  Laplace transforms for integrals of stochastic processes have been known in
analytically closed form for just a handful of Markov processes: namely, the
Ornstein-Uhlenbeck, the Cox-Ingerssol-Ross (CIR) process and the exponential of
Brownian motion. In virtue of their analytical tractability, these processes
are extensively used in modelling applications. In this paper, we construct
broad extensions of these process classes. We show how the known models fit
into a classification scheme for diffusion processes for which Laplace
transforms for integrals of the diffusion processes and transitional
probability densities can be evaluated as integrals of hypergeometric functions
against the spectral measure for certain self-adjoint operators. We also extend
this scheme to a class of finite-state Markov processes related to
hypergeometric polynomials in the discrete series of the Askey classification
tree.
