scholarly journals Numerical Picard Iteration Methods for Simulation of Non-Lipschitz Stochastic Differential Equations

Symmetry ◽  
2020 ◽  
Vol 12 (3) ◽  
pp. 383 ◽  
Author(s):  
Jürgen Geiser
Keyword(s):  
Differential Equations ◽  
Stochastic Differential Equations ◽  
Operator Splitting ◽  
Picard Iteration ◽  
Computational Time ◽  
Iteration Methods ◽  
Exponential Formula ◽  
Exponential Methods ◽  
The One ◽  
Stochastic Phenomena

In this paper, we present splitting approaches for stochastic/deterministic coupled differential equations, which play an important role in many applications for modelling stochastic phenomena, e.g., finance, dynamics in physical applications, population dynamics, biology and mechanics. We are motivated to deal with non-Lipschitz stochastic differential equations, which have functions of growth at infinity and satisfy the one-sided Lipschitz condition. Such problems studied for example in stochastic lubrication equations, while we deal with rational or polynomial functions. Numerically, we propose an approximation, which is based on Picard iterations and applies the Doléans-Dade exponential formula. Such a method allows us to approximate the non-Lipschitzian SDEs with iterative exponential methods. Further, we could apply symmetries with respect to decomposition of the related matrix-operators to reduce the computational time. We discuss the different operator splitting approaches for a nonlinear SDE with multiplicative noise and compare this to standard numerical methods.

Comparison Theorem for any Solutions of Backward Stochastic Differential Equations and its Application

2012 ◽  
Vol 524-527 ◽  
pp. 3801-3804
Author(s):  
Shi Yu Li ◽  
Wu Jun Gao ◽  
Jin Hui Wang
Keyword(s):  
Differential Equations ◽  
Stochastic Differential Equations ◽  
Comparison Theorem ◽  
Backward Stochastic Differential Equations ◽  
Stochastic Equations ◽  
Local Martingale ◽  
One Dimensional ◽  
Lipschitz Continuous ◽  
The One ◽  
Uniformly Lipschitz

ƒIn this paper, we study the one-dimensional backward stochastic equations driven by continuous local martingale. We establish a generalized the comparison theorem for any solutions where the coefficient is uniformly Lipschitz continuous in z and is equi-continuous in y.

An introduction to numerical methods for stochastic differential equations

Acta Numerica ◽  
1999 ◽  
Vol 8 ◽  
pp. 197-246 ◽  
Author(s):  
Eckhard Platen
Keyword(s):  
Numerical Analysis ◽  
Numerical Methods ◽  
Differential Equations ◽  
Stochastic Differential Equations ◽  
The Other ◽  
Weak Approximation ◽  
Approximation Methods ◽  
Unified Framework ◽  
Stochastic Nature ◽  
The One

This paper aims to give an overview and summary of numerical methods for the solution of stochastic differential equations. It covers discrete time strong and weak approximation methods that are suitable for different applications. A range of approaches and results is discussed within a unified framework. On the one hand, these methods can be interpreted as generalizing the well-developed theory on numerical analysis for deterministic ordinary differential equations. On the other hand they highlight the specific stochastic nature of the equations. In some cases these methods lead to completely new and challenging problems.

scholarly journals Improving multilevel Monte Carlo for stochastic differential equations with application to the Langevin equation

2015 ◽  
Vol 471 (2176) ◽  
pp. 20140679 ◽  
Author(s):  
Eike H. Müller ◽  
Rob Scheichl ◽  
Tony Shardlow
Keyword(s):  
Monte Carlo ◽  
Differential Equations ◽  
Stochastic Differential Equations ◽  
Langevin Equation ◽  
Operator Splitting ◽  
Random Variables ◽  
Weak Approximation ◽  
Multilevel Monte Carlo ◽  
Discrete Random Variables ◽  
Modified Equations

This paper applies several well-known tricks from the numerical treatment of deterministic differential equations to improve the efficiency of the multilevel Monte Carlo (MLMC) method for stochastic differential equations (SDEs) and especially the Langevin equation. We use modified equations analysis as an alternative to strong-approximation theory for the integrator, and we apply this to introduce MLMC for Langevin-type equations with integrators based on operator splitting. We combine this with extrapolation and investigate the use of discrete random variables in place of the Gaussian increments, which is a well-known technique for the weak approximation of SDEs. We show that, for small-noise problems, discrete random variables can lead to an increase in efficiency of almost two orders of magnitude for practical levels of accuracy.

A general model system related to affine stochastic differential equations

2020 ◽  
Vol 21 (01) ◽  
pp. 2150001
Author(s):  
Enrico Bernardi ◽  
Vinayak Chuni ◽  
Alberto Lanconelli
Keyword(s):  
Differential Equations ◽  
Interest Rates ◽  
Stochastic Differential Equations ◽  
Factor Model ◽  
Direct Proof ◽  
Auxiliary Process ◽  
The One ◽  
Yield Factor ◽  
General Method

We link a general method for modeling random phenomena using systems of stochastic differential equations (SDEs) to the class of affine SDEs. This general construction emphasizes the central role of the Duffie–Kan system [Duffie and Kan, A yield-factor model of interest rates, Math. Finance 6 (1996) 379–406] as a model for first-order approximations of a wide class of nonlinear systems perturbed by noise. We also specialize to a two-dimensional framework and propose a direct proof of the Duffie–Kan theorem which does not passes through the comparison with an auxiliary process. Our proof produces a scheme to obtain an explicit representation of the solution once the one-dimensional square root process is assigned.

scholarly journals Averaging Principle for Backward Stochastic Differential Equations

2021 ◽  
Vol 2021 ◽  
pp. 1-10
Author(s):  
Yuanyuan Jing ◽  
Zhi Li
Keyword(s):  
Differential Equations ◽  
Stochastic Differential Equations ◽  
Stochastic Systems ◽  
Backward Stochastic Differential Equations ◽  
Averaging Principle ◽  
Mean Square ◽  
The One

The averaging principle for BSDEs and one-barrier RBSDEs, with Lipschitz coefficients, is investigated. An averaged BSDEs for the original BSDEs is proposed, as well as the one-barrier RBSDEs, and their solutions are quantitatively compared. Under some appropriate assumptions, the solutions to original systems can be approximated by the solutions to averaged stochastic systems in the sense of mean square.

scholarly journals The Fractional View Analysis of Polytropic Gas, Unsteady Flow System

2021 ◽  
Vol 2021 ◽  
pp. 1-17
Author(s):  
Hassan Khan ◽  
Saeed Islam ◽  
Muhammad Arif
Keyword(s):  
Exact Solution ◽  
Differential Equations ◽  
Unsteady Flow ◽  
Fractional Order ◽  
Physical System ◽  
Computational Time ◽  
Integer Order ◽  
Polytropic Gas ◽  
Variational Iteration ◽  
Iteration Methods

Generally, the differential equations of integer order do not properly model various phenomena in different areas of science and engineering as compared to differential equations of fractional order. The fractional-order differential equations provide the useful dynamics of the physical system and thus provide the innovative and effective information about the given physical system. Keeping in view the above properties of fractional calculus, the present article is related to the analytical solution of the time-fractional system of equations which describe the unsteady flow of polytropic gas dynamics. The present method provides the series form solution with easily computable components and a higher rate of convergence towards the targeted problem’s exact solution. The present techniques are straightforward and effective for dealing with the solutions of fractional-order problems. The fractional derivatives are expressed in terms of the Caputo operator. The targeted problems’ solutions are calculated using the Adomian decomposition method and variational iteration methods along with Shehu transformation. In the current procedures, we first applied the Shehu transform to reduce the problems into a more straightforward form and then implemented the decomposition and variational iteration methods to achieve the problems’ comprehensive results. The solution of the nonlinear equations of unsteady flow of a polytropic gas at various fractional orders of the derivative is the core point of the present study. The solution of the proposed fractional model is plotted via two- and three-dimensional graphs. It is investigated that each problem’s solution-graphs are best fitted with each other and with the exact solution. The convergence of fractional-order problems can be observed towards the solution of integer-order problems. Less computational time is the major attraction of the suggested methods. The present work will be considered a useful tool to handle the solution of fractional partial differential equations.

scholarly journals Convergence and stability of the one-leg θ method for stochastic differential equations with piecewise continuous arguments

Filomat ◽  
2019 ◽  
Vol 33 (3) ◽  
pp. 945-960
Author(s):  
Yulan Lu ◽  
Minghui Song ◽  
Mingzhu Liu
Keyword(s):  
Differential Equations ◽  
Exponential Stability ◽  
Stochastic Differential Equations ◽  
Step Size ◽  
Almost Sure Exponential Stability ◽  
Exponentially Stable ◽  
Piecewise Continuous ◽  
Piecewise Continuous Arguments ◽  
The Stability ◽  
The One

The equivalent relation is established here about the stability of stochastic differential equations with piecewise continuous arguments(SDEPCAs) and that of the one-leg ? method applied to the SDEPCAs. Firstly, the convergence of the one-leg ? method to SDEPCAs under the global Lipschitz condition is proved. Secondly, it is proved that the SDEPCAs are pth(p 2 (0; 1)) moment exponentially stable if and only if the one-leg ? method is pth moment exponentially stable for some sufficiently small step-size. Thirdly, the corollaries that the pth moment exponential stability of the SDEPCAs (the one-leg ? method) implies the almost sure exponential stability of the SDEPCAs (the one-leg ? method) are given. Finally, numerical simulations are provided to illustrate the theoretical results.

Normal form method in search for periodic solutions of ordinary differential equations. Case of the fourth order

2018 ◽  
pp. 24-34
Author(s):  
V. F. Edneral ◽  
O. D. Timofeevskaya
Keyword(s):  
Normal Form ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Periodic Solutions ◽  
Fourth Order ◽  
Numerical Solutions ◽  
Numerical Calculations ◽  
Computational Time ◽  
Resonant Normal Form ◽  
Normal Form Method

Introduction:The method of resonant normal form is based on reducing a system of nonlinear ordinary differential equations to a simpler form, easier to explore. Moreover, for a number of autonomous nonlinear problems, it is possible to obtain explicit formulas which approximate numerical calculations of families of their periodic solutions. Replacing numerical calculations with their precalculated formulas leads to significant savings in computational time. Similar calculations were made earlier, but their accuracy was insufficient, and their complexity was very high.Purpose:Application of the resonant normal form method and a software package developed for these purposes to fourth-order systems in order to increase the calculation speed.Results:It has been shown that with the help of a single algorithm it is possible to study equations of high orders (4th and higher). Comparing the tabulation of the obtained formulas with the numerical solutions of the corresponding equations shows good quantitative agreement. Moreover, the speed of calculation by prepared approximating formulas is orders of magnitude greater than the numerical calculation speed. The obtained approximations can also be successfully applied to unstable solutions. For example, in the Henon — Heyles system, periodic solutions are surrounded by chaotic solutions and, when numerically integrated, the algorithms are often unstable on them.Practical relevance:The developed approach can be used in the simulation of physical and biological systems.

Stochastic Differential Equations for Power Law Behaviors

2012 ◽  
Author(s):  
Bo Jiang ◽  
Roger Brockett ◽  
Weibo Gong ◽  
Don Towsley
Keyword(s):  
Differential Equations ◽  
Stochastic Differential Equations ◽  
Power Law
Sign in / Sign up

Export Citation Format

Share Document