Stochastic Gronwall inequality

Stochastic Gronwall inequality is a generalization of Gronwall's inequality and has been used for proving the well-posedness of path-dependent stochastic differential equations with local monotonicity and coercivity assumption with respect to supremum norm.[1][2]

Statement

Let be a non-negative right-continuous -adapted process. Assume that is a deterministic non-decreasing càdlàg function with and let be a non-decreasing and càdlàg adapted process starting from . Further, let be an - local martingale with and càdlàg paths.

Assume that for all ,

where .

and define . Then the following estimates hold for and :[1][2]

  • If and is predictable, then ;
  • If and has no negative jumps, then ;
  • If then ;

Proof

It has been proven by Lenglart's inequality.[1]

References

  1. Mehri, Sima; Scheutzow, Michael (2021). "A stochastic Gronwall lemma and well-posedness of path-dependent SDEs driven by martingale noise". Latin Americal Journal of Probability and Mathematical Statistics. 18: 193-209. doi:10.30757/ALEA.v18-09. S2CID 201660248.
  2. von Renesse, Max; Scheutzow, Michael (2010). "Existence and uniqueness of solutions of stochastic functional differential equations". Random Oper. Stoch. Equ. 18 (3): 267-284. arXiv:0812.1726. doi:10.1515/rose.2010.015. S2CID 18595968.
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