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

2016 ◽  
Vol 8 (6) ◽  
pp. 1004-1022 ◽  
Author(s):  
Xu Yang ◽  
Weidong Zhao
Keyword(s):  
Differential Equations ◽  
Theoretical Result ◽  
Strong Convergence ◽  
Stochastic Differential Equations ◽  
Rate Of Convergence ◽  
Numerical Experiments ◽  
Mean Square ◽  
Nonlinear Stochastic Differential Equations ◽  
Mean Square Convergence ◽  
The Mean

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

scholarly journals The StochasticΘ-Method for Nonlinear Stochastic Volterra Integro-Differential Equations

2014 ◽  
Vol 2014 ◽  
pp. 1-13 ◽  
Author(s):  
Peng Hu ◽  
Chengming Huang
Keyword(s):  
Asymptotic Stability ◽  
Differential Equations ◽  
Numerical Experiments ◽  
Sufficient Condition ◽  
Asymptotically Stable ◽  
Mean Square ◽  
True Solution ◽  
Mean Square Exponential Stability ◽  
Mean Square Convergence ◽  
The Mean

The stochasticΘ-method is extended to solve nonlinear stochastic Volterra integro-differential equations. The mean-square convergence and asymptotic stability of the method are studied. First, we prove that the stochasticΘ-method is convergent of order1/2in mean-square sense for such equations. Then, a sufficient condition for mean-square exponential stability of the true solution is given. Under this condition, it is shown that the stochasticΘ-method is mean-square asymptotically stable for every stepsize if1/2≤θ≤1and when0≤θ<1/2, the stochasticΘ-method is mean-square asymptotically stable for some small stepsizes. Finally, we validate our conclusions by numerical experiments.

scholarly journals Euler-type approximation for systems of stochastic differential equations

1989 ◽  
Vol 2 (4) ◽  
pp. 239-249 ◽  
Author(s):  
J. Golec ◽  
G. Ladde
Keyword(s):  
Differential Equations ◽  
Stochastic Differential Equations ◽  
Sufficient Conditions ◽  
Taylor Formula ◽  
Time Varying ◽  
Stochastic Version ◽  
Type Approximation ◽  
Mean Square Convergence ◽  
The Mean ◽  
Time Invariant

By developing a stochastic version of the Taylor formula, the mean-square convergence of the Euler-type approximation for the solution of systems of Itô-type stochastic differential equations is investigated. Sufficient conditions are given to obtain time-varying and time-invariant error estimates.

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 The truncated Milstein method for super-linear stochastic differential equations with Markovian switching

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Weijun Zhan ◽  
Qian Guo ◽  
Yuhao Cong
Keyword(s):  
Markov Chain ◽  
Differential Equations ◽  
Convergence Rate ◽  
Stochastic Differential Equations ◽  
Convergence Result ◽  
Convergence Order ◽  
Mean Square ◽  
Numerical Example ◽  
Milstein Method ◽  
The Mean

<p style='text-indent:20px;'>In this paper, to approximate the super-linear stochastic differential equations modulated by a Markov chain, we investigate a truncated Milstein method with convergence order 1 in the mean-square sense. Under Khasminskii-type conditions, we establish the convergence result by employing a relationship between local and global errors. Finally, we confirm the convergence rate by a numerical example.</p>

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