In this paper, we establish a probabilistic representation for two integration by parts formulas, one being of Bismut-Elworthy-Li’s type, for the marginal law of a one-dimensional diffusion process killed at a given level. These formulas are established by combining a Markovian perturbation argument with a tailor-made Malliavin calculus for the underlying Markov chain structure involved in the probabilistic representation of the original marginal law. Among other applications, an unbiased Monte Carlo path simulation method for both integration by parts formula stems from the previous probabilistic representations.
"Integration by parts formula for killed processes: a point of view from approximation theory." Electron. J. Probab. 24 1 - 44, 2019. https://doi.org/10.1214/19-EJP352