Celebratio Mathematica — Yor — Complete Bibliography

Complete Bibliography

Works connected to Jean-Michel Bismut

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J.-M. Bismut and M. Yor: “An inequality for processes which satisfy Kolmogorov’s continuity criterion: Application to continuous martingales,” J. Funct. Anal. 51 : 2 (1983), pp. 166–173. MR 701054 Zbl 0524.60020 article

Theorem 1 below is a by-product of the continuous martingale inequalities obtained in [Yor 1985], using the technique of enlargment of filtrations. The main purpose of this paper is to give an entirely new proof and some extensions of Theorem 1; some global stability properties of classes of processes which satisfy Kolmogorov’s criterion (\( K^{\phi} \)) play a crucial role in our proof.

Let \( k\in (0,\infty) \), and \( p\in (1,\infty) \). There exists a constant \( C_{k,p} \) such that for any positive, finite variable \( L \), and any continuous local martingale \( (X_t)_{t\geq 0} \), with \( X(0) = 0 \), we have

\( E[(X^*(L))^k]\leq C_{k,p} \bigl\| \langle X\rangle_L^{k/2} \bigr\|_p \),
\( E\bigl[ \langle X\rangle_L^{k/2} \bigr] \leq C_{k,p} \bigl\| (X^*(L))^k \bigr\|_p \), where \( X*(t)\equiv \sup_{s\leq t} |X(s)| \),
and \( (\langle X\rangle_t) \) is the increasing process of \( X \).