Lie symmetry analysis for a parabolic Monge–Ampère equation in the optimal investment theory

It is generally known that Lie symmetries of differential equations can lead to a reduction of the governing equation(s), lead to exact solutions of these equations and, in the best case scenario, lead to a linearization of the original equation. In this paper, we consider a model from optimal investment theory where we show the governing equation possesses an extensive contact symmetry and, through this, we show it is linearizable. Several exact solutions are provided including a solution to a particular terminal value problem.

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“…which recovers the exact solution presented by Yang and Xu [14] by choosing c 1 = 0, c 2 = 1 and s = 1.…”

Section: Exact Solutionssupporting

confidence: 83%

“…A classical symmetry analysis was performed by Yang and Xu [14] who were able to show that (2) admitted the symmetry generator…”

Section: Introductionmentioning

confidence: 99%

Contact Symmetries of a Model in Optimal Investment Theory

Arrigo

Grift

2021

Symmetry

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“…It is well known that it is not easy to get solutions for integrable lattice systems. Currently, there are several useful approaches which can help us obtain explicit solutions, for instance, the binary nonlinearization [16][17][18], Lie symmetry analysis [19,20], and the Darboux transformation [21][22][23][24][25][26][27][28][29]. The Darboux matix method is one of important and effective techniques to obtain explicitly exact solutions of integrable lattice equations [30][31][32][33][34].…”

Section: Introductionmentioning

confidence: 99%

Explicit Solutions and Conservation Laws for a New Integrable Lattice Hierarchy

Yang

Zhao

2019

Complexity

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An integrable lattice hierarchy is derived on the basis of a new matrix spectral problem. Then, some properties of this hierarchy are shown, such as the Liouville integrability, the bi-Hamiltonian structure, and infinitely many conservation laws. After that, the Darboux transformation of the first integrable lattice equation in this hierarchy is constructed. Eventually, the explicitly exact solutions of the integrable lattice equation are investigated via graphs.

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“…Sinkala [38] introduced an invariant solution for the stock price model that prevents arbitrage. The readers can explore [2,3,39,48] for further insight in the use of Lie's symmetry approach for problem-solving in the realm of financial mathematics.…”

mentioning

confidence: 99%

Analyzing the American portfolio options within the CEV model incorporating dividend yield by the Lie symmetry approach

Javaid,

Aziz,

Aziz

2024

DCDS-S

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This study aims to use the classical Lie symmetry algorithm to solve a mathematical model that represents the American put option. The model is based on a constant elasticity of variance (CEV) framework and includes factors like dividend yield. An infinitesimal transformation and symmetry generators are obtained for the governing parabolic (1+1) PDE which is also known as a price equation. Classical symmetries for the Black Scholes, Cox-Ingersoll-Ross, and general CEV models with dividend yield help us understand the mathematical structure of these models and their relationships, enabling us to develop more accurate pricing models and better understand investment risks. Group invariant solutions of governing PDE are obtained by using similarity variables which are evaluated by solving the characteristic equation of the symmetry operator. A graphical representation of solutions with different dividend yields is presented and discussed to identify the new attribute. The results indicate that rising interest rates, volatility, dividends, and expiration time conform to the established trends. The effects of the dividends yield in all three cases show a high premium value.

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Lie symmetry analysis for a parabolic Monge–Ampère equation in the optimal investment theory

Cited by 5 publications

References 8 publications

Contact Symmetries of a Model in Optimal Investment Theory

Contact Symmetries of a Model in Optimal Investment Theory

Explicit Solutions and Conservation Laws for a New Integrable Lattice Hierarchy

Analyzing the American portfolio options within the CEV model incorporating dividend yield by the Lie symmetry approach

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