Lenglart's inequality

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

Statement

Let X be a non-negative right-continuous [math]\displaystyle{ \mathcal{F}_t }[/math]-adapted process and let G be a non-negative right-continuous non-decreasing predictable process such that [math]\displaystyle{ \mathbb{E}[X(\tau)\mid \mathcal{F}_0]\leq \mathbb{E}[G(\tau)\mid \mathcal{F}_0]\lt \infty }[/math] for any bounded stopping time [math]\displaystyle{ \tau }[/math]. Then

1. [math]\displaystyle{ \forall c,d\gt 0, \mathbb{P}\left(\sup_{t\geq 0}X(t)\gt 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). }[/math]
2. [math]\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}. }[/math]

References

General sources

• Geiss, Sarah; Scheutzow, Michael (2021). "Sharpness of Lenglart's domination inequality and a sharp monotone version". Electronic Communications in Probability 26: 1–8. doi:10.1214/21-ECP413. 
• Lenglart, Érik (1977). "Relation de domination entre deux processus". Annales de l'Institut Henri Poincaré B 13 (2): 171−179. 
• Mehri, Sima; Scheutzow, Michael (2021). "A stochastic Gronwall lemma and well-posedness of path-dependent SDEs driven by martingale noise". Latin Americal Journal of Probability and Mathematical Statistics 18: 193−209. doi:10.30757/ALEA.v18-09. 
• Ren, Yaofeng; Schen, Jing (2012). "A note on the domination inequalities and their applications". Statist. Probab. Lett. 82 (6): 1160−1168. doi:10.1016/j.spl.2012.03.002. 
• Revuz, Daniel; Yor, Marc (1999). Continuous Martingales and Brownian Motion. Berlin: Springer. ISBN 3-540-64325-7.