Abstract
If $X$ is a spectrally positive Levy process, $\bar{X}^c$ the continuous part of its maximum process, and $J$ the sum of the jumps of $X$ across its previous maximum, then $X - 2\bar{X}^c - J$ has the same law as $X$ conditioned to stay negative. This extends a result due to Pitman, who links the real Brownian motion and the three-dimensional Bessel process. Several other relations between the Brownian motion and the Bessel process are extended in this setting.
Citation
Jean Bertoin. "An Extension of Pitman's Theorem for Spectrally Positive Levy Processes." Ann. Probab. 20 (3) 1464 - 1483, July, 1992. https://doi.org/10.1214/aop/1176989701
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