Bessel过程的Bismut-Elworthy-Li公式
Bismut-Elworthy-Li formulae for Bessel processes
摘要：
本文主要研究非负维Bessel过程所对应的马氏半群的可微性。更准确地说，对于所有的$δ\geq 0$和$T>0$，我们计算函数$x \mapsto P^δ_{T} F (x) $的导数，其中$(P^δ_{t})_{t \geq 0}$是与$δ$维贝塞尔过程相关联的过渡半群，$F$是$\mathbb{R}_{+}$上的任何有界Borel函数。所得到的表达式显示了一个很好的$δ$和$(δ+ 2)$维贝塞尔过程的过渡半群之间的相互作用。因此，我们推导出Bessel过程满足强Feller性质，具有与维数无关的连续模。此外，我们提供了一个概率解释这个表达式作为一个Bismut-Elworthy-Li公式。
英文摘要：
In this article we are interested in the differentiability property of the Markovian semi-group corresponding to the Bessel processes of nonnegative dimension. More precisely, for all $δ\geq 0$ and $T>0$, we compute the derivative of the function $x \mapsto P^δ_{T} F (x) $, where $(P^δ_{t})_{t \geq 0}$ is the transition semi-group associated to the $δ$ - dimensional Bessel process, and $F$ is any bounded Borel function on $\mathbb{R}_{+}$. The obtained expression shows a nice interplay between the transition semi-groups of the $δ$ - and the $(δ+ 2)$-dimensional Bessel processes. As a consequence, we deduce that the Bessel processes satisfy the strong Feller property, with a continuity modulus which is independent of the dimension. Moreover, we provide a probabilistic interpretation of this expression as a Bismut-Elworthy-Li formula.
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