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.

Statement
Let $$X(t),\, t\geq 0$$ be a non-negative right-continuous $$(\mathcal{F}_t)_{t\ge 0}$$-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 $$(\mathcal{F}_t)_{t\ge 0}$$- local martingale with $$M(0)=0$$ and càdlàg paths.

Assume that for all $$t\geq 0$$,

$$ X(t)\leq \int_0^t X^*(u^-)\,d A(u)+M(t)+H(t),$$ 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 $$T>0$$:


 * 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 $$\displaystyle{\mathbb{E}\left[\left(X^*(T)\right)^p\Big\vert\mathcal{F}_0\right]\leq \frac{c_p}{p}\left(\mathbb{E}\left[ H(T)\big\vert\mathcal{F}_0\right]\right)^p \exp \left\lbrace c_p^{1/p} A(T)\right\rbrace}$$;

Proof
It has been proven by Lenglart's inequality.