Lenglart's inequality

In the mathematical theory of probability, Lenglart's inequality was proved by Èrik Lenglart in 1977.^[1] Later slight modifications are also called Lenglart's inequality.

Statement

Let X be a non-negative right-continuous ${\displaystyle {\mathcal {F}}_{t}}$-adapted process and let G be a non-negative right-continuous non-decreasing predictable process such that ${\displaystyle \mathbb {E} [X(\tau )\mid {\mathcal {F}}_{0}]\leq \mathbb {E} [G(\tau )\mid {\mathcal {F}}_{0}]<\infty }$ for any bounded stopping time ${\displaystyle \tau }$. Then

1. ${\displaystyle \forall c,d>0,\mathbb {P} \left(\sup _{t\geq 0}X(t)>c\,{\Big \vert }{\mathcal {F}}_{0}\right)\leq {\frac {1}{c}}\mathbb {E} \left[\sup _{t\geq 0}G(t)\wedge d\,{\Big \vert }{\mathcal {F}}_{0}\right]+\mathbb {P} \left(\sup _{t\geq 0}G(t)\geq d\,{\Big \vert }{\mathcal {F}}_{0}\right).}$
2. ${\displaystyle \forall p\in (0,1),\mathbb {E} \left[\left(\sup _{t\geq 0}X(t)\right)^{p}{\Big \vert }{\mathcal {F}}_{0}\right]\leq c_{p}\mathbb {E} \left[\left(\sup _{t\geq 0}G(t)\right)^{p}{\Big \vert }{\mathcal {F}}_{0}\right],{\text{ where }}c_{p}:={\frac {p^{-p}}{1-p}}.}$