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**Differential subordination and strong differential subordination for continuous-time martingales and related sharp inequalities.**
*(English)*
Zbl 0832.60055

If \(X\) and \(Y\) are a pair of discrete martingales, suppose the difference sequence of \(X\) satisfies a pointwise domination with the corresponding sequence of \(Y\). Then one says that \(X\) is differentially subordinate to \(Y\). In the continuous parameter case the concept is in terms of their quadratic variations. Thus if \(X = \{X_t, {\mathcal F}_t, t \geq 0\}\) and \(Y = \{Y_t, {\mathcal F}_t, t \geq 0\}\) with values in a (separable) Hilbert space \({\mathcal H}\) are two adapted càdlàg martingales, one says that \(Y\) is differentially subordinate to \(X\) if \([X,X]_t - [Y,Y]_t \geq 0\) and is increasing. These concepts and several of the sharp inequalities on such processes have been the subject of D. L. Burkholder’s work (with \({\mathcal H} = \mathbb{R})\) from the late 1970’s in several papers. Based on the methodology of this work, the author extends some of the results, of which the following is representative.

Theorem. Let \(X\) and \(Y\) be a pair of adapted càdlàg \({\mathcal H}\)- valued martingales with \(Y\) being differentially subordinate to \(X\). Then for each \(1 < p < \infty\), one has \(|Y |_p \leq (p^* - 1) |X |_p\), where the constant \((p^* - 1)\) is the best possible and there is strict inequality if \(p \neq 2\) and \(|X |_p \neq 0\). Here \(p^* = \max (p,p/(p - 1))\).

The extension is nontrivial. The author remarks finally: “When working on sharp martingale inequalities the finite-dimensional discrete-time case is more challenging and more interesting.” With such extensions, it is now natural and possible to consider the corresponding theory in the context of Orlicz spaces in lieu of the present Lebesgue spaces.

Theorem. Let \(X\) and \(Y\) be a pair of adapted càdlàg \({\mathcal H}\)- valued martingales with \(Y\) being differentially subordinate to \(X\). Then for each \(1 < p < \infty\), one has \(|Y |_p \leq (p^* - 1) |X |_p\), where the constant \((p^* - 1)\) is the best possible and there is strict inequality if \(p \neq 2\) and \(|X |_p \neq 0\). Here \(p^* = \max (p,p/(p - 1))\).

The extension is nontrivial. The author remarks finally: “When working on sharp martingale inequalities the finite-dimensional discrete-time case is more challenging and more interesting.” With such extensions, it is now natural and possible to consider the corresponding theory in the context of Orlicz spaces in lieu of the present Lebesgue spaces.

Reviewer: M.M.Rao (Riverside)

### MSC:

60G44 | Martingales with continuous parameter |

60G46 | Martingales and classical analysis |

60G42 | Martingales with discrete parameter |

60E15 | Inequalities; stochastic orderings |