Large deviations for excursions of non-homogeneous Markov processes

Anatoli Mogulskii (Sobolev Institute of Mathematics of Siberian Branch of Russian Academy of Sciences)
Eugene Pechersky (Institute for Information Transmission Problems of Russian Academy of Sciences)
Anatoli Yambartsev (University of São Paulo)


In this paper, the large deviations at the trajectory level for ergodic Markov processes are studied. These processes take values in the non-negative quadrant of the two-dimensional lattice and are concentrated on step-wise functions. The rates of jumps towards the axes (downward jumps) depend on the position of the process - the higher the position, the greater the rate. The rates of jumps going in the same direction as the axes (upward jumps) are constants. Therefore the processes are ergodic. The large deviations are studied under equal scalings of both space and time. The scaled versions of the processes converge to 0. The main result is that the probabilities of excursions far from 0 tend to 0 exponentially fast with an exponent proportional to the square of the scaling parameter. The proportionality coefficient is an integral of a linear combination of path components. A rate function of the large deviation principle is calculated for continuous functions only.

Full Text: Download PDF | View PDF online (requires PDF plugin)

Pages: 1-8

Publication Date: June 22, 2014

DOI: 10.1214/ECP.v19-3289


  • Billingsley, Patrick. Convergence of probability measures. Second edition. Wiley Series in Probability and Statistics: Probability and Statistics. A Wiley-Interscience Publication. John Wiley & Sons, Inc., New York, 1999. x+277 pp. ISBN: 0-471-19745-9 MR1700749
  • Borovkov, A. A.; Mogulʹskiĭ, A. A. Inequalities and principles of large deviations for the trajectories of processes with independent increments. (Russian) Sibirsk. Mat. Zh. 54 (2013), no. 2, 286--297; translation in Sib. Math. J. 54 (2013), no. 2, 217--226 MR3088596
  • Dembo, Amir; Zeitouni, Ofer. Large deviations techniques and applications. Second edition. Applications of Mathematics (New York), 38. Springer-Verlag, New York, 1998. xvi+396 pp. ISBN: 0-387-98406-2 MR1619036
  • Deuschel, Jean-Dominique; Stroock, Daniel W. Large deviations. Pure and Applied Mathematics, 137. Academic Press, Inc., Boston, MA, 1989. xiv+307 pp. ISBN: 0-12-213150-9 MR0997938
  • den Hollander, Frank. Large deviations. Fields Institute Monographs, 14. American Mathematical Society, Providence, RI, 2000. x+143 pp. ISBN: 0-8218-1989-5 MR1739680
  • Olivieri, Enzo; Vares, Maria Eulalia. Large deviations and metastability. Encyclopedia of Mathematics and its Applications, 100. Cambridge University Press, Cambridge, 2005. xvi+512 pp. ISBN: 0-521-59163-5 MR2123364
  • Puhalskii, Anatolii. Large deviations and idempotent probability. Chapman & Hall/CRC Monographs and Surveys in Pure and Applied Mathematics, 119. Chapman & Hall/CRC, Boca Raton, FL, 2001. xiv+500 pp. ISBN: 1-58488-198-4 MR1851048
  • Varadhan, S. R. S. Large deviations and applications. CBMS-NSF Regional Conference Series in Applied Mathematics, 46. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1984. v+75 pp. ISBN: 0-89871-189-4 MR0758258
  • Vvedenskaya, N.; Suhov, Y.; Belitsky, V. A non-linear model of trading mechanism on a financial market. Markov Process. Related Fields 19 (2013), no. 1, 83--98. MR3088424

Creative Commons License
This work is licensed under a Creative Commons Attribution 3.0 License.