scholarly journals Milstein-type semi-implicit split-step numerical methods for nonlinear stochastic differential equations with locally Lipschitz drift terms

2019 ◽  
Vol 23 (Suppl. 1) ◽  
pp. 1-12 ◽  
Author(s):  
Burhaneddin Izgi ◽  
Coskun Cetin
Keyword(s):  
Numerical Methods ◽  
Differential Equations ◽  
Stochastic Differential Equations ◽  
Numerical Solutions ◽  
Rates Of Convergence ◽  
Ginzburg Landau ◽  
Linear Growth Condition ◽  
Locally Lipschitz ◽  
Non Linear ◽  
Ginzburg Landau Equations

We develop Milstein-type versions of semi-implicit split-step methods for numerical solutions of non-linear stochastic differential equations with locally Lipschitz coefficients. Under a one-sided linear growth condition on the drift term, we obtain some moment estimates and discuss convergence properties of these numerical methods. We compare the performance of multiple methods, including the backward Milstein, tamed Milstein, and truncated Milstein procedures on non-linear stochastic differential equations including generalized stochastic Ginzburg-Landau equations. In particular, we discuss their empirical rates of convergence.

scholarly journals Numerical Solutions of Stochastic Differential Equations Driven by Poisson Random Measure with Non-Lipschitz Coefficients

2012 ◽  
Vol 2012 ◽  
pp. 1-17 ◽  
Author(s):  
Hui Yu ◽  
Minghui Song
Keyword(s):  
Numerical Methods ◽  
Differential Equations ◽  
Stochastic Differential Equations ◽  
Linear Growth ◽  
Numerical Solutions ◽  
Random Measure ◽  
Euler Method ◽  
Poisson Random Measure ◽  
Euler’S Method ◽  
Linear Growth Condition

The numerical methods in the current known literature require the stochastic differential equations (SDEs) driven by Poisson random measure satisfying the global Lipschitz condition and the linear growth condition. In this paper, Euler's method is introduced for SDEs driven by Poisson random measure with non-Lipschitz coefficients which cover more classes of such equations than before. The main aim is to investigate the convergence of the Euler method in probability to such equations with non-Lipschitz coefficients. Numerical example is given to demonstrate our results.

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.

The moment exponential stability criterion of nonlinear hybrid stochastic differential equations and its discrete approximations

2016 ◽  
Vol 146 (6) ◽  
pp. 1303-1328 ◽  
Author(s):  
Xiaofeng Zong ◽  
Fuke Wu ◽  
Chengming Huang
Keyword(s):  
Differential Equations ◽  
Exponential Stability ◽  
Stochastic Differential Equations ◽  
Growth Condition ◽  
Stability Criterion ◽  
Linear Growth ◽  
Numerical Solutions ◽  
Large Deviation ◽  
Linear Growth Condition ◽  
Moment Exponential Stability

Based on the martingale theory and large deviation techniques, we investigate the pth moment exponential stability criterion of the exact and numerical solutions to hybrid stochastic differential equations (SDEs) under the local Lipschitz condition. This new stability criterion shows that Markovian switching can serve as a stochastic stabilizing factor by its logarithmic moment-generating function. We also investigate the pth moment exponential stability of Euler–Maruyama (EM), backward EM (BEM) and split-step backward EM (SSBEM) approximations for hybrid SDEs and show that, under the additional linear growth condition, the EM method can share the mean-square exponential stability of the exact solution for sufficiently small step size. However, the BEM method can work without the linear growth condition. We further investigate the SSBEM method under a coupled condition.

Glycemia monitoring

1998 ◽  
Vol 2 ◽  
pp. 23-30
Author(s):  
Igor Basov ◽  
Donatas Švitra
Keyword(s):  
Diabetes Mellitus ◽  
Mathematical Model ◽  
Numerical Methods ◽  
Differential Equations ◽  
Blood Sugar ◽  
Self Regulation ◽  
Physiological System ◽  
Non Linear ◽  
The Mathematical Model

Here a system of two non-linear difference-differential equations, which is mathematical model of self-regulation of the sugar level in blood, is investigated. The analysis carried out by qualitative and numerical methods allows us to conclude that the mathematical model explains the functioning of the physiological system "insulin-blood sugar" in both normal and pathological cases, i.e. diabetes mellitus and hyperinsulinism.

scholarly journals Conditions to Guarantee the Existence of the Solution to Stochastic Differential Equations of Neutral Type

Symmetry ◽  
2020 ◽  
Vol 12 (10) ◽  
pp. 1613
Author(s):  
Mun-Jin Bae ◽  
Chan-Ho Park ◽  
Young-Ho Kim
Keyword(s):  
Differential Equations ◽  
Stochastic Differential Equations ◽  
Existence And Uniqueness ◽  
Neutral Type ◽  
Sufficient Conditions ◽  
Exponential Estimates ◽  
Analysis Theory ◽  
Linear Growth Condition ◽  
Uniqueness Of The Solution ◽  
The Uniqueness Theorem

The main purpose of this study was to demonstrate the existence and the uniqueness theorem of the solution of the neutral stochastic differential equations under sufficient conditions. As an alternative to the stochastic analysis theory of the neutral stochastic differential equations, we impose a weakened Ho¨lder condition and a weakened linear growth condition. Stochastic results are obtained for the theory of the existence and uniqueness of the solution. We first show that the conditions guarantee the existence and uniqueness; then, we show some exponential estimates for the solutions.

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