A Method for Evaluating the Distribution of the Total Cost of a Random Process over its Lifetime

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 method which assumes only that f is non-negative. We characterize both the distribution and the expected value of Γ as extremal solutions of systems of linear equations. Of particular interest in biological applications is the case when there is a single absorbing state, corresponding to population extinction, where we are usually interested in evaluating the cost of the process up to the time of extinction. We will illustrate our results with reference to three important Markovian models: the pure-birth process, the birth-death process, and the linear birth-death and catastrophe process.