Approximating The Functions In Ruin Theory By Frame Duality Projection

Under the classical risk model,the thesis studies and calculates the density function of ruin time and Gerber-Shiu expected discount penalty function,which play a crucial role in the ruin theory.However,under general claim distribution hypothesis,it is difficult to get the analytical expressions of these two functions,so finding the numerical methods is of great sense.The frame duality projection via calculating the Fourier transform of integrable function to complete its approximation.This method does not need to accelerate or reach convergence by strict control parameters and can be calculated by Fast Fourier Transform algorithm,which greatly improves the computational efficiency.So this method is chosen in the thesis.Ruin time describes how long it takes an insurance company to make its surplus reach zero at the first time.Gerber-Shiu function can study various ruin variables through different selection of its parameters.Therefore,researching above two functions has great practical utility.After deriving the Fourier transform of these functions,the approximate calculation of the frame duality projection is carried out.Then the thesis shows the numerical simulation to test the feasibility and effectiveness of the method.In particular,when studying Gerber-Shiu function,the function is firstly extended and the approximate calculation is given to explore the continuation effect.The limitation of previous studies on the two types of ruin functions lies in the dependence to the distribution of special claims.The thesis tries to break this limitation and extend it to non-specific situations.The results show that the approximation effect of the frame duality projection is pretty good and it has a stable performance.The numerical simulation results and the truly function curves in different situations are almost completely coincident.For the GerberShiu expected discount penalty function,the continuation method does achieve an accurate approximation effect and the error decreases with the enhance of continuation.What's more,in the calculation process of two bankruptcy related functions,they are not dependent on the assumption of specific claim distribution,which have universality and extensibility.