Multiplicity results of positive solutions for fourth‐order nonlinear differential equation with singularity

Cheng, Zhibo; Ren, Jingli

doi:10.1002/mma.3481

Cited by 9 publications

(4 citation statements)

References 28 publications

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“…Financial derivatives can be divided into four types of contracts: forwards, futures, options and swaps according to their own product forms [1,2]. If divided according to the relationship between the price of financial derivatives and the price of its underlying assets, financial derivatives can be divided into two categories: linear derivatives and non-linear derivatives: linear derivatives include forward contracts and futures contracts With swap contracts, there is a linear relationship between the price of such derivatives and the price of the underlying asset; nonlinear derivatives include options, structured derivative securities, and exotic derivative securities, and their prices are very close to the price of the underlying asset [3,4]. Complex nonlinear relationship.…”

Section: Introductionmentioning

confidence: 99%

Nonlinear differential equations based on the B-S-M model in the pricing of derivatives in financial markets

Tao

Sulaimani

2021

Applied Mathematics and Nonlinear Sciences

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The pricing and hedging of financial derivatives have become one of the hot research issues in mathematical finance today. In the case of non-risk neutrality, this article uses the martingale method and probability measurement method to study the pricing method and hedging strategy of financial derivatives. This paper also further studies the hedging strategy of financial derivatives in the incomplete market based on the BSM model and converts the solution of this problem into solving a vector on the Hilbert space to its closure. The problem of space projection is to use projection theory to decompose financial derivatives under a given martingale measure. In the imperfect market, the vertical projection theory is used to obtain the approximate pricing method and hedging strategy of financial derivatives in which the underlying asset follows the martingale process; the projection theory is further expanded, and the pricing problem of financial derivatives under the mixed-asset portfolio is obtained. Approximate pricing of financial derivatives; in the discrete state, the hedging investment strategy of financial derivatives H in the imperfect market is found through the method of variance approximation.

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Section: Introductionmentioning

confidence: 99%

Nonlinear differential equations based on the B-S-M model in the pricing of derivatives in financial markets

Tao

Sulaimani

2021

Applied Mathematics and Nonlinear Sciences

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“…The second order differential equations with singularities have a wide range of applications in many subjects, such as physics, engineering, mechanics, and so on (see [1][2][3][4][5]). After the pioneering paper [6] came out, many scholars put their attention to the periodic problems of second order singular differential equations without friction term (see [7][8][9][10][11][12][13] and the references therein). Beginning with the paper of Habets-Sanchez in [14], the interest in the problem of periodic solution for second order singular differential equations with friction term has increased [15][16][17][18][19][20][21].…”

Section: Introductionmentioning

confidence: 99%

Existence of positive periodic solutions for super-linear neutral Liénard equation with a singularity of attractive type

Zhu

2020

Bound Value Probl

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In this paper, the existence of positive periodic solutions is studied for super-linear neutral Liénard equation with a singularity of attractive type $$ \bigl(x(t)-cx(t-\sigma)\bigr)''+f\bigl(x(t) \bigr)x'(t)-\varphi(t)x^{\mu}(t)+ \frac{\alpha(t)}{x^{\gamma}(t)}=e(t), $$ ( x ( t ) − c x ( t − σ ) ) ″ + f ( x ( t ) ) x ′ ( t ) − φ ( t ) x μ ( t ) + α ( t ) x γ ( t ) = e ( t ) , where $f:(0,+\infty)\rightarrow R$ f : ( 0 , + ∞ ) → R , $\varphi(t)>0$ φ ( t ) > 0 and $\alpha(t)>0$ α ( t ) > 0 are continuous functions with T-periodicity in the t variable, c, γ are constants with $|c|<1$ | c | < 1 , $\gamma\geq1$ γ ≥ 1 . Many authors obtained the existence of periodic solutions under the condition $0<\mu\leq1$ 0 < μ ≤ 1 , and we extend the result to $\mu>1$ μ > 1 by using Mawhin’s continuation theorem as well as the techniques of a priori estimates. At last, an example is given to show applications of the theorem.

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“…However, the exponent δ in the power function x δ is required to satisfy δ ∈ [0, 1), and there is no friction term of Liénard type f (x)x in (1.6). For other recent developments and applications in this field, we refer the reader to [16,23,[26][27][28][29][30][31]. Inspired by the papers mentioned, the aim of this paper is to study the periodic problem For convenience, in the end of this introduction, we give some notations used throughout the paper: …”

Section: Introductionmentioning

confidence: 99%

Existence of positive periodic solutions for Liénard equations with an indefinite singularity of attractive type

2018

Bound Value Probl

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In this paper, we study the periodic problem for the Liénard equation with an indefinite singularity of attractive typewhere f : (0, +∞) → R is continuous and may have singularities at zero, r, ϕ : R → R are T-periodic functions, and μ is a positive constant. Using the method of upper and lower functions, we obtain some new results on the existence of positive periodic solutions to the equation.

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Multiplicity results of positive solutions for fourth‐order nonlinear differential equation with singularity

Cited by 9 publications

References 28 publications

Nonlinear differential equations based on the B-S-M model in the pricing of derivatives in financial markets

Nonlinear differential equations based on the B-S-M model in the pricing of derivatives in financial markets

Existence of positive periodic solutions for super-linear neutral Liénard equation with a singularity of attractive type

Existence of positive periodic solutions for Liénard equations with an indefinite singularity of attractive type

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