scholarly journals Implicit Numerical Solutions for Solving Stochastic Differential Equations with Jumps

2014 ◽  
Vol 2014 ◽  
pp. 1-11
Author(s):  
Ying Du ◽  
Changlin Mei
Keyword(s):  
Differential Equations ◽  
Stochastic Differential Equations ◽  
Numerical Solutions ◽  
Implicit Methods ◽  
Mean Square Stability ◽  
Test Equation ◽  
The Mean ◽  
Taylor Method ◽  
Strong Order ◽  
Linear Scalar

To realize the applications of stochastic differential equations with jumps, much attention has recently been paid to the construction of efficient numerical solutions of the equations. Considering the fact that the use of the explicit methods often results in instability and inaccurate approximations in solving stochastic differential equations, we propose two implicit methods, theθ-Taylor method and the balancedθ-Taylor method, for numerically solving the stochastic differential equation with jumps and prove that the numerical solutions are convergent with strong order 1.0. For a linear scalar test equation, the mean-square stability regions of the methods are derived. Finally, numerical examples are given to evaluate the performance of the methods.

scholarly journals A Compensated Numerical Method for Solving Stochastic Differential Equations with Variable Delays and Random Jump Magnitudes

2015 ◽  
Vol 2015 ◽  
pp. 1-11
Author(s):  
Ying Du ◽  
Changlin Mei
Keyword(s):  
Differential Equations ◽  
Stochastic Differential Equations ◽  
Numerical Method ◽  
Analytical Solutions ◽  
Numerical Solutions ◽  
Mean Square ◽  
Variable Delays ◽  
Mean Square Stability ◽  
The Mean ◽  
Random Jump

Stochastic differential equations with jumps are of a wide application area especially in mathematical finance. In general, it is hard to obtain their analytical solutions and the construction of some numerical solutions with good performance is therefore an important task in practice. In this study, a compensated split-stepθmethod is proposed to numerically solve the stochastic differential equations with variable delays and random jump magnitudes. It is proved that the numerical solutions converge to the analytical solutions in mean-square with the approximate rate of 1/2. Furthermore, the mean-square stability of the exact solutions and the numerical solutions are investigated via a linear test equation and the results show that the proposed numerical method shares both the mean-square stability and the so-called A-stability.

scholarly journals Mean-Square Stability of Split-Step Theta Milstein Methods for Stochastic Differential Equations

2018 ◽  
Vol 2018 ◽  
pp. 1-13 ◽  
Author(s):  
Mahmoud A. Eissa ◽  
Haiying Zhang ◽  
Yu Xiao
Keyword(s):  
Numerical Methods ◽  
Differential Equations ◽  
Stochastic Differential Equations ◽  
Mean Square ◽  
Step Size ◽  
Mean Square Stability ◽  
Test Equation ◽  
The Mean ◽  
The Stability ◽  
And Diffusion

The fundamental analysis of numerical methods for stochastic differential equations (SDEs) has been improved by constructing new split-step numerical methods. In this paper, we are interested in studying the mean-square (MS) stability of the new general drifting split-step theta Milstein (DSSθM) methods for SDEs. First, we consider scalar linear SDEs. The stability function of the DSSθM methods is investigated. Furthermore, the stability regions of the DSSθM methods are compared with those of test equation, and it is proved that the methods with θ≥3/2 are stochastically A-stable. Second, the nonlinear stability of DSSθM methods is studied. Under a coupled condition on the drifting and diffusion coefficients, it is proved that the methods with θ>1/2 can preserve the MS stability of the SDEs with no restriction on the step-size. Finally, numerical examples are given to examine the accuracy of the proposed methods under the stability conditions in approximation of SDEs.

Itô-Taylor Schemes for Solving Mean-Field Stochastic Differential Equations

2017 ◽  
Vol 10 (4) ◽  
pp. 798-828 ◽  
Author(s):  
Yabing Sun ◽  
Jie Yang ◽  
Weidong Zhao
Keyword(s):  
Differential Equations ◽  
Stochastic Differential Equations ◽  
Taylor Expansion ◽  
Mean Field ◽  
Strong Convergence Theorem ◽  
The Mean ◽  
New Formula ◽  
Strong Order ◽  
Taylor Scheme ◽  
Theoretical Results

AbstractThis paper is devoted to numerical methods for mean-field stochastic differential equations (MSDEs). We first develop the mean-field Itô formula and mean-field Itô-Taylor expansion. Then based on the new formula and expansion, we propose the Itô-Taylor schemes of strong order γ and weak order η for MSDEs, and theoretically obtain the convergence rate γ of the strong Itô-Taylor scheme, which can be seen as an extension of the well-known fundamental strong convergence theorem to the mean-field SDE setting. Finally some numerical examples are given to verify our theoretical results.

scholarly journals Numerical Spline Method for Simulation of Stochastic Differential Equations systems: طريقة شرائحية عددية لمحاكاة حل نظم من المعادلات التفاضلية العشوائية

2021 ◽  
Vol 5 (4) ◽  
pp. 130-111
Author(s):  
Suliman M. Mahmoud, Ahmad Al-Wassouf, Ali S. Ehsaan Suliman M. Mahmoud, Ahmad Al-Wassouf, Ali S. Ehsaan
Keyword(s):  
Differential Equations ◽  
Stochastic Differential Equations ◽  
Continuous Process ◽  
Mean Square ◽  
Spline Method ◽  
Third Order ◽  
Mean Square Stability ◽  
The Mean ◽  
Linear Problems ◽  
Two Parameters

In this paper, numerical spline method is presented with collocation two parameters for solving systems of multi-dimensional stochastic differential equations (SDEs). Multi-Wiener's time-continuous process is simulated as a discrete process, and then the mean-square stability of proposed method when applied to a system of two-dimensional linear SDEs is studied. The study shows that the method is mean-square stability and third-order convergent when applied to a system of linear and nonlinear SDEs. Moreover, the effectiveness of our method was tested by solving two test linear and non-linear problems. The numerical results show that the accuracy and applicability of the proposed method are worthy of attention.

scholarly journals Stability of a Class of Nonlinear Neutral Stochastic Differential Equations with Variable Time Delays

2012 ◽  
Vol 20 (1) ◽  
pp. 467-488 ◽  
Author(s):  
Meng Wu ◽  
Nanjing Huang ◽  
Changwen Zhao
Keyword(s):  
Differential Equations ◽  
Stochastic Differential Equations ◽  
Time Delays ◽  
Fixed Point Theory ◽  
Point Theory ◽  
Mean Square ◽  
Variable Time ◽  
Mean Square Stability ◽  
The Mean ◽  
Necessary And Sufficient

Abstract In this paper, we study the mean square asymptotic stability of a class of generalized nonlinear neutral stochastic differential equations with variable time delays by using fixed point theory. An asymptotic mean square stability theorem with a necessary and sufficient condition is proved which improves and generalizes some well-known results. Finally, two examples are given to illustrate our results.

scholarly journals The Mean Stability Criteria in terms of Two Measures for Stochastic Differential Equations with Coefficient’s Uncertainty

2015 ◽  
Vol 2015 ◽  
pp. 1-7
Author(s):  
Rui Zhang ◽  
Yinjing Guo ◽  
Xiangrong Wang ◽  
Xueqing Zhang
Keyword(s):  
Optimal Control ◽  
Differential Equations ◽  
Stochastic Differential Equations ◽  
Stochastic Stability ◽  
Stochastic Systems ◽  
Stability Criteria ◽  
Two Measures ◽  
The Mean ◽  
The Stability

This paper extends the stochastic stability criteria of two measures to the mean stability and proves the stability criteria for a kind of stochastic Itô’s systems. Moreover, by applying optimal control approaches, the mean stability criteria in terms of two measures are also obtained for the stochastic systems with coefficient’s uncertainty.

scholarly journals Exponential Mean-Square Stability of Numerical Solutions to Stochastic Differential Equations

2003 ◽  
Vol 6 ◽  
pp. 297-313 ◽  
Author(s):  
Desmond J. Higham ◽  
Xuerong Mao ◽  
Andrew M. Stuart
Keyword(s):  
Differential Equations ◽  
Stochastic Differential Equations ◽  
Numerical Method ◽  
Finite Time ◽  
Mean Square ◽  
Convergence Conditions ◽  
Finite Time Convergence ◽  
Mean Square Stability ◽  
Exponential Mean Square Stability ◽  
Positive Results

AbstractPositive results are proved here about the ability of numerical simulations to reproduce the exponential mean-square stability of stochastic differential equations (SDEs). The first set of results applies under finite-time convergence conditions on the numerical method. Under these conditions, the exponential mean-square stability of the SDE and that of the method (for sufficiently small step sizes) are shown to be equivalent, and the corresponding second-moment Lyapunov exponent bounds can be taken to be arbitrarily close. The required finite-time convergence conditions hold for the class of stochastic theta methods on globally Lipschitz problems. It is then shown that exponential mean-square stability for non-globally Lipschitz SDEs is not inherited, in general, by numerical methods. However, for a class of SDEs that satisfy a one-sided Lipschitz condition, positive results are obtained for two implicit methods. These results highlight the fact that for long-time simulation on nonlinear SDEs, the choice of numerical method can be crucial.

scholarly journals Numerical Solutions of Stochastic Differential Equations with Piecewise Continuous Arguments under Khasminskii-Type Conditions

2012 ◽  
Vol 2012 ◽  
pp. 1-21 ◽  
Author(s):  
Minghui Song ◽  
Ling Zhang
Keyword(s):  
Differential Equations ◽  
Stochastic Differential Equations ◽  
Growth Condition ◽  
Linear Growth ◽  
Numerical Solutions ◽  
Type Theorem ◽  
Global Existence Of Solutions ◽  
Linear Growth Condition ◽  
Piecewise Continuous ◽  
Piecewise Continuous Arguments

The main purpose of this paper is to investigate the convergence of the Euler method to stochastic differential equations with piecewise continuous arguments (SEPCAs). The classical Khasminskii-type theorem gives a powerful tool to examine the global existence of solutions for stochastic differential equations (SDEs) without the linear growth condition by the use of the Lyapunov functions. However, there is no such result for SEPCAs. Firstly, this paper shows SEPCAs which have nonexplosion global solutions under local Lipschitz condition without the linear growth condition. Then the convergence in probability of numerical solutions to SEPCAs under the same conditions is established. Finally, an example is provided to illustrate our theory.

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