Runge–Kutta convolution quadrature methods with convergence and stability analysis for nonlinear singular fractional integro-differential equations

Zhang, Gengen; Zhu, Rui

doi:10.1016/j.cnsns.2019.105132

Cited by 13 publications

(6 citation statements)

References 28 publications

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“…Recently, the fractional models have aroused numerous research interests in various fields, ranging from finance, 28,29 neuroscience, 30,31 physics, 32,33 and so on. [34][35][36][37][38][39] We here focus on the fractional model in option pricing. The finite difference method is a widely used method in option pricing.…”

Section: Monte Carlo and Finite Difference Of Fpdementioning

confidence: 99%

Total value adjustment of Bermudan option valuation under pure jump Lévy fluctuations

Yuan¹,

Ding²,

Duan³

et al. 2021

Preprint

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During the COVID-19 pandemic, many institutions have announced that their counterparties are struggling to fulfill contracts. Therefore, it is necessary to consider the counterparty default risk when pricing options. After the 2008 financial crisis, a variety of value adjustments have been emphasized in the financial industry. The total value adjustment (XVA) is the sum of multiple value adjustments, which is also investigated in many stochastic models such as Heston 1 and Bates 2 models. In this work, a widely used pure jump Lévy process, the CGMY process has been considered for pricing a Bermudan option with various value adjustments. Under a pure jump Lévy process, the value of derivatives satisfies a fractional partial differential equation (FPDE). Therefore, we construct a method which combines Monte Carlo with finite difference of FPDE (MC-FF) to find the numerical approximation of exposure, and compare it with the benchmark Monte Carlo-COS (MC-COS) method. We use the discrete energy estimate method, which is different with the existing works, to derive the convergence of the numerical scheme. Based on the numerical results, the XVA is computed by the financial exposure of the derivative value.

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Section: Monte Carlo and Finite Difference Of Fpdementioning

confidence: 99%

Total value adjustment of Bermudan option valuation under pure jump Lévy fluctuations

Yuan¹,

Ding²,

Duan³

et al. 2021

Preprint

View full text Add to dashboard Cite

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“…Fractional integro-differential equations are sometimes difficult to solve analytically, requiring the construction of effective approximation solutions. The Jacobi spectral method [20], Runge Kutta method [24], Chebyshev collocation method [3], Laplace Power Series Method [1], rationalized Haar functions method [15], Galerkin methods with hybrid functions [14] and Laguerre collocation method [5] are just a few of the numerical techniques that have been used to solve such equations.…”

Section: Introductionmentioning

confidence: 99%

A Spectral Collocation Method for Solving Caputo-Liouville Fractional Order Fredholm Integro-differential Equations

Saad,

Khirallah

2024

Eur. J. Pure Appl. Math.

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In this paper, a numerical method for solving the fractional order Fredholm integro-differential equations via the Caputo-Liouville derivative is presented. The method uses the well-known shifted Chebyshev expansion and a truncated series to represent the unknown function. It also incorporates numerical integration techniques like the Trapezoidal, Simpson’s 1/3, and Simpson’s 8/3 methods. The paper also provides an approximation for the derivative of an integer. The procedure converts the provided problem into a system of algebraic equations using shifted Chebyshev coefficients and collocation points. The coefficients are found by solving this system using well-known techniques like Newton’s method. Numerical results are presented graphycally to illustrate the applicability, efficacy, and accuracy of the approach presented in this work. All calculations in this study were performed using the MATHEMATICA software program.

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“…Consider the following initial value problem of d-dimensional nonlinear SFNIDEs in Ito's sense (1), stimulated by the non-negligible noise source of some actual problems modeled by fractional integrodifferential equations with Abel-type singular kernels [28].…”

Section: Introductionmentioning

confidence: 99%

Euler-Maruyama approximation for stochastic fractional neutral integro-differential equations with weakly singular kernel

Asadzade,

Mahmudov

2024

Phys. Scr.

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This manuscript examines the problem of nonlinear stochastic fractional neutral integro-differential equations with weakly singular kernels. Our focus is on obtaining precise estimates to cover all possible cases of Abel-type singular kernels. Initially, we establish the existence, uniqueness, and continuous dependence on the initial value of the true solution, assuming a local Lipschitz condition and linear growth condition. Additionally, we develop the Euler-Maruyama method for numerical solution of the equation and prove its strong convergence under the same conditions as the well-posedness. Moreover, we determine the accurate convergence rate of this method under global Lipschitz conditions and linear growth conditions.

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Runge–Kutta convolution quadrature methods with convergence and stability analysis for nonlinear singular fractional integro-differential equations

Cited by 13 publications

References 28 publications

Total value adjustment of Bermudan option valuation under pure jump Lévy fluctuations

Total value adjustment of Bermudan option valuation under pure jump Lévy fluctuations

A Spectral Collocation Method for Solving Caputo-Liouville Fractional Order Fredholm Integro-differential Equations

Euler-Maruyama approximation for stochastic fractional neutral integro-differential equations with weakly singular kernel

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