Abstract: In this paper,a risk model driven by Markovian environment process that affects both the claim sizes and rates is described.The expression of Laplace transform of the Gerber-Shiu function is obtained.By means ofthe general Lundberg fundamental equation,the explicit expressions for the Gerber-Shiu functions with zero initial capital,the given state of environment,the discounted joint density functions and the density marginal density of the surplus prior to and after ruin are derived,respectively.Meanwhile,the methods to compute the arbitrary moments of the time to ruin,surplus before ruin and the deficit at ruin are also given.

Key words: risk process, Gerber-Shiu function, general Lundbergfundamental equation, Markovian environment