Grant: Discovery Grants Program - Individual

The proposed research program concerns the application of probability methods to three areas of financial and actuarial mathematics. First, we consider the problem of finding an optimal portfolio under new combinations of trading constraints. Our approach is to apply the duality idea, which means that an auxiliary problem (the dual problem) is introduced to help solving the given problem. We will work in both the classical framework where the constraints are convex, and the less explored framework where the constraints are not necessarily convex. Second, nonnegative stochastic continuous-time processes are often used in financial modeling, for instance, to model the (nonnegative) interest rates or the volatility of a stock. In many practical applications, namely for approximating the price of an option, one has to run simulations on a computer. To do so, the continuous-time process has to be discretized, a problem complicated by its nonnegativity. We will work on formalizing such a practice by proposing new ways to achieve this goal, while making sure that the approximate prices are theoretically correct. Third, ruin theory's goals are mainly to model the surplus of an insurance company, and quantify the risk of ruin (ruin occurs when the company's surplus becomes negative). In addition to claims which are random in nature, there are situations where there is also some randomness in the company's income. We propose to develop new ways to model the evolution of the company's surplus over time, allowing for dependence between random claims and random income, with the aim of assessing the company's risk of ruin.