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Stochastic Gronwall inequality

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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.

Statement

Let X ( t ) , t ≥ 0 {\displaystyle X(t),\,t\geq 0} be a non-negative right-continuous ( F t ) t ≥ 0 {\displaystyle ({\mathcal {F}}_{t})_{t\geq 0}} -adapted process. Assume that A : [ 0 , ∞ ) → [ 0 , ∞ ) {\displaystyle A:[0,\infty )\to [0,\infty )} is a deterministic non-decreasing càdlàg function with A ( 0 ) = 0 {\displaystyle A(0)=0} and let H ( t ) , t ≥ 0 {\displaystyle H(t),\,t\geq 0} be a non-decreasing and càdlàg adapted process starting from H ( 0 ) ≥ 0 {\displaystyle H(0)\geq 0} . Further, let M ( t ) , t ≥ 0 {\displaystyle M(t),\,t\geq 0} be an ( F t ) t ≥ 0 {\displaystyle ({\mathcal {F}}_{t})_{t\geq 0}} - local martingale with M ( 0 ) = 0 {\displaystyle M(0)=0} and càdlàg paths.

Assume that for all t ≥ 0 {\displaystyle t\geq 0} ,

X ( t ) ≤ ∫ 0 t X ∗ ( u − ) d A ( u ) + M ( t ) + H ( t ) , {\displaystyle X(t)\leq \int _{0}^{t}X^{*}(u^{-})\,dA(u)+M(t)+H(t),} where X ∗ ( u ) := sup r ∈ [ 0 , u ] X ( r ) {\displaystyle X^{*}(u):=\sup _{r\in }X(r)} .

and define c p = p − p 1 − p {\displaystyle c_{p}={\frac {p^{-p}}{1-p}}} . Then the following estimates hold for p ∈ ( 0 , 1 ) {\displaystyle p\in (0,1)} and T > 0 {\displaystyle T>0} :

  • If E ( H ( T ) p ) < ∞ {\displaystyle \mathbb {E} {\big (}H(T)^{p}{\big )}<\infty } and H {\displaystyle H} is predictable, then E [ ( X ∗ ( T ) ) p | F 0 ] ≤ c p p E [ ( H ( T ) ) p | F 0 ] exp ⁡ { c p 1 / p A ( T ) } {\displaystyle \mathbb {E} \left\leq {\frac {c_{p}}{p}}\mathbb {E} \left\exp \left\lbrace c_{p}^{1/p}A(T)\right\rbrace } ;
  • If E ( H ( T ) p ) < ∞ {\displaystyle \mathbb {E} {\big (}H(T)^{p}{\big )}<\infty } and M {\displaystyle M} has no negative jumps, then E [ ( X ∗ ( T ) ) p | F 0 ] ≤ c p + 1 p E [ ( H ( T ) ) p | F 0 ] exp ⁡ { ( c p + 1 ) 1 / p A ( T ) } {\displaystyle \mathbb {E} \left\leq {\frac {c_{p}+1}{p}}\mathbb {E} \left\exp \left\lbrace (c_{p}+1)^{1/p}A(T)\right\rbrace } ;
  • If E H ( T ) < ∞ , {\displaystyle \mathbb {E} H(T)<\infty ,} then E [ ( X ∗ ( T ) ) p | F 0 ] ≤ c p p ( E [ H ( T ) | F 0 ] ) p exp ⁡ { c p 1 / p A ( T ) } {\displaystyle \displaystyle {\mathbb {E} \left\leq {\frac {c_{p}}{p}}\left(\mathbb {E} \left\right)^{p}\exp \left\lbrace c_{p}^{1/p}A(T)\right\rbrace }} ;

Proof

It has been proven by Lenglart's inequality.

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. arXiv:1908.10646. 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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