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SCHWARZ METHOD FOR FINANCIAL ENGINEERING

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Journal of Computational Mathematics 2021年第4期39卷 538-555页

作者： Guangbao Guo weidong zhao Department of Statistics Shandong University of TechnologyZibo 255000China School of Mathematics Shandong UniversityJinan 250000China

Schwarz method is put forward to solve second order backward stochastic di erential equations(2BSDEs)in this *** will analyze uniqueness,convergence,stability and optimality of the proposed ***,several simulation results are presented to demonstrate the e ectiveness;several applications of the 2BSDEs are *** is concluded from these results that the proposed the method is powerful to calculate the 2BSDEs listing from the nancial engineering.

关键词： 2BSDE Schwarz method Domain decomposition Viscosity solution Stochas-tic volatility models

Euler-type schemes for weakly coupled forward-backward stochastic differential equations and optimal convergence analysis

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Frontiers of Mathematics in China 2015年第2期10卷 415-434页

作者： Wei ZHANG weidong zhao School of Mathematics Finance Institute Shandong University Jinan 250100 China

We introduce a new Euler-type scheme and its iterative algorithm for solving weakly coupled forward-backward stochastic differential equations （FBSDEs）. Although the schemes share some common features with the ones proposed by C. Bender and J. Zhang [Ann. Appl. Probab., 2008, 18： 143-177], less computational work is needed for our method. For both our schemes and the ones proposed by Bender and Zhang, we rigorously obtain first-order error estimates, which improve the half-order error estimates of Bender and Zhang. Moreover, numerical tests are given to demonstrate the first-order accuracy of the schemes.

关键词： Weakly coupled forward-backward stochastic differential equations (FBSDEs) Euler-type scheme time discretization first-order error estimate

ODE-Based Multistep Schemes for Backward Stochastic Differential Equations

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Numerical Mathematics(Theory,Methods and Applications) 2023年第4期16卷 1053-1086页

作者： Shuixin Fang weidong zhao School of Mathematics&Institute of Finance Shandong UniversityJinanShandong 250100P.R.China

In this paper,we explore a new approach to design and analyze numerical schemes for backward stochastic differential equations(BSDEs).By the nonlinear Feynman-Kac formula,we reformulate the BSDE into a pair of reference ordinary differential equations(ODEs),which can be directly discretized by many standard ODE solvers,yielding the corresponding numerical schemes for *** particular,by applying strong stability preserving(SSP)time discretizations to the reference ODEs,we can propose new SSP multistep schemes for *** analyses are rigorously performed to prove the consistency,stability and convergency of the proposed SSP multistep *** experiments are further carried out to verify our theoretical results and the capacity of the proposed SSP multistep schemes for solving complex associated problems.

关键词： Backward stochastic differential equation parabolic partial differential equation strong stability preserving linear multistep scheme high order discretization

Revisiting GM-CSF as an adjuvant for therapeutic vaccines

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Cellular & Molecular Immunology 2018年第2期15卷 187-189页

作者： weidong zhao Gan zhao Bin Wang Key Laboratory of Medical Molecular Virology of the Ministry of Health and Ministry of Education School of Basic Medical SciencesFudan UniversityShanghai200032China Department of Laboratory Medicine Clinical Medicine CollegeDali UniversityDali 200032China. PhD Key Laboratory of Medical Molecular Virology of the Ministry of Health and Ministry of EducationSchool of Basic Medical SciencesFudan University131 Dong An RoadShanghai200032China

Granulocyte macrophage colony stimulating factor(GM-CSF)is generally considered to be a proinflammatory cytokine;GM-CSF-based vaccines have been shown to elicit potent antitumor and antiviral immune responses in preclinical *** clinical trials,however,these effects were not as robust and sometimes even contradicted the findings from animal *** this commentary,we aim to discuss the latest advances in this field,hoping that an upto-date and comprehensive understanding of this molecule will help others design effective GM-CSF-based strategies for antiviral and anticancer vaccines.

关键词： vaccines clinical stimulating

Finite Element Methods for Nonlinear Backward Stochastic Partial Differential Equations and Their Error Estimates

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Advances in Applied Mathematics and Mechanics 2020年第6期12卷 1457-1480页

作者： Xu Yang weidong zhao School of Mathematics China University of Mining and TechnologyXuzhouJiangsu 221116China School of Mathematics Shandong UniversityJinanShandong 250100China

In this paper,we consider numerical approximation of a class of nonlinear backward stochastic partial differential equations(BSPDEs).By using finite element methods in the physical space domain and the Euler method in the time domain,we propose a spatial finite element semi-discrete scheme and a spatio-temporal full discrete scheme for solving the *** of the schemes are rigorously analyzed and theoretical error estimates with convergence rates are obtained.

关键词： Backward stochastic partial differential equations finite element method error estimate.

Strong Convergence Analysis of Split-Step q-Scheme for Nonlinear Stochastic Differential Equations with Jumps

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Advances in Applied Mathematics and Mechanics 2016年第6期8卷 1004-1022页

作者： Xu Yang weidong zhao School of Mathematics Shandong UniversityJinan 250100China

In this paper,we investigate the mean-square convergence of the split-step q-scheme for nonlinear stochastic differential equations with *** some standard assumptions,we rigorously prove that the strong rate of convergence of the splitstep q-scheme in strong sense is one *** numerical experiments are carried out to assert our theoretical result.

关键词： Split-step scheme strong convergence stochastic differential equation jumpdiffusion,one-side Lipschitz condition

Sinc-Multistep Schemes for Forward Backward Stochastic Differential Equations

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Advances in Applied Mathematics and Mechanics 2023年第3期15卷 737-768页

作者： Xu Wang weidong zhao School of Mathematics&Institute of Finance Shandong UniversityJinanShandong 250100China

In this work,by combining the multistep discretization in time and the Sinc quadrature rule for approximating the conditional mathematical expectations,we will propose new fully discrete multistep schemes called“Sinc-multistep schemes”for forward backward stochastic differential equations(FBSDEs).The schemes avoid spatial interpolations and admit high order of *** stability and the K-th order error estimates in time for the K-step Sinc multistep schemes are theoretically proved(1≤K≤6).This seems to be the first time for analyzing fully time-space discrete multistep schemes for *** examples are also presented to demonstrate the effectiveness,stability,and high order of convergence of the proposed schemes.

关键词： Forward backward stochastic differential equations multistep schemes Sinc quadrature rule error estimates

A FIRST-ORDER NUMERICAL SCHEME FOR FORWARD-BACKWARD STOCHASTIC DIFFERENTIAL EQUATIONS IN BOUNDED DOMAINS

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Journal of Computational Mathematics 2018年第2期36卷 237-258页

作者： Jie Yang Guannan Zhang weidong zhao School of Mathematics and Statistics Shandong University Weihai Shandong 264209 China Computer Science and Mathematics Division Oak Ridge National Laboratory Oak Ridge Tennessee 37831 United States School of Mathematics Shandong University Jinan Shandong 250100 China

We propose a novel numerical scheme for decoupled forward-backward stochastic differ- ential equations （FBSDEs） in bounded domains, which corresponds to a class of nonlinear parabolic partial differential equations with Dirichlet boundary conditions. The key idea is to exploit the regularity of the solution （Yt,Zt） with respect to Xt to avoid direct ap- proximation of the involved random exit time. Especially, in the one-dimensional case, we prove that the probability of Xt exiting the domain within At is on the order of O（（△t）ε exp（--1/（△t）2ε））, if the distance between the start point X0 and the boundary is 1 g at least on the order of O（△t）^1/2-ε ） for any fixed c 〉 0. Hence, in spatial discretization, we set the mesh size △x - （9（（At）^1/2-ε ）, so that all the interior grid points are sufficiently far from the boundary, which makes the error caused by the exit time decay sub-exponentially with respect to △t. The accuracy of the approximate solution near the boundary can be guaranteed by means of high-order piecewise polynomial interpolation. Our method is developed using the implicit Euler scheme and cubic polynomial interpolation, which leads to an overall first-order convergence rate with respect to △t.

关键词： Forward-backward stochastic differential equations Exit time Dirichlet bound-ary conditions Implicit Euler scheme.

A SPARSE-GRID METHOD FOR MULTI-DIMENSIONAL BACKWARD STOCHASTIC DIFFERENTIAL EQUATIONS

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Journal of Computational Mathematics 2013年第3期31卷 221-248页

作者： Guannan Zhang Max Gunzburger weidong zhao Department of Scientific Computing Florida State University Tallahassee FL 32306Computer Science and Mathematics Division Oak Ridge National Lab Oak Ridge TN 37831 Department of Scientific Computing Florida State University Tallahassee FL 32306 School of Mathematics Shandong University Jinan Shandong 250100 China

A sparse-grid method for solving multi-dimensional backward stochastic differential equations （BSDEs） based on a multi-step time discretization scheme [31] is presented. In the multi-dimensional spatial domain, i.e. the Brownian space, the conditional mathe- matical expectations derived from the original equation are approximated using sparse-grid Gauss-Hermite quadrature rule and （adaptive） hierarchical sparse-grid interpolation. Error estimates are proved for the proposed fully-discrete scheme for multi-dimensional BSDEs with certain types of simplified generator functions. Finally, several numerical examples are provided to illustrate the accuracy and efficiency of our scheme.

关键词： Backward stochastic differential equations Multi-step scheme Gauss-Hermite quadrature rule Adaptive hierarchical basis Sparse grids.