  The ruin probability in the classical Brownian risk model can be explicitly
calculated for both finite and infinite-time horizon. This is not the case for
the simultaneous ruin probability in two-dimensional Brownian risk model.
Resorting on asymptotic theory, we derive in this contribution approximations
of both simultaneous ruin probability and simultaneous ruin time for the
two-dimensional Brownian risk model when the initial capital increases to
infinity. Given the interest in proportional reinsurance, we consider in some
details the case where the correlation is 1. This model is tractable allowing
for explicit formulas for the simultaneous ruin probability for linearly
dependent spectrally positive L\'evy processes. Examples include perturbed
Brownian and gamma L\'evy processes.
