  We consider the problem of minimizing the probability of ruin by purchasing
reinsurance whose premium is computed according to the mean-variance premium
principle, a combination of the expected-value and variance premium principles.
We derive closed-form expressions of the optimal reinsurance strategy and the
corresponding minimum probability of ruin under the diffusion approximation of
the classical Cram\'er-Lundberg risk process perturbed by a diffusion. We find
an explicit expression for the reinsurance strategy that maximizes the
adjustment coefficient for the classical risk process perturbed by a diffusion.
Also, for this risk process, we use stochastic Perron's method to prove that
the minimum probability of ruin is the unique viscosity solution of its
Hamilton-Jacobi-Bellman equation with appropriate boundary conditions. Finally,
we prove that, under an appropriate scaling of the classical risk process, the
minimum probability of ruin converges to the minimum probability of ruin under
the diffusion approximation.

Optimization and Control