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

Abstract

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.

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Acknowledgements

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

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Correspondence to János Engländer.

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This research was supported in part by Simons Foundation Grant 579110. The support is gratefully acknowledged.

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Engländer, J. A Generalization of the Submartingale Property: Maximal Inequality and Applications to Various Stochastic Processes. J Theor Probab 33, 506–521 (2020). https://doi.org/10.1007/s10959-019-00880-6

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Keywords

  • 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