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Lenglart's inequality

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

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

References

Citations

  1. Lenglart 1977, Théorème I and Corollaire II, pp. 171−179

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. arXiv:1908.10646. 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.
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