Polynomial Processes and Applications

Abstract

The goal of this thesis is to introduce polynomial processes and to present some of the most important findings and applications in a clear and pedagogical way. A polynomial process is a particular type of Markov process. Polynomial processes are defined by being in a sense polynomial preserving in expectation. This enables us to show that the calculation of moments can be done using matrix exponentials. Furthermore, the matrix is easily obtained from the extend generator of the process. As pricing of options and hedging depends often on the computation of moments, polynomial processes are attractable in financial modeling. The class of polynomial processes contains many of the processes already used in application, for example Ornstein-Uhlenbeck processes and Jacobi processes. Electricity markets provide an interesting application for polynomial processes. A case of hedging long-term electricity commitments with a risk-minimizing rolling hedge is introduced. Polynomial processes can also be applied in many other areas from interest rate models to computing life insurance liabilities.