Abstract
In this paper, a mathematical formalism for defining the J-Curve phenomenon is set up in the form of a corresponding differential equation with practical application. An explicit form of the differential equation to describe the J-Curve, as well as the solution of the differential equations in terms of polynomials with coefficients satisfying a particular property was presented for the first time. The J-Curve and S-Curve are modelled as Riccati's nonlinear differential equation of the 1<sup>st</sup> order. A mathematical form of the differential equation is proposed that corresponds in structure to the Laguerre Polynomials (linear differential equation of the 2<sup>nd</sup> order), resulting in J-Curve as a solution. To confirm this, it is necessary to fulfil two main criteria for the mathematical validation of the J-Curve - confirmation of the structure of Laguerre polynomials (a polynomial equation with coefficients of alternating sign) and an R-squared score greater than 0.6. Two case studies are presented to validate the theoretical concepts and demonstrate the application of J-Curve mathematical modelling - Returns on Venture Investments (where returns from start-up investments are analysed over a 12-year period) and Returns on Long-Term Stock Investments (based on actual financial data for the period from 2016 to 2020 for the NIFTY 500 Index of the National Stock Exchange of India). A direct connection between the mathematical formulation and the graphs obtained is shown, which corresponds to the mathematical validation of the J-Curve phenomenon. It is mathematically shown how the financial data manifest the J-Curve behaviour, satisfying the initial assumptions of such a model. The mathematical model set up in this way can be verified in practice on different other types of data, which could create interest in an interdisciplinary approach in such research. This could include studies particularly from other economic fields such as micro- or macroeconomics.
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