Forward–backward stochastic differential games for optimal investment and dividend problem of an insurer under model uncertainty

Abstract

• Model uncertainty is incorporated in investment and dividend problem of an insurer. • Insurer’s control problem is formulated as forward–backward insurer-market games. • Insurer-market games are generalized to regular-singular forward–backward games. • Maximum principles are derived for regular-singular forward–backward games. We consider optimal investment and dividend problem of an insurer, where the insurer decides dividend payment policy and invests his surplus into the financial market to manage his risk exposure. The insurer’s control problem, with the presence of model uncertainty, is formulated as zero-sum, forward–backward games between insurer and market. In the framework of game theory, we develop the games between insurer and market to the more general forward–backward stochastic differential games, where the system is governed by forward–backward stochastic differential equations; the control processes are regular-singular controls; and the informations available to the two players are asymmetric partial informations. Then the maximum principles are established to give sufficient and necessary optimality conditions for the saddle points of the general forward–backward games. Finally, we apply the maximum principles to solve the optimal investment and dividend problem of an insurer under model uncertainty.