. The proposed approach allows the derivation of a new FP model that accommodates the presence of the jumps and guarantees conservation of total probability in the case of reflecting barriers, which are modelled by appropriate operators. Results of numerical experiments are presented that successfully validate the proposed control framework.