Level-Crossing Properties of the Risk Process

For the classical risk process R(t) that is linear increasing with slope 1 between downward jumps of i.i.d. random sizes at the points of a homogeneous Poisson process we consider the level-crossing process C(x) = (L(x), (A_i(x), B_i(x))_1≤i≤L(x)), where L(x) is the number of jumps from (x, ∞) to (−∞, x] and A_i(x) (B_i(x)) are the distances from x to R(t) after (before) the ith jump of this kind. It is shown that if R(·) has a drift toward infinity, C(·) is a stationary Markov process; its transition probabilities are determined. As an application we derive the expected value E(L(x)L(x + y)).