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Cox's factoring of regression model likelihoods for continuous-time processes

Cox showed that the likelihood of regression models for discrete-time processes factors into a partial likelihood and a product of conditional laws for the covariates, given the history. Jacod constructed a partial likelihood for continuous-time regression models in terms of the predictable characteristics of the response process. Here we prove a factorization of the likelihood, analogous to Cox's, assuming both the response and the covariates to be semimartingales. The result is useful for counting process regression modelling and inference, and also for regression involving continuous processes and diffusions with jumps.