Modern Mathematical Methods for Actuarial Sciences

In the ruin theory, premium income and outgoing claims play an important role. We introduce several ruin type mathematical models and apply various mathematical methods to find optimal premium price for the insurance companies. Quantum theory is one of the significant novel approaches to compute the finite time non-ruin probability. More exactly, we apply the discrete space Quantum mechanics formalism (see main thesis for formalism) and continuous space Quantum mechanics formalism (see main thesis for formalism) with the appropriately chosen Hamiltonians. Several particular examples are treated via the traditional basis and quantum mechanics formalism with the different eigenvector basis. The numerical results are also obtained using the path calculation method and compared with the stochastic modeling results. In addition, we also construct various models with interest rate. For these models, optimal premium prices are stochastically calculated for independent and dependent claims with different dependence levels by using the Frank copula method.