A Generalization of the Submartingale Property: Maximal Inequality and Applications to Various Stochastic Processes

  • János EngländerEmail author


We generalize the notion of the submartingale property and Doob’s inequality. Furthermore, we show how the latter leads to new inequalities for several stochastic processes: certain time series, Lévy processes, random walks, processes with independent increments, branching processes and continuous state branching processes, branching diffusions and superdiffusions, as well as some Markov processes, including geometric Brownian motion.


a-achieving process Doob’s inequality Maximal inequality Time series Lévy process Processes with independent increments Submartingale Random walk Branching process Branching diffusion Superprocess Continuous state branching process Markov process Geometric Brownian motion Approximate convexity 

Mathematics Subject Classification (2010)

60E15 60G45 60G48 60G51 60J80 



The author is grateful to an anonymous referee for his/her close reading of the manuscript and for pointing out some glitches.


  1. 1.
    Engländer, J.: Spatial Branching in Random Environments and with Interaction. Advanced Series on Statistical Science and Applied Probability, vol. 20. World Scientific, Singapore (2015)zbMATHGoogle Scholar
  2. 2.
    Engländer, J., Ren, Y.-X., Song, R.: Weak extinction versus global exponential growth of total mass for superdiffusions. Ann. Inst. Henri Poincaré Probab. Stat. 52(1), 448–482 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Graversen, S.E., Peskir, G.: Optimal stopping and maximal inequalities for geometric Brownian motion. J. Appl. Probab. 35(4), 856–872 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Hyers, D.H., Ulam, S.M.: Approximately convex functions. Proc. Am. Math. Soc. 3, 821–828 (1952)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Kingman, J.F.C.: Ergodic properties of continuous-time Markov processes and their discrete skeletons. Proc. Lond. Math. Soc. (3) 13, 593–604 (1963)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Kyprianou, A.E.: Fluctuations of Lévy Processes with Applications. Introductory Lectures. Universitext, 2nd edn. Springer, Heidelberg (2014)CrossRefzbMATHGoogle Scholar
  7. 7.
    Páles, Zs: On approximately convex functions. Proc. Am. Math. Soc. 131(1), 243–252 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Revuz, D., Yor, M.: Continuous Martingales and Brownian Motion. Grundlehren der Mathematischen Wissenschaften, vol. 293, 3rd edn. Springer, Berlin (1999)CrossRefzbMATHGoogle Scholar
  9. 9.
    Stroock, D.W.: Probability Theory. An Analytic View, 2nd edn. Cambridge University Press, Cambridge (2011)zbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of ColoradoBoulderUSA

Personalised recommendations