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We provide a probabilistic representation for the derivative of the semigroupcorresponding to a diffusion process killed at the boundary of a half interval.In particular, we show that the derivative of the semi-group can be expressedas the expected value of a functional of a reflected diffusion process.Furthermore, as an application, we obtain a Bismut-Elworthy-Li formula which isalso valid at the boundary.

A probabilistic representation of the derivative of a one dimensional killed diffusion semigroup and associated Bismut-Elworthy-Li formula