Exact conditional simulation of Point processes: Application to pathwise market impact estimation

Market impact is defined as the difference between the observed price trajectory under a given execution strategy and the counterfactual trajectory that would have prevailed without it. Since this counterfactual is unobservable, estimating market impact requires simulating alternative paths under the same realized market randomness. We address this by studying the conditional simulation of point processes under perturbed intensities. Given an observed counting process whose intensity is determined by its own history, we characterize the conditional law of the latent Poisson random measure in a thinning representation. This yields an exact, event-driven algorithm that reconstructs counterfactual paths on a common randomness source, enabling rigorous pathwise market impact estimation for aggressive, passive, and mixed strategies.