Lenglart's inequality

In the mathematical theory of probability, Lenglart's inequality was proved by Èrik Lenglart in 1977.{{harvnb|Lenglart|1977|loc = Théorème I and Corollaire II, pp. 171−179}} Later slight modifications are also called Lenglart's inequality.

Statement

Let {{math|X}} be a non-negative right-continuous $\mathcal{F}_t$ -adapted process and let {{math|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

{{ordered list | list-style-type = lower-roman

| $\forall c,d>0, \mathbb{P}\left(\sup_{t\geq 0}X(t)>c\,\Big\vert\mathcal{F}_0\right)\leq \frac{1}{c}\mathbb{E} \left[\sup_{t\geq 0}G(t)\wedge d\,\Big\vert\mathcal{F}_0\right]+\mathbb{P}\left(\sup_{t\geq 0}G(t)\geq d\,\Big\vert\mathcal{F}_0\right).$

| $\forall p\in(0,1), \mathbb{E}\left[\left(\sup_{t\geq 0}X(t)\right)^p\Big\vert \mathcal{F}_0 \right]\leq c_p\mathbb{E}\left[\left(\sup_{t\geq 0}G(t)\right)^p\Big\vert \mathcal{F}_0\right], \text{ where } c_p:=\frac{p^{-p}}{1-p}.$

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References

= Citations =

= General sources =

{{cite journal |title=Sharpness of Lenglart's domination inequality and a sharp monotone version|journal=Electronic Communications in Probability|first1=Sarah|last1=Geiss|first2=Michael |last2=Scheutzow |volume=26 | year=2021 | pages=1–8 | doi=10.1214/21-ECP413|arxiv=2101.10884|s2cid=231709277}}
{{cite journal |title=Relation de domination entre deux processus|journal=Annales de l'Institut Henri Poincaré B| first1=Érik |last1=Lenglart|volume=13|issue=2|year=1977|pages=171−179}}
{{cite journal |title=A stochastic Gronwall lemma and well-posedness of path-dependent SDEs driven by martingale noise | journal=Latin Americal Journal of Probability and Mathematical Statistics|first1=Sima |last1=Mehri | first2=Michael |last2=Scheutzow |volume=18|year=2021|pages=193−209|doi=10.30757/ALEA.v18-09|s2cid=201660248|arxiv=1908.10646}}
{{cite journal |title= A note on the domination inequalities and their applications|journal=Statist. Probab. Lett.|first1=Yaofeng |last1=Ren|first2=Jing |last2=Schen | volume=82| issue=6| year=2012| pages=1160−1168| doi=10.1016/j.spl.2012.03.002}}
{{cite book |title=Continuous Martingales and Brownian Motion|first1=Daniel |last1=Revuz|first2=Marc |last2=Yor| publisher=Springer| location = Berlin| year=1999| isbn=3-540-64325-7}}

Category:Stochastic differential equations

Category:Articles containing proofs

Category:Probabilistic inequalities