Some characteristics of a surplus process in the presence of an upper barrier

For the classical model of risk theory, we consider the time until the insurer’s surplus reaches an upper barrier a. We consider some properties of hitting times, first unconditionally and then conditional on the event that ruin does not occur before the surplus exceeds a. For the latter case, we obtain an upper bound for the expected time to reach a given that ruin has not occurred. We also study the more general case where the insurer’s target profit is not constant but varies with time. By considering the random walk associated with the surplus process, we show that the distribution of the overshoot over a general barrier is always exponential and we obtain expressions for the moments of first passage times for a non-constant barrier.