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

Assume that for all $t\ge 0$,

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

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

• If $\mathbb{𝔼}(H(T{)}^p)<\mathrm{\infty }$ and $H$ is predictable, then $\mathbb{𝔼}\left[{(X^{*}(T))}^p\right|{{ℱ}}_0]\le \frac{c_p}{p}\mathbb{𝔼}[(H(T){)}^p\left|{{ℱ}}_0\right]\exp \{c_p^{1/p}A(T)\}$;

• If $\mathbb{𝔼}(H(T{)}^p)<\mathrm{\infty }$ and $M$ has no negative jumps, then $\mathbb{𝔼}\left[{(X^{*}(T))}^p\right|{{ℱ}}_0]\le \frac{c_p+1}{p}\mathbb{𝔼}[(H(T){)}^p\left|{{ℱ}}_0\right]\exp \{(c_p+1{)}^{1/p}A(T)\}$;

• If $\mathbb{𝔼}H(T)<\mathrm{\infty },$ then $\mathbb{𝔼}\left[{(X^{*}(T))}^p\right|{{ℱ}}_0]\le \frac{c_p}{p}{\left(\mathbb{𝔼}[H(T)|{{ℱ}}_0]\right)}^p\exp \{c_p^{1/p}A(T)\}$;

Proof

It has been proven by Lenglart's inequality.