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

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

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

and define ${\displaystyle c_{p}={\frac {p^{-p}}{1-p}}}$. Then the following estimates hold for ${\displaystyle p\in (0,1)}$ and ${\displaystyle T>0}$:^[1][2]

• If ${\displaystyle \mathbb {E} {\big (}H(T)^{p}{\big )}<\infty }$ and ${\displaystyle H}$ is predictable, then ${\displaystyle \mathbb {E} \left[\left(X^{*}(T)\right)^{p}{\Big \vert }{\mathcal {F}}_{0}\right]\leq {\frac {c_{p}}{p}}\mathbb {E} \left[(H(T))^{p}{\big \vert }{\mathcal {F}}_{0}\right]\exp \left\lbrace c_{p}^{1/p}A(T)\right\rbrace }$;
• If ${\displaystyle \mathbb {E} {\big (}H(T)^{p}{\big )}<\infty }$ and ${\displaystyle M}$ has no negative jumps, then ${\displaystyle \mathbb {E} \left[\left(X^{*}(T)\right)^{p}{\Big \vert }{\mathcal {F}}_{0}\right]\leq {\frac {c_{p}+1}{p}}\mathbb {E} \left[(H(T))^{p}{\big \vert }{\mathcal {F}}_{0}\right]\exp \left\lbrace (c_{p}+1)^{1/p}A(T)\right\rbrace }$;
• If ${\displaystyle \mathbb {E} H(T)<\infty ,}$ then ${\displaystyle \displaystyle {\mathbb {E} \left[\left(X^{*}(T)\right)^{p}{\Big \vert }{\mathcal {F}}_{0}\right]\leq {\frac {c_{p}}{p}}\left(\mathbb {E} \left[H(T){\big \vert }{\mathcal {F}}_{0}\right]\right)^{p}\exp \left\lbrace c_{p}^{1/p}A(T)\right\rbrace }}$;

Proof

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

References