Abstract: In this paper, we consider the two-dimensional optimal dividend problem in the context of two insurance companies with compound Poisson surplus processes. The surplus process is a piecewise deterministic Markov process (PDMP), so the optimal dividend problem can be studied by virtue of use the PDMP theory. After getting the properties of the optimal value function, we give the definition of admissible strategy and Markov strategy, and prove the necessary and sufficient condition for a strategy to be a stationary Markov strategy. Using the theory of measure-valued generators, we derive the associated measure-valued dynamic programming equation (measure-valued DPE). Finally we prove the verification theorem under the existence of the optimal strategy.

Key words: Optimal dividend problem ,  PDMP ,  Measure-valued DPE ,  Markov strategy ,  Verification theorem