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), (Ai(x), B-i (x))(1 less than or equal to i less than or equal to L(x))), where L(x) is the number of jumps from (X, infinity) to (-infinity, x) and A(i)(x) (B-i(x)) are the distances from x to R(t) after (before) the i th 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)).