Appl Math Optim 53:101–119 (2006)
2005 Springer Science+Business Media, Inc.
A System of Poisson Equations for a Nonconstant Varadhan
Functional on a Finite State Space
and Daniel Hern´andez-Hern´andez
Departamento de Estad´ıstica y C´alculo,
Universidad Aut´onoma Agraria Antonio Narro,
Buenavista, Saltillo, COAH 25315, M´exico
Centro de Investigaci´on en Matem´aticas,
Apartado Postal 402, Guanajuato,
GTO 36000, M´exico
Abstract. Given a discrete-time Markov chain with ﬁnite state space and a sta-
tionary transition matrix, a system of “local” Poisson equations characterizing the
(exponential) Varadhan’s functional J (·) is given. The main results, which are de-
rived for an arbitrary transition structure so that J (·) may be nonconstant, are as
follows: (i) Any solution to the local Poisson equations immediately renders Varad-
han’s functional, and (ii) a solution of the system always exist. The proof of this latter
result is constructive and suggests a method to solve the local Poisson equations.
Key Words. Local Poisson equations, Exponential grow rate, Closed and com-
municating sets, Risk-sensitive long-run average cost.
AMS Classiﬁcation. Primary 60J05, 60F10, Secondary 93C55.
This work concerns Varadhan’s functional J (·) associated to an arbitrary ﬁnite-state
Markov chain endowed with a cost structure, and the main objective of the paper is to
provide a characterization of J (·) in terms of a system of Poisson equations. To describe
This work was supported by the PSF Organization under Grant No. 2003-1, and by the Consejo Nacional
de Ciencia y Tecnolog´ıa under Grant 37643-E.