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

Assume that for all [math]\displaystyle{ t\geq 0 }[/math],

[math]\displaystyle{ X(t)\leq \int_0^t X^*(u^-)\,d A(u)+M(t)+H(t), }[/math] where [math]\displaystyle{ X^*(u):=\sup_{r\in[0,u]}X(r) }[/math].

and define [math]\displaystyle{ c_p=\frac{p^{-p}}{1-p} }[/math]. Then the following estimates hold for [math]\displaystyle{ p\in (0,1) }[/math] and [math]\displaystyle{ T\gt 0 }[/math]:

• If [math]\displaystyle{ \mathbb{E} \big(H(T)^p\big)\lt \infty }[/math] and [math]\displaystyle{ H }[/math] is predictable, then [math]\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 }[/math];

• If [math]\displaystyle{ \mathbb{E} \big(H(T)^p\big)\lt \infty }[/math] and [math]\displaystyle{ M }[/math] has no negative jumps, then [math]\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 }[/math];

• If [math]\displaystyle{ \mathbb{E} H(T)\lt \infty, }[/math] then [math]\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} }[/math];

Proof

It has been proven by Lenglart's inequality.