scholarly journals A High Order Accurate and Effective Scheme for Solving Markovian Switching Stochastic Models

Mathematics ◽  
2021 ◽  
Vol 9 (6) ◽  
pp. 588
Author(s):  
Yang Li ◽  
Taitao Feng ◽  
Yaolei Wang ◽  
Yifei Xin
Keyword(s):  
Markov Chain ◽  
Convergence Rate ◽  
Stochastic Models ◽  
Stochastic Analysis ◽  
Numerical Experiments ◽  
Numerical Scheme ◽  
Special Property ◽  
Weak Order ◽  
Markovian Switching ◽  
Theoretical Results

In this paper, we propose a new weak order 2.0 numerical scheme for solving stochastic differential equations with Markovian switching (SDEwMS). Using the Malliavin stochastic analysis, we theoretically prove that the new scheme has local weak order 3.0 convergence rate. Combining the special property of Markov chain, we study the effects from the changes of state space on the convergence rate of the new scheme. Two numerical experiments are given to verify the theoretical results.

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 Analysis of a Two-Level Algorithm for HDG Methods for Diffusion Problems

2016 ◽  
Vol 19 (5) ◽  
pp. 1435-1460 ◽  
Author(s):  
Binjie Li ◽  
Xiaoping Xie ◽  
Shiquan Zhang
Keyword(s):  
Convergence Rate ◽  
Numerical Experiments ◽  
Sharp Estimate ◽  
Model Problem ◽  
The Other ◽  
Extended Version ◽  
Weak Galerkin ◽  
Hybridizable Discontinuous Galerkin ◽  
Theoretical Results ◽  
Diffusion Problems

AbstractThis paper analyzes an abstract two-level algorithm for hybridizable discontinuous Galerkin (HDG) methods in a unified fashion. We use an extended version of the Xu-Zikatanov (X-Z) identity to derive a sharp estimate of the convergence rate of the algorithm, and show that the theoretical results also are applied to weak Galerkin (WG) methods. The main features of our analysis are twofold: one is that we only need the minimal regularity of the model problem; the other is that we do not require the triangulations to be quasi-uniform. Numerical experiments are provided to confirm the theoretical results.

Error Estimates for the Time Discretization of a Semilinear Integrodifferential Parabolic Problem with Unknown Memory Kernel

2017 ◽  
Vol 10 (1) ◽  
pp. 116-144
Author(s):  
Marijke Grimmonprez ◽  
Karel Van Bockstal ◽  
Marián Slodička
Keyword(s):  
Numerical Experiments ◽  
Numerical Scheme ◽  
Parabolic Problem ◽  
Time Discretization ◽  
Integral Form ◽  
Robin Boundary Condition ◽  
Memory Kernel ◽  
Euler’S Method ◽  
Robin Boundary ◽  
Theoretical Results

AbstractThis paper is devoted to the study of an inverse problem containing a semilinear integrodifferential parabolic equation with an unknown memory kernel. This equation is accompanied by a Robin boundary condition. The missing kernel can be recovered from an additional global measurement in integral form. In this contribution, an error analysis is performed for a time-discrete numerical scheme based on Backward Euler's Method. The theoretical results are supported by some numerical experiments.

scholarly journals A Reliable numerical scheme for a computer virus propagation model

2021 ◽  
Author(s):  
Hoai Thu Pham ◽  
Manh Tuan Hoang
Keyword(s):  
Numerical Experiments ◽  
Numerical Scheme ◽  
Computer Virus ◽  
Propagation Model ◽  
Virus Propagation ◽  
Essential Properties ◽  
Computer Virus Propagation Model ◽  
Nsfd Scheme ◽  
Theoretical Results ◽  
Good Agreement

In this paper, we apply the Mickens’methodology to construct a dynamically consistentnonstandard finite difference (NSFD) scheme for acomputer virus propagation model. It is proved thatthe constructed NSFD scheme correctly preservesessential mathematical features of the continuous-timemodel, which are positivity, boundedness and asymptotic stability. Consequently, we obtain an effectivenumerical scheme that can provide reliable approximations for the computer virus propagation model.Meanwhile, some typical standard finite differenceschemes fail to preserve the essential properties ofthe computer virus propagation model; hence, theycan generate numerical approximations which arenot only negative but also unstable. Finally, a setof numerical experiments is performed to supportthe theoretical results as well as to demonstrate theadvantage of the NSFD scheme over standard ones.As we expected, there is a good agreement betweenthe numerical results and theoretical assertions.

scholarly journals A New GMRES(m) Method for Markov Chains

2013 ◽  
Vol 2013 ◽  
pp. 1-7 ◽  
Author(s):  
Bing-Yuan Pu ◽  
Ting-Zhu Huang ◽  
Chun Wen
Keyword(s):  
Markov Chain ◽  
Convergence Rate ◽  
Numerical Experiments ◽  
Gmres Method ◽  
Stationary Probability ◽  
Probability Vector ◽  
Vector Extrapolation ◽  
Restarted Gmres ◽  
Stationary Probability Vector ◽  
Irreducible Markov Chain

This paper presents a class of new accelerated restarted GMRES method for calculating the stationary probability vector of an irreducible Markov chain. We focus on the mechanism of this new hybrid method by showing how to periodically combine the GMRES and vector extrapolation method into a much efficient one for improving the convergence rate in Markov chain problems. Numerical experiments are carried out to demonstrate the efficiency of our new algorithm on several typical Markov chain problems.

scholarly journals A numerical solution for a class of time fractional diffusion equations with delay

2017 ◽  
Vol 27 (3) ◽  
pp. 477-488 ◽  
Author(s):  
Vladimir G. Pimenov ◽  
Ahmed S. Hendy
Keyword(s):  
Numerical Solution ◽  
Numerical Experiments ◽  
Numerical Scheme ◽  
Fixed Time ◽  
Fractional Diffusion ◽  
Diffusion Equations ◽  
Discrete Energy ◽  
Fractional Diffusion Equations ◽  
Equations With Delay ◽  
Theoretical Results

AbstractThis paper describes a numerical scheme for a class of fractional diffusion equations with fixed time delay. The study focuses on the uniqueness, convergence and stability of the resulting numerical solution by means of the discrete energy method. The derivation of a linearized difference scheme with convergence orderO(τ2−α+h4) in L∞-norm is the main purpose of this study. Numerical experiments are carried out to support the obtained theoretical results.

scholarly journals An Intuitive Introduction to Fractional and Rough Volatilities

Mathematics ◽  
2021 ◽  
Vol 9 (9) ◽  
pp. 994
Author(s):  
Elisa Alòs ◽  
Jorge A. León
Keyword(s):  
Malliavin Calculus ◽  
Numerical Experiments ◽  
Implied Volatility ◽  
Natural Approach ◽  
Volatility Models ◽  
Implied Volatility Surface ◽  
Short Memory ◽  
White Type ◽  
Volatility Surface ◽  
Theoretical Results

Here, we review some results of fractional volatility models, where the volatility is driven by fractional Brownian motion (fBm). In these models, the future average volatility is not a process adapted to the underlying filtration, and fBm is not a semimartingale in general. So, we cannot use the classical Itô’s calculus to explain how the memory properties of fBm allow us to describe some empirical findings of the implied volatility surface through Hull and White type formulas. Thus, Malliavin calculus provides a natural approach to deal with the implied volatility without assuming any particular structure of the volatility. The aim of this paper is to provides the basic tools of Malliavin calculus for the study of fractional volatility models. That is, we explain how the long and short memory of fBm improves the description of the implied volatility. In particular, we consider in detail a model that combines the long and short memory properties of fBm as an example of the approach introduced in this paper. The theoretical results are tested with numerical experiments.

scholarly journals An efficient algorithm for estimating the parameters of superimposed exponential signals in multiplicative and additive noise

2013 ◽  
Vol 23 (1) ◽  
pp. 117-129 ◽  
Author(s):  
Jiawen Bian ◽  
Huiming Peng ◽  
Jing Xing ◽  
Zhihui Liu ◽  
Hongwei Li
Keyword(s):  
Parameter Estimation ◽  
Convergence Rate ◽  
Efficient Algorithm ◽  
Numerical Experiments ◽  
Additive Noise ◽  
Mean Squared Errors ◽  
Least Squares Estimators ◽  
The Mean ◽  
Newton Raphson ◽  
Raphson Algorithm

This paper considers parameter estimation of superimposed exponential signals in multiplicative and additive noise which are all independent and identically distributed. A modified Newton-Raphson algorithm is used to estimate the frequencies of the considered model, which is further used to estimate other linear parameters. It is proved that the modified Newton- Raphson algorithm is robust and the corresponding estimators of frequencies attain the same convergence rate with Least Squares Estimators (LSEs) under the same noise conditions, but it outperforms LSEs in terms of the mean squared errors. Finally, the effectiveness of the algorithm is verified by some numerical experiments.

scholarly journals On the numerical solutions for nonlinear Volterra-Fredholm integral equations

2022 ◽  
Vol 40 ◽  
pp. 1-11
Author(s):  
Parviz Darania ◽  
Saeed Pishbin
Keyword(s):  
Nonlinear System ◽  
Integral Equations ◽  
Numerical Experiments ◽  
Numerical Solutions ◽  
Coefficient Matrix ◽  
Collocation Methods ◽  
Fredholm Integral Equations ◽  
Stability Estimates ◽  
Lower Triangular ◽  
Theoretical Results

In this note, we study a class of multistep collocation methods for the numerical integration of nonlinear Volterra-Fredholm Integral Equations (V-FIEs). The derived method is characterized by a lower triangular or diagonal coefficient matrix of the nonlinear system for the computation of the stages which, as it is known, can beexploited to get an efficient implementation. Convergence analysis and linear stability estimates are investigated. Finally numerical experiments are given, which confirm our theoretical results.

The Study on Convergence and Convergence Rate of Genetic Algorithm Based on an Absorbing Markov Chain

2012 ◽  
Vol 239-240 ◽  
pp. 1511-1515 ◽  
Author(s):  
Jing Jiang ◽  
Li Dong Meng ◽  
Xiu Mei Xu
Keyword(s):  
Genetic Algorithm ◽  
Markov Chain ◽  
Convergence Rate ◽  
Hitting Time ◽  
Sufficient Condition ◽  
First Hitting Time ◽  
Absorbing Markov Chain ◽  
Expected Hitting Time ◽  
Optimization Efficiency ◽  
Theoretical Issues

The study on convergence of GA is always one of the most important theoretical issues. This paper analyses the sufficient condition which guarantees the convergence of GA. Via analyzing the convergence rate of GA, the average computational complexity can be implied and the optimization efficiency of GA can be judged. This paper proposes the approach to calculating the first expected hitting time and analyzes the bounds of the first hitting time of concrete GA using the proposed approach.

Sign in / Sign up

Export Citation Format

Share Document