  A new derivation method of duality relations in stochastic processes is
proposed. The current focus is on the duality between stochastic differential
equations and birth-death processes. Although previous derivation methods have
been based on the viewpoint of time-evolution operators, the current derivation
is based on basis expansions. In addition, only the tool needed for the
derivation is the integration by parts, which is rather simple and
understandable. The viewpoint of basis expansions enables us to obtain various
dual stochastic processes. As a demonstration, dual processes based on
Taylor-type and Hermite polynomials are derived.
