Stochastic Gronwall inequality explained

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

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

Assume that for all

t\geq 0 ,X(t)\leq \int ^{t} _{0} X ^{*(u} ^{-)d} A(u) + M(t) + H(t), where X ^{*(u): = \sup} _{r\in[ 0 ,u]} X(r) .

and define

c _{p = (p ^{-p})/(1- p)} . Then the following estimates hold for p\in ( 0 , 1 ) and T> 0 :

• If

E ( H(T) ^{p)<infty} and H is predictable, then E \left[\left(X ^{*(T)\right)} ^{p \ vert l{F}} ^{0 \right]\leq} (c _{p})/(p) E \left[(H(T)) ^{p \ vert l{F}} _{0 \right]} \exp \left\lbrace c ^{1 /p} _{p} A(T)\right\rbrace ;

• If

E ( H(T) ^{p)<infty} and M has no negative jumps, then E \left[\left(X ^{*(T)\right)} ^{p \ vert l{F}} ^{0 \right]\leq} (c _{p+1})/(p) E \left[(H(T)) ^{p \ vert l{F}} _{0 \right]} \exp \left\lbrace (c ^{1 /p} _{p +1 )} A(T)\right\rbrace ;

• If

E H(T)< inf ty, then \displaystyle{ E \left[\left(X ^{*(T)\right)} ^{p \ vert l{F}} ^{0 \right]\leq} (c _{p})/(p) \left( E \left[ H(T) \ vert l{F} ^{p} _{0 \right]\right)} \exp \left\lbrace c ^{1 /p} _{p} A(T)\right\rbrace} ;

Proof

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

Notes and References

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