The modern mathematical models in economics and finance

With this article, we open a new section in this journal: the application of mathematical methods in economics and finance. A few topics we would like to discuss to get started are corporate finance, investments, business valuation, taxation, and ratings. We describe shortly mathematical models in these areas. In the field of corporate finance, we discuss the foundations of two main theories of capital structure, the Modigliani-Miller and the modern theory of Brusov-Filatova-Orekhova (BFO theory). We compare them and describe the differences between them and their results. In the field of investments, we describe two modern investment models: (1) with debt repayment at the end of the project and (2) with uniform debt repayment and discuss their properties and applications. In business valuation, we discuss the problems that exist in this area and ways to solve them. In rating methodology, a new approach is devoted to the rating of non–financial issuers, as well as to long–term and arbitrary duration project rating. The key factors of a new approach are the adequate use of discounting of financial flows virtually not used in existing rating methodologies, and the incorporation of rating parameters (financial "ratios") into the modern theory of capital structure (Brusov–Filatova–Orekhova (BFO) theory). This article is devoted to the analysis of the theoretical mathematical methods and models based on first principles. The novelty of this consideration is due to the fact that we are considering and discussing recently developed mathematical models in economics and finance.