Generalization of the Feynman-Kac formula for Markov processes

The main goal of this paper is twofold. First, we want to demonstrate how the forward and backward Feynman-Kac equations can be extended to account for the general form of Markov processes described by the differential Chapman-Kolmogorov equations. Second, we demonstrate specialization of these equations for different models of Markov processes, ranging from classical diffusion to more computationally challenging jump diffusion with drift, regime switching, and hybrid systems with resetting. To illustrate the power of the extended Feynman-Kac formalism, we consider different settings corresponding to various models of the stochastic process. We provide the closed-form solutions to the respective problems for an arbitrary initial (final) condition. For deterministic initial (final) conditions, the obtained results reduce to the known solutions to the respective special problems derived by different methods. We study the model with mean reversion and uncorrelated pre- and postjump states. This model is antipodal to both continuous diffusion and random walks. It is most appropriate when physical reality involves strong exogenous shocks with exponential relaxation to the mean value between these shocks, leading to a significant departure from conventional modeling assumptions. Finally, we demonstrate how naturally the generalized Feynman-Kac equations can be specialized to treat resetting (as a particular case of the uncorrelated states model) in stochastic hybrid systems.