Stochastic processes with applications in physics and insurance

Abstract

Stochastic processes have been applied to described various phenomena that evolve in a random manner, such as anomalous dynamics observed in physical or biological systems and risk surplus in insurance mathematics. Modeling such phenomena via stochastic processes and further evaluating the relevant quantities, including mean square displacements that are widely used to detect anomalous diffusion, and risk quantities such as the ruin probability, dividend paid until ruin, and the Gerber-Shiu function, have been attracting a great amount of interests in the recent years. This thesis is a collection of five papers contributing to two applications of stochastic processes. For anomalous diffusion in biophysical systems, we investigate a number of stochastic models along with their properties through the lens of the probability density function, second-order structure, mean square displacements, and sample paths, in order to deduce practical implications and precautions from modeling and inference perspectives. For insurance mathematics, we establish a novel numerical quantification method based upon mathematical programming for the ruin-related quantities, and provide a survey of a variety of evaluation methods for the Gerber-Shiu function.