Fractional Interaction of Financial Agents in a Stock Market Network

Abstract

In this study, we present a model which represents the interaction of financial companies in their network. Since the long time series have a global memory effect, we present our model in the terms of fractional integro-differential equations. This model characterize the behavior of the complex network where vertices are the financial companies operating in XU100 and edges are formed by distance based on Pearson correlation coefficient. This behavior can be seen as the financial interactions of the agents. Hence, we first cluster the complex network in the terms of high modularity of the edges. Then, we give a system of fractional integro-differential equation model with two parameters. First parameter defines the strength of the connection of agents to their cluster. Hence, to estimate this parameter we use vibrational potential of each agent in their cluster. The second parameter in our model defines how much agents in a cluster affect each other. Therefore, we use the disparity measure of PMFGs of each cluster to estimate second parameter. To solve model numerically we use an efficient algorithmic decomposition method and concluded that those solutions are consistent with real world data. The model and the solutions we present with fractional derivative show that the real data of Borsa Istanbul Stock Exchange Market always seek for an equilibrium state.

The global sliding mode tracking control for a class of variable order fractional differential systems

“…Theorem 4.1. For the variable order fractional sliding mode dynamics (3) , if it's applied the global sliding mode surface (12) into the system, then, its error state trajectories converge to zero asymptotically.…”

“…Many complicate phenomena in practical problems can be described by the fractional order mathematical formulation, which stimulates the rapid development of the basic mathematical theory of fractional calculus, see [1][2][3][4][5][6][7] . In 1998, Lorenzo and Hartley [8] proposed a kind of definition of the variable order fractional derivative and gave the basic properties, whereafter, some kinds of definitions of variable order fractional derivative, such as the Riemann-Liouville type [9] , Caputo type and Marchaud type [10] , were obtained.…”

Nonlinear Differential Equations in Preventing Financial Risks

“…This makes the fractional Brownian motion neither a Markov process nor a semimartingale. This has brought huge difficulties to avoid arbitrage opportunities in the financial market under fractional Brownian motion [3]. Some scholars have proposed using the fractional Brownian jump-diffusion model to avoid the existence of arbitrage, and the results are still not ideal.…”

Adomian Decomposition and Fractional Power Series Solution of a Class of Nonlinear Fractional Differential Equations

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