A decomposition of the ruin probability for the risk process perturbed by diffusion

In this paper, we consider the ruin probabilities (caused by oscillation or by a claim) of the classical risk process perturbed by diffusion and the r isk process with return on investments. We will prove their twice continuou s differentiability and derive the integro-differential equations satisfied by them. We will present the explicit expressions for them when the claims are exponentially distributed. (C) 2001 Elsevier Science B.V. All rights r eserved.