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 be a non-negative right-continuous $\mathcal_t$- adapted process and let be a non-negative right-continuous non-decreasing predictable process such that \mathbb (\tau)\mid \mathcal_0leq \mathbb (\tau)\mid \mathcal_0 \infty for any bounded stopping time $\tau$. Then

(i) \forall c,d>0, \mathbb\left(\sup_X(t)>c\,\Big\vert\mathcal_0\right)\leq \frac\mathbb \left sup_G(t)\wedge d\,\Big\vert\mathcal_0\right\mathbb\left(\sup_G(t)\geq d\,\Big\vert\mathcal_0\right).

(ii) \forall p\in(0,1), \mathbb\left left(\sup_X(t)\right)^p\Big\vert \mathcal_0 \rightleq c_p\mathbb\left left(\sup_G(t)\right)^p\Big\vert \mathcal_0\right \text c_p:=\frac.

Notes

: 1.See ''Théorème'' I and ''Corollaire ''II of

Bibliography

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Stochastic differential equations
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