Efficient sequential estimation in multiplicative counting processes

The problem of efficient sequential estimation is counting processes with multiplicative intensity processes is considered. A sequential version of Cramer-Rao type information inequality is obtained and all the 'efficient' triples (S, f, g) are characterized: the variance of an unbiased estimator f for g attains the lower bound under a sampling plan S. Applications to Poisson processes, Markov processes, birth and death processes and Markov branching processes with immigration are also considered.