Abstract
Topological methods are rapidly developing and are becoming more used in physics, biology and chemistry. One area of topology has showed its immense potential in explaining potential financial contagion and financial crisis in financial markets. The aforementioned method is knot theory. The movement of stock price has been marked and braids and knots have been noted. By analysing the knots and braids using Jones polynomial, it is tried to find if there exists an untrivial knot equal to unknot? After thorough analysis, possible financial contagion and financial crisis prediction are analysed by using instruments of knot theory pertaining in that sense to Jones, Laurent and Alexander polynomial. It is proved that it is possible to predict financial disruptions by observing possible knots in the graphs and finding appropriate polynomials. In order to analyse knot formation, the following approach is used: “Knot formation in three-dimensional space is considered and the equations about knot forming and its disentangling are considered”. After having defined the equations in three-dimensional space, the definition of Brownian bridge concerning formation of knots in three-dimensional space is defined. Using analogy method, the notion of Brownian bridge is translated into 2-dimensional space and the foundations for the application of knot theory in 2-dimensional space have been set up. At the same time, the aforementioned approach is innovative and it could be used in accordance with stochastic analysis and quantum finance.
Highlights
IntroductionRandom dynamical systems are considered. It is assumed that financial time series exhibit fractional Brownian motion and knot theory is used in order to analyse the formation of knots in financial time series
In this paper, random dynamical systems are considered
One question was posed and it stated: “What would happen if the time series pertaining in that sense to major financial indices follow fractional Brownian motion and are forming knots? Can the financial crisis be predicted by observing knot formation?” Afterwards, we proceed to analysis and formation of knots in threedimensional space, we present the equations that represent the formation of 3 dimensional knots by using formulas from quantum physics and afterwards and we make the Brownian bridge hypothesis in three-dimensional space and translate it to two-dimensional space
Summary
Random dynamical systems are considered. It is assumed that financial time series exhibit fractional Brownian motion and knot theory is used in order to analyse the formation of knots in financial time series. The foundations are set up to further the analysis of the financial time series using quantum physics, knot theory and topology. We will define mathematically random system, afterwards Wiener process via stochastic differential equation is defined and ordinary Brownian motion is at the same time defined. One question was posed and it stated: “What would happen if the time series pertaining in that sense to major financial indices follow fractional Brownian motion and are forming knots? Can the financial crisis be predicted by observing knot formation?” Afterwards, we proceed to analysis and formation of knots in threedimensional space, we present the equations that represent the formation of 3 dimensional knots by using formulas from quantum physics and afterwards and we make the Brownian bridge hypothesis in three-dimensional space and translate it to two-dimensional space. The following papers of the author will approach the further analysis and development of equations in the formation of knots in two-dimensional space
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