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 ${{ℱ}}_t$-adapted process and let G be a non-negative right-continuous non-decreasing predictable process such that $\mathbb{𝔼}[X(\tau )\mid {{ℱ}}_0]\le \mathbb{𝔼}[G(\tau )\mid {{ℱ}}_0]<\mathrm{\infty }$ for any bounded stopping time $\tau$. Then

1. $\mathrm{\forall }c,d>0,\mathbb{\mathbb{P}}(\underset{t\ge 0}{\sup }X(t)>c|{{ℱ}}_0)\le \frac{1}{c}\mathbb{𝔼}[\underset{t\ge 0}{\sup }G(t)\wedge d|{{ℱ}}_0]+\mathbb{\mathbb{P}}(\underset{t\ge 0}{\sup }G(t)\ge d|{{ℱ}}_0).$
2. $\mathrm{\forall }p\in (0,1),\mathbb{𝔼}\left[{(\underset{t\ge 0}{\sup }X(t))}^p\right|{{ℱ}}_0]\le c_p\mathbb{𝔼}\left[{(\underset{t\ge 0}{\sup }G(t))}^p\right|{{ℱ}}_0],\text{ where }{c}_p:=\frac{p^{-p}}{1-p}.$

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.