Graduation date: 2007This thesis considers one of the classical problems in the actuarial mathematics literature, the decay of the probability of ruin in the collective risk model. The claim number process N(t) is assumed to be a renewal process, the resulting model being referred as the Sparre Andersen risk model. The inter-claim times form a sequence of independent identically distributed random variables. The additional non-classical feature is that the company invests in an asset with stochastic returns. A very general integro-differential equation is derived for expected values of functions of this renewal risk model with stochastic returns. Moreover, as a particular case, an integro-differential equation is derived for the probability of ruin, under very general conditions regarding the claim sizes, claim arrivals and the returns from investment. Through this unified approach, specific integro-differential equations of the ruin probability may be written for various risk model scenarios, allowing the asymptotic analysis of the ruin probabilities.