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manuscript No. (will be inserted by the editor)
 

Summary: manuscript No.
(will be inserted by the editor)
Claudio Albanese Stephan Lawi
Laplace Transforms for Integrals of Markov Processes
February 6, 2005
Abstract. Laplace transforms for integrals of stochastic processes have been known in analyt-
ically 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.
1. Introduction
Let (Xt)t0 be a time-homogenous, real-valued Markov process on the filtered probability
space (, {Ft}t0, P) and consider the Laplace transform LT -t(Xt, ) defined as follows:
LT -t(Xt, ) = EP
e- T

  

Source: Albanese, Claudio - Department of Mathematics, King's College London

 

Collections: Mathematics