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

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

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. arXiv:2101.10884. doi:10.1214/21-ECP413. S2CID 231709277.
• 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. S2CID 201660248.
• 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.