Lenglart's inequality

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

(i) ${\displaystyle \forall c,d>0,\mathbb {P} \left(\sup _{t\geq 0}X(t)>c\,{\Big \vert }{\mathcal {F}}_{0}\right)\leq {\frac {1}{c}}\mathbb {E} \left[\sup _{t\geq 0}G(t)\wedge d\,{\Big \vert }{\mathcal {F}}_{0}\right]+\mathbb {P} \left(\sup _{t\geq 0}G(t)\geq d\,{\Big \vert }{\mathcal {F}}_{0}\right).}$

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

1.^ See Théorème I and Corollaire II of Lenglart, Érik (1977). "Relation de domination entre deux processus". Annales de l'Institut Henri Poincaré, Section B. 13 (2): 171−179.

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