Abstract
A simple parametrization is introduced which represents the insurance market's response to an insurer adopting a pricing strategy determined via optimal control theory. Claims are modeled using a lognormally distributed mean claim size rate and the market average premium is determined via the expected value principle. If the insurer maximizes its expected wealth then the resulting Bellman equation has a moving boundary in state space, which determines when it is optimal to stop selling insurance. Three finite difference schemes are used to verify the existence of a solution to the Bellman equation when there is market reaction. All of the schemes use a front-fixing transformation. If the market reacts then it is found that the optimal strategy is altered, so that premiums are raised if the strategy is of loss-leading type and lowered if it is optimal for the insurer to set a relatively high premium and sell little insurance.
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