On logarithmic Sobolev inequalities for normal martingales (Q5952892)

Let \(Z_{t}\in L^{4}\) be an \(({\mathcal F}_{t})\)-martingale with \(\langle Z_{t},Z_{t}\rangle=t\) and having the chaos representation property. It follows that \(d[Z_{t},Z_{t}]=dt+\varphi_{t-} dZ_{t}\), let \(i_{t}= 1_{(\varphi_{t}=0)}\) and \(j_{t}=1_{(\varphi_{t}\neq 0)}\). \(D_{t}F=\sum_{n\geq 1}nI_{n-1}(f_{n}(\cdot,t))\) for \(F=\sum_{n\geq 0}I_{n}(f_{n})\) in the chaos representation; denote NEWLINE\[NEWLINE\Psi (u,v)=(u+v)\log (u+v)-u\log u-(1+\log u)v,NEWLINE\]NEWLINE NEWLINE\[NEWLINEEnt F= E(F\log F)-E(F)\log E(F).NEWLINE\]NEWLINE The author proves NEWLINE\[NEWLINEE(F)\leq E \left[\int_0^T j_{t}\varphi_{t}^{- 2}\Psi (F,\varphi_{t}D_{t}F) dt+\frac 12\int_0^T i_{t}F^{-1}(D_{t}F)^{2} dt \right]NEWLINE\]NEWLINE for a bounded \({\mathcal F}_{T}\)-measurable \(F\in \operatorname {Dom}D\). This result has four corollaries. The first is obtained by using \(\Psi (u,v)\leq |v|^{2}/u\), for the second \(\Psi (u,v)\leq v(\log (u+v)-\log u)\), the third and the fourth are stated for a deterministic \(\varphi\), \(F>\eta >0\), the third being based on \(\varphi_{t}D_{t}e^{F}=e^{F}(e^{\varphi_{t}D_{t}F}-1)\) and the fourth on \(F+\varphi_{t}D_{t}F=\log (e^{F}+\varphi_{t}D_{t}e^{F})=\log (e^{F}e^{\varphi_{t}D_{t}F})\). The case of the Azéma martingales is discussed. A frequent quotation is \textit{L. Wu} [Probab. Theory Relat. Fields 118, No. 3, 427-438 (2000; Zbl 0970.60093)].