 Bichteler-Dellacherie theorem

An adapted càdlàg process is a semimartingale exactly when it is a good integrator: on each finite horizon, for every sequence of bounded elementary predictable processes with stopping-time intervals $H^n$ whose deterministic uniform bounds tend to zero, the elementary terminal integrals $(H^n\cdot X{)}_T$ tend to zero in probability. The elementary integral is a finite sum of measurable coefficients times subsequent increments; the characterization says precisely when this operation extends continuously to stochastic integration. The theorem is used here as a standard characterization, not proved from first principles.