On logarithmic Sobolev inequalities for normal martingales

 

Nicolas Privault
Université de la Rochelle
Avenue Michel Crépeau
17042 La Rochelle Cedex 1
France

 

Abstract:

Let $ (Z_t)_{t\in {\mathord{{\rm I\kern-2.8pt R}}}_+}$ be a martingale in $ L^4$ having the chaos representation property and angle bracket $ d\langle Z_t ,Z_t\rangle = dt$. We show that the positive functionals $ F$ of $ (Z_t)_{t\in {\mathord{{\rm I\kern-2.8pt R}}}_+}$ satisfy the modified logarithmic Sobolev inequality

$\displaystyle E[F\log F] - E[F]\log E[F] \leq {1\over 2} E\left[ {1\over F} \int_0^\infty (2-i_t) (D_tF)^2 dt \right], $

where $ D$ is the gradient operator defined by lowering the degree of multiple stochastic integrals with respect to $ (Z_t)_{t\in {\mathord{{\rm I\kern-2.8pt R}}}_+}$ and $ (i_t)_{t\in {\mathord{{\rm I\kern-2.8pt R}}}_+}\subset \{0,1\}$ is a process given by the structure equation satisfied by $ (Z_t)_{t\in {\mathord{{\rm I\kern-2.8pt R}}}_+}$.

Key words: Logarithmic Sobolev inequalities, normal martingales, Azéma martingales, Poisson random measures.
Mathematics Subject Classification: 60G44, 60G60, 46E35, 46E39.

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