Quarterly of Applied Mathematics

Quarterly of Applied Mathematics

Online ISSN 1552-4485; Print ISSN 0033-569X



Moment growth bounds on continuous time Markov processes on non-negative integer lattices

Author: Muruhan Rathinam
Journal: Quart. Appl. Math. 73 (2015), 347-364
MSC (2010): Primary 60J27
DOI: https://doi.org/10.1090/S0033-569X-2015-01372-7
Published electronically: March 17, 2015
MathSciNet review: 3357498
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Abstract: We consider time homogeneous Markov processes in continuous time with state space $ \mathbb{Z}_{+}^{N}$ and provide two sufficient conditions and one necessary condition for the existence of moments $ E(\Vert X(t)\Vert^r)$ of all orders $ r \in \mathbb{N}$ for all $ t \geq 0$. The sufficient conditions also guarantee an exponential in time growth bound for the moments. The class of processes studied has finitely many state independent jumpsize vectors $ \nu _1,\dots ,\nu _M$. This class of processes arises naturally in many applications such as stochastic models of chemical kinetics, population dynamics and epidemiology for example. We also provide a necessary and sufficient condition for stoichiometric boundedness of species in terms of $ \nu _j$.

References [Enhancements On Off] (What's this?)

  • [1] Pierre Brémaud, Markov chains, Texts in Applied Mathematics, vol. 31, Springer-Verlag, New York, 1999. Gibbs fields, Monte Carlo simulation, and queues. MR 1689633
  • [2] A. Gupta, C. Briat, and M. Khammash, A scalable computational framework for establishing long-term behavior of stochastic reaction networks, PLoS Comput. Biol., 10(6): e1003669, 2014.
  • [3] Jim G. Dai and Sean P. Meyn, Stability and convergence of moments for multiclass queueing networks via fluid limit models, IEEE Trans. Automat. Control 40 (1995), no. 11, 1889–1904. MR 1358006, https://doi.org/10.1109/9.471210
  • [4] Stewart N. Ethier and Thomas G. Kurtz, Markov processes, Wiley Series in Probability and Mathematical Statistics: Probability and Mathematical Statistics, John Wiley & Sons, Inc., New York, 1986. Characterization and convergence. MR 838085
  • [5] Monique Florenzano and Cuong Le Van, Finite dimensional convexity and optimization, Studies in Economic Theory, vol. 13, Springer-Verlag, Berlin, 2001. In cooperation with Pascal Gourdel. MR 1878374
  • [6] Daniel T. Gillespie, Exact stochastic simulation of coupled chemical reactions, Journal of Physical Chemistry 81 (1977), no. 25, 2340-2361.
  • [7] Desmond J. Higham, Xuerong Mao, and Andrew M. Stuart, Strong convergence of Euler-type methods for nonlinear stochastic differential equations, SIAM J. Numer. Anal. 40 (2002), no. 3, 1041–1063. MR 1949404, https://doi.org/10.1137/S0036142901389530
  • [8] Xuerong Mao, Stochastic differential equations and their applications, Horwood Publishing Series in Mathematics & Applications, Horwood Publishing Limited, Chichester, 1997. MR 1475218
  • [9] N. G. van Kampen, Stochastic processes in physics and chemistry, Lecture Notes in Mathematics, vol. 888, North-Holland Publishing Co., Amsterdam-New York, 1981. MR 648937
  • [10] Robert J. Vanderbei, Linear programming, 2nd ed., International Series in Operations Research & Management Science, vol. 37, Kluwer Academic Publishers, Boston, MA, 2001. Foundations and extensions. MR 1845638
  • [11] David Williams, Probability with martingales, Cambridge Mathematical Textbooks, Cambridge University Press, Cambridge, 1991. MR 1155402

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Additional Information

Muruhan Rathinam
Affiliation: University of Maryland Baltimore County, Department of Mathematics and Statistics, Baltimore, Maryland 21250
Email: muruhan@umbc.edu

DOI: https://doi.org/10.1090/S0033-569X-2015-01372-7
Received by editor(s): May 28, 2013
Published electronically: March 17, 2015
Additional Notes: The research of this author was supported by grant NSF DMS-0610013
Article copyright: © Copyright 2015 Brown University
The copyright for this article reverts to public domain 28 years after publication.

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