Stochastic differential portfolio games for an insurer in a jump‐diffusion risk process

We discuss an optimal portfolio selection problem of an insurer who faces model uncertainty in a jump‐diffusion risk model using a game theoretic approach. In particular, the optimal portfolio selection problem is formulated as a two‐person, zero‐sum, stochastic differential game between the insurer and the market. There are two leader‐follower games embedded in the game problem: (i) The insurer is the leader of the game and aims to select an optimal portfolio strategy by maximizing the expected utility of the terminal surplus in the ‘worst‐case’ scenario; (ii) The market acts as the leader of the game and aims to choose an optimal probability scenario to minimize the maximal expected utility of the terminal surplus. Using techniques of stochastic linear‐quadratic control, we obtain closed‐form solutions to the game problems in both the jump‐diffusion risk process and its diffusion approximation for the case of an exponential utility.