scholarly journals A New Simplified Weak Second-Order Scheme for Solving Stochastic Differential Equations with Jumps

Mathematics ◽  
2021 ◽  
Vol 9 (3) ◽  
pp. 224
Author(s):  
Yang Li ◽  
Yaolei Wang ◽  
Taitao Feng ◽  
Yifei Xin
Keyword(s):  
Differential Equations ◽  
Convergence Rate ◽  
Stochastic Differential Equations ◽  
Numerical Scheme ◽  
Second Order ◽  
Trapezoidal Rule ◽  
Order Convergence ◽  
Integration By Parts Formula ◽  
Order Scheme ◽  
Second Order Convergence

In this paper, we propose a new weak second-order numerical scheme for solving stochastic differential equations with jumps. By using trapezoidal rule and the integration-by-parts formula of Malliavin calculus, we theoretically prove that the numerical scheme has second-order convergence rate. To demonstrate the effectiveness and the second-order convergence rate, three numerical experiments are given.

Convergence of Recent Multistep Schemes for a Forward-Backward Stochastic Differential Equation

2015 ◽  
Vol 5 (4) ◽  
pp. 387-404 ◽  
Author(s):  
Jie Yang ◽  
Weidong Zhao
Keyword(s):  
Differential Equation ◽  
Exact Solution ◽  
Differential Equations ◽  
Convergence Rate ◽  
Stochastic Differential Equation ◽  
Stochastic Differential Equations ◽  
Convergence Analysis ◽  
Stochastic Control ◽  
Backward Stochastic Differential Equation ◽  
Order Convergence

AbstractConvergence analysis is presented for recently proposed multistep schemes, when applied to a special type of forward-backward stochastic differential equations (FB-SDEs) that arises in finance and stochastic control. The corresponding k-step scheme admits a k-order convergence rate in time, when the exact solution of the forward stochastic differential equation (SDE) is given. Our analysis assumes that the terminal conditions and the FBSDE coefficients are sufficiently regular.

An Explicit Second-Order Numerical Scheme to Solve Decoupled Forward Backward Stochastic Equations

2014 ◽  
Vol 4 (4) ◽  
pp. 368-385 ◽  
Author(s):  
Yu Fu ◽  
Weidong Zhao
Keyword(s):  
Integral Equation ◽  
Differential Equations ◽  
Stochastic Differential Equations ◽  
Numerical Experiments ◽  
Numerical Scheme ◽  
High Accuracy ◽  
Stochastic Equations ◽  
Explicit Scheme ◽  
Second Order ◽  
General Error

AbstractAn explicit numerical scheme is proposed for solving decoupled forward backward stochastic differential equations (FBSDE) represented in integral equation form. A general error inequality is derived for this numerical scheme, which also implies its stability. Error estimates are given based on this inequality, showing that the explicit scheme can be second-order. Some numerical experiments are carried out to illustrate the high accuracy of the proposed scheme.

scholarly journals Second-Order Symmetric Smoothed Particle Hydrodynamics Method for Transient Heat Conduction Problems with Initial Discontinuity

Processes ◽  
2018 ◽  
Vol 6 (11) ◽  
pp. 215 ◽  
Author(s):  
Zhanjie Song ◽  
Yaxuan Xing ◽  
Qingzhi Hou ◽  
Wenhuan Lu
Keyword(s):  
Heat Conduction ◽  
Convergence Rate ◽  
Smoothed Particle Hydrodynamics ◽  
Transient Heat Conduction ◽  
Second Order ◽  
Order Convergence ◽  
Particle Hydrodynamics ◽  
Smoothed Particle ◽  
Initial Discontinuity ◽  
Second Order Convergence

To eliminate the numerical oscillations appearing in the first-order symmetric smoothed particle hydrodynamics (FO-SSPH) method for simulating transient heat conduction problems with discontinuous initial distribution, this paper presents a second-order symmetric smoothed particle hydrodynamics (SO-SSPH) method. Numerical properties of both SO-SSPH and FO-SSPH are analyzed, including truncation error, numerical accuracy, convergence rate, and stability. Experimental results show that for transient heat conduction with initial smooth distribution, both FO-SSPH and SO-SSPH can achieve second order convergence rate, which is consistent with the theoretical analysis. However, for one- and two-dimensional conduction with initial discontinuity, the FO-SSPH method suffers from serious unphysical oscillations, which do not disappear over time, and hence it only achieves a first-order convergence rate; while the present SO-SSPH method can avoid unphysical oscillations and has second-order convergence rate. Therefore, the SO-SSPH method is a feasible tool for solving transient heat conduction problems with both smooth and discontinuous distributions, and it is easy to be extended to high dimensional cases.

A Multistep Scheme for Decoupled Forward-Backward Stochastic Differential Equations

2016 ◽  
Vol 9 (2) ◽  
pp. 262-288 ◽  
Author(s):  
Weidong Zhao ◽  
Wei Zhang ◽  
Lili Ju
Keyword(s):  
Differential Equations ◽  
Convergence Rate ◽  
Error Estimate ◽  
Stochastic Differential Equations ◽  
Numerical Experiments ◽  
Numerical Scheme ◽  
Backward Stochastic Differential Equations ◽  
Multistep Scheme ◽  
Theoretical Results ◽  
General Error

AbstractUpon a set of backward orthogonal polynomials, we propose a novel multi-step numerical scheme for solving the decoupled forward-backward stochastic differential equations (FBSDEs). Under Lipschtiz conditions on the coefficients of the FBSDEs, we first get a general error estimate result which implies zero-stability of the proposed scheme, and then we further prove that the convergence rate of the scheme can be of high order for Markovian FBSDEs. Some numerical experiments are presented to demonstrate the accuracy of the proposed multi-step scheme and to numerically verify the theoretical results.

scholarly journals Verification of Convergence Rates of Numerical Solutions for Parabolic Equations

2019 ◽  
Vol 2019 ◽  
pp. 1-10
Author(s):  
Darae Jeong ◽  
Yibao Li ◽  
Chaeyoung Lee ◽  
Junxiang Yang ◽  
Yongho Choi ◽  
...  
Keyword(s):  
Parabolic Equations ◽  
Convergence Rates ◽  
Numerical Solutions ◽  
Second Order ◽  
Order Convergence ◽  
Numerical Convergence ◽  
Verification Method ◽  
First Order ◽  
Convergence Test ◽  
Second Order Convergence

In this paper, we propose a verification method for the convergence rates of the numerical solutions for parabolic equations. Specifically, we consider the numerical convergence rates of the heat equation, the Allen–Cahn equation, and the Cahn–Hilliard equation. Convergence test results show that if we refine the spatial and temporal steps at the same time, then we have the second-order convergence rate for the second-order scheme. However, in the case of the first-order in time and the second-order in space scheme, we may have the first-order or the second-order convergence rates depending on starting spatial and temporal step sizes. Therefore, for a rigorous numerical convergence test, we need to perform the spatial and the temporal convergence tests separately.

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