Garcia, Nancy Lopes, Birth and death processes as projections of higher-dimensional Poisson processes, Advances in applied probability , 27(4), 1995, pp. 911-930
Birth and death processes can be constructed as projections of higher-dimensional Poisson processes. The existence and uniqueness in the strong sense of the solutions of the time change problem are obtained. It is shown that the solution of the time change problem is equivalent to the solution of the corresponding martingale problem. Moreover, the processes obtained by the projection method are ergodic under translations.