Integrability, Variational Principle, Bifurcation, and New Wave Solutions for the Ivancevic Option Pricing Model

Abstract

The Ivancevic option pricing model comes as an alternative to the Black-Scholes model and depicts a controlled Brownian motion associated with the nonlinear Schrodinger equation. The applicability and practicality of this model have been studied by many researchers, but the analytical approach has been virtually absent from the literature. This study intends to examine some dynamic features of this model. By using the well-known ARS algorithm, it is demonstrated that this model is not integrable in the Painlevé sense. He’s variational method is utilized to create new abundant solutions, which contain the bright soliton, bright-like soliton, kinky-bright soliton, and periodic solution. The bifurcation theory is applied to investigate the phase portrait and to study some dynamical behavior of this model. Furthermore, we introduce a classification of the wave solutions into periodic, super periodic, kink, and solitary solutions according to the type of the phase plane orbits. Some 3D-graphical representations of some of the obtained solutions are displayed. The influence of the model’s parameters on the obtained wave solutions is discussed and clarified graphically.

Integrability and Dynamic Behavior of a Piezoelectro-Magnetic Circular Rod

“…(b) Determining the intervals of real solutions, which are sometimes named the intervals of real wave propagation, is significant because it implies that there are different types of solutions that are completely different from mathematical and physical points of view. For clarification, when ( f , g, h) ∈ R + × R + ×]h 1 , 0[, we have two solutions (26) and ( 29) that are periodic and unbounded. The two solutions are obtained with the same conditions on the parameters f , g, and h but with different intervals of real solutions.…”

“…Soliton solutions are one of the important aspects of nonlinear equations [15][16][17][18]. There are diverse methods for solving them, such as Lie symmetry analysis [19,20], the bilinear formalism method, the first integral method [21], the bifurcation theory [22][23][24][25][26], the direct method of the Hirota and the linear superposition principle [27], and a combination of the complete discriminate method and bifurcation theory [28]. The Painlevé analysis is a powerful approach employed to detect the integrability of both ordinary and partial differential equations [29].…”

“…Consequently, the periodic solution corresponding to this family will also degenerate to a solitary solution corresponding to the homoclinic orbit. Thus, we have ψ 1 = ψ 2 = 0 and ψ 3 = f g , and solution (26) becomes…”

Qualitative analysis and new exact solutions for the extended space-fractional stochastic (3 + 1)-dimensional Zakharov-Kuznetsov equation

“…For more details about qualitative analysis utilizing the bifurcation theory, you can see, e.g. [26][27][28][29][30][31][32][33].…”

Bifurcation of Traveling Wave Solution of Sakovich Equation with Beta Fractional Derivative

“…The solutions for several fractional nonlinear equations were investigated in the literature [24,25]. Bifurcation is one of the methods used to describe the solutions of many fractional and classical differential equations [26][27][28][29][30][31][32][33][34][35]. In this work, we use bifurcation methods to investigate the dynamical behavior of Equation (2).…”