Level-Crossing Properties of the Risk Process

Abstract

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), Bi (x))1≤ i ≤ L ( x )), where L(x) is the number of jumps from (x, ∞) to (−∞, x] and Ai (x) (Bi (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)).

On the zero crossings of a generalized shot noise process

“…[7], modelling that underpins risk theory, e.g. [8], etc. To this end a generalized shot noise process, comprising a summation of pulses, which occur at a uniform rate of per second and with random amplitudes, is considered.…”