Stochastic risk processes with economic factors

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In this research program I plan to follow some recent works to model the surplus process of an insurance business as a compound of two semimartingales. Among them, one process is regarded as a risk process in a world without economic factors and the other describes returns on risky and/or risk-free investments. This proposed stochastic model covers most of the useful insurance risk models. Also, it builds a broad platform for many results and methods from statistics and probability.First I plan to investigate the quantitative impacts of different types of taxation on insurance business. My study will be conducted mainly via the ruin probabilities of the surplus process with and without tax payment. I will study periodic taxation, loss-carry-forward taxation, as well as premium tax payment on their influences on the ruin probability of the surplus process.I will also investigate the right tail behavior of the stochastic discounted net loss of the surplus process, where the stochastic discounted net loss is the present value of the aggregate loss in a period of time by a stochastic discount factor. This study is important for us to grasp the influence of financial risks on insurance surplus process. Also, the right tail probability of loss distribution has its applications in risk measures, such as Value-at-Risk and Tail-Value-at-Risk, as well as default risk estimation.Since except for some very specific cases it is impossible to find out explicit solutions to my questions, I will use bounds and asymptotics as main tools to approximate the desired probabilities. To do so, it will be very helpful to impose some conditions (such as subexponentiality, exponentiality, and reliability properties) on the distributions of the insurance and/or financial risks.