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In this paper, we build on the techniques developed in Albrecher et al.(2013), to generate initial-boundary value problems for ruin probabilities ofsurplus-dependent premium risk processes, under a renewal case scenario, Erlang(2) claim arrivals, and an exponential claims scenario, Erlang (2) claim sizes.Applying the approximation theory of solutions of linear ordinary differentialequations developed in Fedoryuk (1993), we derive the asymptotics of the ruinprobabilities when the initial reserve tends to infinity. When consideringpremiums that are {\it linearly} dependent on reserves, representing forinstance returns on risk-free investments of the insurance capital, we firstlyderive explicit formulas for the ruin probabilities, from which we can easilydetermine their asymptotics, only to match the ones obtained for generalpremiums dependent on reserves. We compare them with the asymptotics of theequivalent ruin probabilities when the premium rate is fixed over time, tomeasure the gain generated by this additional mechanism of binding the premiumrates with the amount of reserve own by the insurance company.

How much we gain by surplus-dependent premiums -- asymptotic analysis of ruin probability