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Minimizing the Discounted Probability of Exponential Parisian Ruin via Reinsurance

We study the problem of minimizing the discounted probability of exponential Parisian ruin, that is, the discounted probability that an insurer's surplus exhibits an excursion below zero in excess of an exponentially distributed clock. The insurer controls its surplus via reinsurance priced according to the mean-variance premium principle, as in Liang, Liang, and Young (2019). We consider the classical risk model and apply stochastic Perron's method, as introduced by Bayraktar and Sirbu (2012,2013,2014), to show that the minimum discounted probability of exponential Parisian ruin is the unique viscosity solution of its Hamilton-Jacobi-Bellman equation with boundary conditions at $\pm \infty$. A major difficulty in proving the comparison principle arises from the discontinuity of the Hamiltonian.