The Koopman operator is defined as the pullback of a dynamical system on a function space. Koopman operators have been investigated as one of the most promising approaches for analyzing time series data generated by a nonlinear dynamical system. It is important to find data-driven methods to estimate the mathematical invariants of Koopman operators. Consequently, in this study, we explain the motivation and idea behind applying the Koopman operator theory to data analysis and introduce three topics pertinent to our recent progress on the theoretical aspect of Koopman operators with function space theory. We consider several types of function spaces in which Koopman operators act, for example, reproducing kernel Hilbert spaces and Besov spaces, and reveal the relationship between the boundedness of a Koopman operator and the behavior of the dynamical system. In addition, we explicitly compute the generalized spectrum of the Koopman operator of the one-sided full 2-shift.

Counterparty risk refers to the possibility that a counterparty in an over-the-counter financial derivatives contract may default before the contract matures, resulting in a loss without the exposure being paid according to the contract. The 2008 financial crisis brought counterparty risk into the spotlight. Specifically, research on computation methods for credit valuation adjustment (CVA), regarded as the “market price of counterparty risk,” flourished in the first half of the 2010s. In recent years, the exchange of margin has been employed as a standard measure in practice for counterparty risk management due to international regulations — specifically, the importance of initial margin (IM) to cover the risk of fluctuations in the contract value between the counterparty’s default and final settlement has been recognized. This study considers CVA and IM as fundamental tools in counterparty risk management from a mathematical perspective.

The method of fundamental solutions is known as a mesh-free numerical method for solving potential problems. It has been actively used in numerous fields, especially in engineering. In this paper, two problems are treated as part of an extension of the application of the method of fundamental solutions. The first is a moving boundary problem for viscous fluids, known as the Hele–Shaw problem. It is shown that the method of fundamental solutions can be used for a spatial discretization mechanism that satisfies the geometric variational structures in an asymptotic sense. The second is the numerical analysis of conformal mappings, an essential research subject in complex analysis. By combining the dipole simulation method, which is an improvement of the method of fundamental solutions from the viewpoint of potential theory, with the complex dipole simulation method, which is an extension of the method of fundamental solutions to approximation theory of holomorphic functions, we show that a simple and mathematically natural bidirectional numerical method can be obtained to analyze conformal mappings.