Two optimal investment problems for an insurer with two business lines are considered

where each business line's risk process is modeled by twodimensional Lévy process. It is assumed that the insurer can invest its surplus in a riskfree asset and two risky assets

where the risky assets' price processes are described by a two-dimensional Lévy process. A benchmark problem and a mean-variance problem are discussed. The first problem is to choose the optimal investment strategy to minimize the expected quadratic distance of the risk reserve to a given benchmark; the second problem is to minimize the variance of the terminal wealth when the expected terminal reserve is given. By employing stochastic dynamic programming approach

the explicit expressions of the optimal investment strategy and the optimal value function are derived for the first problem; with the results of the first problem and the duality theory

the optimal investment strategy and the efficient frontier for the second problem are derived.