Numerical solution of Itô-Volterra integral equation by least squares method

Ahmadinia, M.; Heydari, Mohammad

doi:10.1007/s11075-019-00770-2

Cited by 16 publications

(8 citation statements)

References 23 publications

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“…Lemma 3 (see [29]) (continuous module). f with respect to ϱ in [0, T] is the definition of continuous modulus ω(f, ϱ) ω(f, ϱ)…”

Section: Error Analysismentioning

confidence: 99%

“…In contrast to the articles [13,29], this paper's difference is to study linear SIVIEs driven by multiple independent Brownian motions. In addition, the numerical solution based on the least squares method and BPFs is more accurate than other methods in the reference [30].…”

Section: Introductionmentioning

confidence: 99%

“…In [28], Jiang et al used BPFs to obtain the numerical solution of two-dimensional nonlinear SIVIEs. Ahmadinia et al presented a new method that applies the least squares method to study SIVIEs [29]. Moreover, Maleknejad et al simplified m-dimensional SIVIEs into linear lower trigonometric equations by using BPFs and stochastic integral operation matrix, and then the equations were solved [30].…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Numerical Solution of Multidimensional Stochastic Itô-Volterra Integral Equation Based on the Least Squares Method and Block Pulse Function

Jiang

Deng

2021

Mathematical Problems in Engineering

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In this paper, a method based on the least squares method and block pulse function is proposed to solve the multidimensional stochastic Itô-Volterra integral equation. The Itô-Volterra integral equation is transformed into a linear algebraic equation. Furthermore, the error analysis is given by the isometry property and Doob’s inequality. Numerical examples verify the effectiveness and precision of this method.

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“…Lemma 3 (see [29]) (continuous module). f with respect to ϱ in [0, T] is the definition of continuous modulus ω(f, ϱ) ω(f, ϱ)…”

Section: Error Analysismentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Numerical Solution of Multidimensional Stochastic Itô-Volterra Integral Equation Based on the Least Squares Method and Block Pulse Function

Jiang

Deng

2021

Mathematical Problems in Engineering

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“…Because Equation is usually difficult to be solved analytically, several numerical methods are used; here, we refer to previous studies 4 . For more details, a variable transformation method 4–21 is applied for the numerical solution of (VFEs) of the second kind. Also, Du and Chen 5 proposed a high order reproducing kernel method for solving linear (VFEs) in the form of ().…”

Section: Introductionmentioning

confidence: 99%

A new least‐squares‐based reproducing kernel method for solving regular and weakly singular Volterra‐Fredholm integral equations with smooth and nonsmooth solutions

Jing

Tohidi

et al. 2021

Math Methods in App Sciences

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Based on the least‐squares method, we proposed a new algorithm to obtain the solution of the second kind of regular and weakly singular Volterra‐Fredholm integral equations in reproducing kernel spaces. The stability and uniform convergence of the algorithm are investigated in detail. Numerical experiments verify the theoretical findings. Meanwhile, this method is also applicable to the nonlinear Volterra integral equations. Test problems which have non‐smooth solutions are also considered, and our proposed method is efficient as some recent Krylov subspace methods such as LSQR and LSMR.

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“…Recently, many research has been carried out on solving the stochastic Itô‐Volterra integral equation. In these researches, the numerical methods based on the least squares, stochastic operational matrix, radial basis functions (RBFs), Euler polynomial, orthonormal Bernoulli polynomials, Haar wavelets, and cubic B‐spline approximation are introduced. In Saffarzadeh et al, an iterative numerical algorithm to approximate the solution of stochastic Itô‐Volterra integral equations with

m

‐dimensional Brownian motion process is provided.…”

Section: Introductionmentioning

confidence: 99%

Convergence analysis of an iterative algorithm to solve system of nonlinear stochastic Itô‐Volterra integral equations

Saffarzadeh

Heydari

Loghmani

2020

Math Methods in App Sciences

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In this paper, an efficient and accurate numerical iterative algorithm based on the linear spline interpolation for solving the system of nonlinear stochastic Itô-Volterra integral equations is presented. The most important merit of this method is that it does not need to solve any system of nonlinear algebraic equations. An upper bound for the linear spline approximation of the stochastic function is provided. Using this upper bound and under the Lipschitz and linear growth conditions, the convergence analysis of the suggested method is studied.Finally, to verify the efficiency of the proposed scheme, some problems in the finance, physics, and biology are investigated, and the obtained results are compared with the stochastic -method. KEYWORDSlinear spline interpolation, multiple stock models, pendulum problem, predator-prey models, successive approximations method, system of stochastic Volterra integral equations MSC CLASSIFICATION 60H10; 60H20; 65C30Math Meth Appl Sci. 2020;43:5212-5233. wileyonlinelibrary.com/journal/mma

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Numerical solution of Itô-Volterra integral equation by least squares method

Cited by 16 publications

References 23 publications

Numerical Solution of Multidimensional Stochastic Itô-Volterra Integral Equation Based on the Least Squares Method and Block Pulse Function

Numerical Solution of Multidimensional Stochastic Itô-Volterra Integral Equation Based on the Least Squares Method and Block Pulse Function

A new least‐squares‐based reproducing kernel method for solving regular and weakly singular Volterra‐Fredholm integral equations with smooth and nonsmooth solutions

Convergence analysis of an iterative algorithm to solve system of nonlinear stochastic Itô‐Volterra integral equations

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