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

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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

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

References

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  1. ^ a b c 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. arXiv:1908.10646. doi:10.30757/ALEA.v18-09. S2CID 201660248.
  2. ^ a b 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.