Continuous-time optimal reporting with full insurance under the mean-variance criterion

We study a continuous-time, loss-reporting problem for an insured with full insurance under the mean-variance (MV) criterion. When a loss occurs, the insured faces two options: she can report it to the insurer for full reimbursement but will pay a higher premium rate; or she can hide it from the insurer by paying it herself and enjoy a lower premium rate. The insured follows a barrier strategy for loss reporting and seeks an optimal barrier to maximize her MV preferences over a random horizon. We show that this problem yields an optimal barrier that is not necessarily decreasing with respect to the insured's risk aversion, as intuition suggests it should. To address this non-monotonicity, we propose two solutions: in the first solution, we restrict the feasible strategies to a bounded interval; in the second, we modify the MV criterion by replacing the variance of the insured's wealth with the variance of the insured's retained losses. We obtain the optimal barrier strategy in semiclosed form—as a unique positive zero of a nonlinear function—for both modified models, and we show that it is a decreasing function of the insured's risk aversion, as expected.