Within this thesis, the problem of dividend payments for companies whose value is modelled by Brownian motion with a positive drift is investigated. The first introductory chapter presents the problem and the motives for which the modelling of company values was required to change. In the second chapter, we initially defined Brownian motion; then, more importantly, exhibited a model of Brownian motion with drift. Subsequently, the probability of ruin and the time of ruin was introduced. The third chapter encompasses the introduction of dividend payment strategies, as well as examining in further detail the barrier strategy. We also analysed the consequences that result from the effect that the dividend strategy has on defining the time of ruin, and introduced the notion of the expected present value of all dividends paid until the moment of ruin when dividends are paid by the barrier strategy. Furthermore, we observed what happens to the stated expectation when the interest rate is equal to 0. The fourth chapter gave us more detailed results on the time of ruin under the barrier strategy. We described the expected present value of the dividend payment of 1 at the time of ruin, which ultimately led to more specific results on the distribution of the time of ruin. We then further analysed the seemingly unobvious relationship between the derivation of the expected present value of all dividends paid up to the time of failure, provided that the initial value was 0 and the expected present value of the payment of dividends of 1 at the time of ruin provided that the initial value of the company was a barrier b. We then considered the results of a case where volatility tends to infinity. In the fifth chapter, we depict the moment-generating function of the present value of dividends paid until the moment of ruin and the results that ensue when volatility tends to infinity. The sixth chapter pertains to the moment-generating function of accumulated dividends paid up to the time of ruin. For such a function, we also perceived the case when the drift tends to 0. The final chapter is based on determining the optimal barrier strategy, i.e. the optimal barrier b∗, which maximizes the expected present value of all dividends paid until the moment of ruin.