Handbook of Mathematical Functions.

Florentin, J. J.; Abramowitz, Milton; Stegun, Irene A.

doi:10.2307/2314682

Cited by 217 publications

(410 citation statements)

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“…where ρ and τ are two stopping times satisfying 0 ≤ ρ(w) ≤ τ (w) ≤ T , a.s.. In particular, I α [1] 0,τ will be written as I α,τ , and I α [1] ρ,τ as I α,ρ,τ . iii) Itô coefficient functions.…”

Section: 2mentioning

confidence: 99%

“…where {a i } M i=1 are the roots of the Hermite polynomial of degree M and {w i } M i=1 are the corresponding weights [1]. In general, x + √ 2∆t n a i is not on the grid points.…”

Section: Error Analysismentioning

confidence: 99%

“…where t ∈ [0, T ], ϕ : R m → R p is the terminal condition and f : [0, T ]×R p ×R p×m → R p is the generator. A pair (Y t , Z t ) is called an L 2 -adapted solution of the BSDE (1) if it is F t -adapted, square integrable and satisfies the equation (1).…”

mentioning

confidence: 99%

“…In this paper, we aim at designing a class of explicit high order one-step numerical schemes for solving BSDEs including a fifth order scheme and a sixth order scheme. Based on the nonlinear Feynman-Kac formula, by using the Itô-Taylor expansions and high order finite difference approximations for derivatives, we give the discretizations of the solutions of the BSDE (1). Then by using the predictioncorrection method based on the fourth order one-step scheme in [40], we propose an explicit fifth order one-step scheme for solving the BSDE (1).…”

mentioning

confidence: 99%

“…Based on the nonlinear Feynman-Kac formula, by using the Itô-Taylor expansions and high order finite difference approximations for derivatives, we give the discretizations of the solutions of the BSDE (1). Then by using the predictioncorrection method based on the fourth order one-step scheme in [40], we propose an explicit fifth order one-step scheme for solving the BSDE (1). Based on the fifth order scheme, we use the prediction-correction method again to further propose an explicit sixth order one-step scheme.…”

mentioning

confidence: 99%

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High order one-step methods for backward stochastic differential equations via Itô-Taylor expansion

Zhou¹,

Sun²

2022

DCDS-B

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<p style='text-indent:20px;'>In this work, by combining the Feynman-Kac formula with an Itô-Taylor expansion, we propose a class of high order one-step schemes for backward stochastic differential equations, which can achieve at most six order rate of convergence and only need the terminal conditions on the last one step. Numerical experiments are carried out to show the efficiency and high order accuracy of the proposed schemes.</p>

show abstract

Section: 2mentioning

confidence: 99%

“…where {a i } M i=1 are the roots of the Hermite polynomial of degree M and {w i } M i=1 are the corresponding weights [1]. In general, x + √ 2∆t n a i is not on the grid points.…”

Section: Error Analysismentioning

confidence: 99%

mentioning

confidence: 99%

mentioning

confidence: 99%

mentioning

confidence: 99%

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High order one-step methods for backward stochastic differential equations via Itô-Taylor expansion

Zhou¹,

Sun²

2022

DCDS-B

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Longest Relaxation Time of Relaxation Processes for Classical and Quantum Brownian Motion in a Potential: Escape Rate Theory Approach

Coffey¹,

Kalmykov²,

Титов³

et al. 2013

Advances in Chemical Physics

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Model‐free based adaptive finite time control with multilayer perceptron neural network estimation for a 10 DOF lower limb exoskeleton

Kenas,

Saadia,

Ababou

et al. 2023

Adaptive Control & Signal

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SummaryThis article presents a Model‐Free Adaptive Nonsingular Fast Terminal Sliding Mode Controller with Super Twisting and Multi‐Layer Perceptron (MLP) neural network for motion control of a 10 DOFs lower limb exoskeleton used in rehabilitation. The proposed controller employs a second‐order ultra‐local model to replace the complex dynamics of the exoskeleton and uses an MLP neural network to estimate the lumped disturbance of the ultra‐local model. To ensure accurate tracking of the desired trajectory and address the estimation errors of the MLP, an Adaptive Nonsingular Fast Terminal Sliding Mode Controller is introduced. Moreover, a Super Twisting approach is employed to eliminate the chattering phenomenon. The system's stability is analyzed using Lyapunov theory, and the desired trajectories are obtained from surface electromyography (EMG) signal measurements. The effectiveness of the proposed controller is validated through co‐simulation experiments using SolidWorks, Simscape Multibody, and MATLAB/Robotics Toolbox. Results demonstrate significant improvements in stability and precision compared to existing model‐free controllers.

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Handbook of Mathematical Functions.

Cited by 217 publications

References 0 publications

High order one-step methods for backward stochastic differential equations via Itô-Taylor expansion

High order one-step methods for backward stochastic differential equations via Itô-Taylor expansion

Longest Relaxation Time of Relaxation Processes for Classical and Quantum Brownian Motion in a Potential: Escape Rate Theory Approach

Model‐free based adaptive finite time control with multilayer perceptron neural network estimation for a 10 DOF lower limb exoskeleton

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