A birth-death process is a continuous-time Markov chain that counts the number of particles in a system over time. In the general process with n current particles, a new particle is born with instantaneous rate λ(n) and a particle dies with instantaneous rate μ(n). Currently no robust and efficient method exists to evaluate the finite-time transition probabilities in a general birth-death proce...

For a random process (X(t), t ≥ 0), suppose that there is a cost fx associated with being in state x. This paper is concerned with evaluating the distribution and the expected value of the total cost Γ over the life of the process. The existing literature contains results for particular classes of process and particular choices of f , usually linear functions of the state. We will describe a me...

This paper presents a method of evaluating the expected value of a path integral for a general Markov chain on a countable state space. We illustrate the method with reference to several models, including birth-death processes and the birth, death and catastrophe process.