This chapter examines a random process (X(t):t ≥ 0) taking values in R, that is governed by the events of an independent renewal process N(t), as follows: whenever an event of N(t) occurs, the process X(t) is restarted and runs independently of the past with initial value that has the same distribution as X(0). The case when each segment of the process between consecutive events of N(t) is a diffusion is studied, and expressions for the characteristic function of X(t) and its stationary distribution as t → ∞ are presented. An expression is derived for the expected first-passage time of X(t) to any value a, and several explicit examples of interest are considered. The chapter presents two approaches: first, it uses Wald's equation which supplies the mean in quite general circumstances; second, it explores possibilities for use of the moment-generating function of the first-passage time.
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