Optimal Dividend Strategies For Surplus-dependent Premiums Model

In this paper,we consider the optimal dividend problem with bounded dividend rate for the risk model with surplus-dependent premiums and surplus-dependent claim arrivals.The model is a general model,and surplus process is piecewise deterministic Markov process.the optimal dividend problem is able to use means of the piecewise deterministic Markov process theory.We have to give the maximum of dividends rate l₀ when we discuss the restricted dividends.We use the expected cumulative discounted dividends represent the value function which payment up to the time of ruin.After proving the properties of valued functions,We give the definition of feasible strategy and Markov strategy,and prove that the necessary and sufficient condition for a strategy to be a stationary Markov strategy.Using the the theory of measure-valued generators,we derive the associated measure-valued dynamic programming equation（measure-valued DPE）.Finally,We discuss the relationship between the measure-valued DPE and the QVI,and show that the optimal dividend strategy is a stationary Markov strategy with a band structure.