Laplace Transforms for Integrals of Markov Processes

TL;DR

This paper extends the class of Markov processes for which Laplace transforms of integrals can be analytically computed, using spectral methods and hypergeometric functions, broadening their applicability in stochastic modeling.

Contribution

It introduces a classification scheme for diffusion and finite-state Markov processes enabling explicit Laplace transform calculations using hypergeometric functions.

Findings

Extended the class of processes with known Laplace transforms

Unified diffusion and finite-state Markov process analysis

Provided spectral integral representations for these transforms

Abstract

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.