EXISTENCE, UNIQUENESS AND ASYMPTOTIC PROPERTIES OF A CLASS OF NONLINEAR STOCHASTIC DIFFERENTIAL DELAY EQUATIONS WITH MARKOVIAN SWITCHING

2009 ◽  
Vol 09 (02) ◽  
pp. 253-275 ◽  
Author(s):  
LIN WANG ◽  
FUKE WU
Keyword(s):  
Asymptotic Properties ◽  
Delay Equations ◽  
Markovian Switching ◽  
Differential Delay Equation ◽  
Differential Delay ◽  
Stochastic Differential Delay Equation ◽  
Differential Delay Equations ◽  
Stochastic Differential Delay Equations ◽  
Moment Boundedness ◽  
The Moment

The main aim of this paper is to give some new conditions under which a class of nonlinear stochastic differential delay equations with Markovian switching admit a unique solution and this solution has nice asymptotic properties, including the moment boundedness and the moment boundedness average in time of this solution. These new conditions show that the coefficients of the nonlinear stochastic differential delay equation with Markovian switching are polynomial or controlled by the polynomial functions or functionals. Two examples are also given for illustration.

scholarly journals Weak approximation of stochastic differential delay equations for bounded measurable function

2013 ◽  
Vol 16 ◽  
pp. 319-343
Author(s):  
Hua Zhang
Keyword(s):  
Approximation Problem ◽  
Delay Equations ◽  
Weak Approximation ◽  
Euler Scheme ◽  
Differential Delay Equation ◽  
Differential Delay ◽  
Stochastic Differential Delay Equation ◽  
Differential Delay Equations ◽  
Stochastic Differential Delay Equations ◽  
Degeneracy Condition

AbstractIn this paper we study the weak approximation problem of $E[\phi (x(T))] $ by $E[\phi (y(T))] $, where $x(T)$ is the solution of a stochastic differential delay equation and $y(T)$ is defined by the Euler scheme. For $\phi \in { C}_{b}^{3} $, Buckwar, Kuske, Mohammed and Shardlow (‘Weak convergence of the Euler scheme for stochastic differential delay equations’, LMS J. Comput. Math. 11 (2008) 60–69) have shown that the Euler scheme has weak order of convergence $1$. Here we prove that the same results hold when $\phi $ is only assumed to be measurable and bounded under an additional non-degeneracy condition.

scholarly journals Euler-Maruyama approximation of backward doubly stochastic differential delay equations

2016 ◽  
Vol 5 (3) ◽  
pp. 146
Author(s):  
Falah Sarhan ◽  
LIU JICHENG
Keyword(s):  
Numerical Solutions ◽  
Delay Equations ◽  
Delay Equation ◽  
Differential Delay Equation ◽  
True Solution ◽  
Differential Delay ◽  
Doubly Stochastic ◽  
Stochastic Differential Delay Equation ◽  
Differential Delay Equations ◽  
Stochastic Differential Delay Equations

In this paper, we attempt to introduce a new numerical approach to solve backward doubly stochastic differential delay equation ( shortly-BDSDDEs ). In the beginning, we present some assumptions to get the numerical scheme for BDSDDEs, from which we prove important theorem. We use the relationship between backward doubly stochastic differential delay equations and stochastic controls by interpreting BDSDDEs as some stochastic optimal control problems, to solve the approximated BDSDDEs and we prove that the numerical solutions of backward doubly stochastic differential delay equation converge to the true solution under the Lipschitz condition.

On the Partial Stochastic Stability of Stochastic Differential Delay Equations with Markovian Switching

2013 ◽  
Vol 765-767 ◽  
pp. 709-712 ◽  
Author(s):  
De Zhi Liu ◽  
Wei Qun Wang
Keyword(s):  
Asymptotic Stability ◽  
Stochastic Stability ◽  
Sufficient Conditions ◽  
Delay Equations ◽  
Markovian Switching ◽  
Differential Delay ◽  
Stability In Probability ◽  
Differential Delay Equations ◽  
Stochastic Differential Delay Equations ◽  
Asymptotic Stability In Probability

In the paper, we are concerned with the partial asymptotic stochastic stability (stability in probability) of stochastic differential delay equations with Markovian switching (SDDEwMSs), the sufficient conditions for partial asymptotic stability in probability have been given and we have generalized some results of Sharov and Ignatyev to cover a class of much more general SDDEwMSs.

scholarly journals Analytic Approximation of the Solutions of Stochastic Differential Delay Equations with Poisson Jump and Markovian Switching

2012 ◽  
Vol 2012 ◽  
pp. 1-14 ◽  
Author(s):  
Hua Yang ◽  
Feng Jiang
Keyword(s):  
Diffusion Coefficients ◽  
Numerical Solutions ◽  
Delay Equations ◽  
Markovian Switching ◽  
Analytic Approximation ◽  
Differential Delay ◽  
Differential Delay Equations ◽  
Stochastic Differential Delay Equations ◽  
Poisson Jump ◽  
And Diffusion

We are concerned with the stochastic differential delay equations with Poisson jump and Markovian switching (SDDEsPJMSs). Most SDDEsPJMSs cannot be solved explicitly as stochastic differential equations. Therefore, numerical solutions have become an important issue in the study of SDDEsPJMSs. The key contribution of this paper is to investigate the strong convergence between the true solutions and the numerical solutions to SDDEsPJMSs when the drift and diffusion coefficients are Taylor approximations.

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