  Duality relations between continuous-state and discrete-state stochastic
processes with continuous-time have already been studied and used in various
research fields. We propose extended duality relations, which enable us to
derive discrete-state stochastic processes from arbitrary diffusion-type
partial differential equations. The derivation is based on the Doi-Peliti
formalism and the algebraic probability theory, and it will be clarified that
additional states for the discrete-state stochastic processes must be
considered in some cases.
