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Robust optimal investment and risk control for an insurer with general insider information

In this paper, we study the robust optimal investment and risk control problem for an insurer who owns the insider information about the financial market and the insurance market under model uncertainty. Both financial risky asset process and insurance risk process are assumed to be very general jump diffusion processes. The insider information is of the most general form rather than the initial enlargement type. We use the theory of forward integrals to give the first half characterization of the robust optimal strategy and transform the anticipating stochastic differential game problem into the nonanticipative stochastic differential game problem. Then we adopt the stochastic maximum principle to obtain the total characterization of the robust strategy. We discuss the two typical situations when the insurer is `small' and `large' by Malliavin calculus. For the `small' insurer, we obtain the closed-form solution in the continuous case and the half closed-form solution in the case with jumps. For the `large' insurer, we reduce the problem to the quadratic backward stochastic differential equation (BSDE) and obtain the closed-form solution in the continuous case without model uncertainty. We discuss some impacts of the model uncertainty, insider information and the `large' insurer on the optimal strategy.