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 $X(t),\,t\geq 0$ be a non-negative right-continuous $({\mathcal {F}}_{t})_{t\geq 0}$ -adapted process. Assume that $A:[0,\infty )\to [0,\infty )$ 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 $({\mathcal {F}}_{t})_{t\geq 0}$ - local martingale with $M(0)=0$ and càdlàg paths.

Assume that for all $t\geq 0$ ,

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

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

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