Integration by Parts Formula for Exit Times of One Dimensional Diffusions

In line with the methodology introduced in Frikha et al. (Electron J Probab 24(95):1–44, 2019) for formulating probabilistic representations of integration by parts involving killed diffusion, we establish an integration by parts formula for the first exit time of one-dimensional diffusion processes. However, our approach diverges from the conventional differential calculus applied to the associated space Markov chain; instead, we employ calculus techniques that focus on the underlying time variables.