Compound Poisson processes, as modified by Ornstein-Uhlenbeck processes

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Abstract
Assume that {C(t), 0 ⩽ t < ∞} is a compound Poisson stochastic process, which models a collective risk situation. Let {I(t), 0 ⩽ t < ∞} be a stochastic process describing the investment performance deviations (from the expected) over time. It is assumed that the I(t) process is an Ornstein-Uhlenbeck (O.U.) process. Such a process is Gaussian (normal) and Markovian. Its conditional mean function reflects the stabilizing effects needed in a model for an economic process in which excessive movements are rare. Let (t), 0 ⩽ t < ∞ and {L(t), 0 ⩽ t < ∞} be random processes representing the deviations from the operating and lapse expense assumptions. It is assumed that they are O.U. processes, and that the four processes are independent of each other. A risk process (t), 0 ⩽ t < ∞ is formed by a linear combination of the four processes. For the risk process, probabilities of ruin are discussed. A detailed example is provided. References to the recent papers by Harald Bohman, and Olof Thorin are given.