Moment and polynomial bounds for ruin-related quantities in risk theory

We establish a novel numerical quantification method based upon mathematical programming for the ruin-related quantities on collective risk models with fairly general features, such as reflecting, non-homogeneous and multidimensional dynamics with non-smooth coefficients. Rather than seeking approximations with a great deal of computing effort as in the existing numerical approximation methods, the proposed method aims to instantly provide deterministic upper and lower bounds for such quantities as linear combinations of moments or in the explicit polynomial form, without random number generation or sophisticated numerical algorithms. Given that closed-form solutions are rarely available especially in finite-time problems, the upper and lower bounds have great potential to be useful information that can complement approximate solutions obtained from existing numerical methods. We present numerical results throughout to justify the theoretical developments and convergence results so as to illustrate the effectiveness of the proposed method with low computational complexity. We also demonstrate further improvements of the bounds by scaling the problem domain, employing piecewise polynomial test functions and the exponential tempering of the polynomial bases.