Usage of the homotopy analysis method for solving the nonlinear and linear integral equations of the second kind

Hetmaniok, Edyta�; Słota, Damian; Trawiński, T.; Wituła, Roman

doi:10.1007/s11075-013-9781-0

Cited by 31 publications

(17 citation statements)

References 54 publications

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“…Rather, we search an approximate homotopy series solution 27) where the deformation equations (3.16) for m > 1 are approximated by 28) and where the (m − 1)-th homotopy derivative is evaluated on the grid points t i , s j…”

Section: A Discrete Homotopy Analysis Methodsmentioning

confidence: 99%

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Optimal execution with non-linear transient market impact

Curato

Gatheral

Lillo

2016

Quantitative Finance

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We study the problem of the optimal execution of a large trade in the presence of nonlinear transient impact. We propose an approach based on homotopy analysis, whereby a well behaved initial strategy is continuously deformed to lower the expected execution cost. We find that the optimal solution is front loaded for concave impact and that its expected cost is significantly lower than that of conventional strategies. We then consider brute force numerical optimization of the cost functional; we find that the optimal solution for a buy program typically features a few short intense buying periods separated by long periods of weak selling. Indeed, in some cases we find negative expected cost. We show that this undesirable characteristic of the nonlinear transient impact model may be mitigated either by introducing a bid-ask spread cost or by imposing convexity of the instantaneous market impact function for large trading rates.

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Section: A Discrete Homotopy Analysis Methodsmentioning

confidence: 99%

“…As suggested in [27], we choose the linear operator L to be the identity, that is 14) and the auxiliary function to be H (t) = 1. The zero-order deformation equation is then…”

Section: Homotopy For Nonlinear Transient Market Impactmentioning

confidence: 99%

Optimal execution with non-linear transient market impact

Curato

Gatheral

Lillo

2016

Quantitative Finance

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“…Hetmaniok et al [23] show influence of this parameter on the convergence rate. In a book, current advances of the HAM to solve strongly nonlinear problems were edited by Liao [24].…”

Section: Homotopy Analysis Methods (Ham)mentioning

confidence: 99%

A Mathematical Approach Based on the Homotopy Analysis Method: Application to Solve the Nonlinear Harry-Dym (HD) Equation

Ghiasi¹,

Saleh²

2017

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In this paper, the homotopy analysis method (HAM) has been employed to obtain the approximate analytical solution of the nonlinear Harry-Dym (HD) equation, which is one of the most important soliton equations. Utilizing the HAM, thereby employing the initial approximation, variations of the 7th-order approximation of the Harry-Dym equation is obtained. It is found that effect of the nonzero auxiliary parameter on convergence rate of the series solution is undeniable. It is also shown that, to some extent, order of the fractional derivative plays a fundamental role in the prediction of convergence. The final results reported by the HAM have been compared with the exact solution as well as those obtained through the other methods.

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“…Most often there is given only a theorem showing that if the series converges its sum is the solution of the equation. More theoretical results are contained in papers [1,10,11,25].…”

Section: Introductionmentioning

confidence: 99%

Application of the Homotopy Analysis Method for Solving the Systems of Linear and Nonlinear Integral Equations

Brociek

Hetmaniok

Matlak

et al. 2016

Mathematical Modelling and Analysis

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In this paper we indicate some applications of homotopy analysis method for solving the systems of linear and nonlinear integral equations. The method is based on the concept of creating function series. If the series converges, its sum is the solution of this system of equations. The paper presents conditions to ensure the convergence of this series and estimation of the error of approximate solution obtained when the partial sum of the series is used. Application of the method will be illustrated by examples.Keywords: homotopy analysis method, convergence, error estimation, system of Fredholm integral equations, system of Volterra integral equations.

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Usage of the homotopy analysis method for solving the nonlinear and linear integral equations of the second kind

Cited by 31 publications

References 54 publications

Optimal execution with non-linear transient market impact

Optimal execution with non-linear transient market impact

A Mathematical Approach Based on the Homotopy Analysis Method: Application to Solve the Nonlinear Harry-Dym (HD) Equation

Application of the Homotopy Analysis Method for Solving the Systems of Linear and Nonlinear Integral Equations

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