The paper develops exponential stability of the analytic solution and convergence in probability of the numerical method for highly nonlinear hybrid stochastic pantograph equation. The classical linear growth conditio...The paper develops exponential stability of the analytic solution and convergence in probability of the numerical method for highly nonlinear hybrid stochastic pantograph equation. The classical linear growth condition is replaced by polynomial growth conditions, under which there exists a unique global solution and the solution is almost surely exponentially stable. On the basis of a series of lemmas, the paper establishes a new criterion on convergence in probability of the Euler-Maruyama approximate solution. The criterion is very general so that many highly nonlinear stochastic pantograph equations can obey these conditions. A highly nonlinear example is provided to illustrate the main theory.展开更多
In this paper,we present the semi-implicit Euler（SIE）numerical solution for stochastic pantograph equations with jumps and prove that the SIE approximation solution converges to the exact solution in the mean-square...In this paper,we present the semi-implicit Euler（SIE）numerical solution for stochastic pantograph equations with jumps and prove that the SIE approximation solution converges to the exact solution in the mean-square sense under the Local Lipschitz condition.展开更多
The paper introduces a new class of numerical schemes for the approximate solutions of stochastic pantograph equations. As an effective technique to implement implicit stochastic methods, strong predictor-corrector me...The paper introduces a new class of numerical schemes for the approximate solutions of stochastic pantograph equations. As an effective technique to implement implicit stochastic methods, strong predictor-corrector methods （PCMs） are designed to handle scenario simulation of solutions of stochastic pantograph equations. It is proved that the PCMs are strong convergent with order 1/2.Linear M^-stabiiity of stochastic pantograph equationsand the PCMs are researched in the paper. Sufficient conditions of MS-unstability of stochastic pantograph equations and MS-stability of the PCMs are obtained, respectively. Numerical experiments demonstrate these theoretical results.展开更多
In this paper, a deterministic delay differential pantograph equation （DDPE） with an unbounded memory is stochastically perturbed by an Ito-type noise. The contribution of white noise to the oscillatory behaviour of...In this paper, a deterministic delay differential pantograph equation （DDPE） with an unbounded memory is stochastically perturbed by an Ito-type noise. The contribution of white noise to the oscillatory behaviour of the new stochastic delay differential pantograph equation （SDDPE） is investigated. It is established that under certain conditions and with a highly positive probability, the new stochastic delay differential pantograph equation has an oscillatory solution influenced by the presence of the noise. This is not possible with the original deterministic system which has a non-oscillatory solution due to the absence of noise.展开更多
In this paper, we derive the stochastic maximum principle for optimal control problems of the forward-backward Markovian regime-switching system. The control system is described by an anticipated forward-backward stoc...In this paper, we derive the stochastic maximum principle for optimal control problems of the forward-backward Markovian regime-switching system. The control system is described by an anticipated forward-backward stochastic pantograph equation and modulated by a continuous-time finite-state Markov chain. By virtue of classical variational approach, duality method, and convex analysis, we obtain a stochastic maximum principle for the optimal control.展开更多
基金support from the National Natural Science Foundation of China(70871046,71171091,71191091)Fundamental Research Funds for the Central Universities(2011QN167)
文摘The paper develops exponential stability of the analytic solution and convergence in probability of the numerical method for highly nonlinear hybrid stochastic pantograph equation. The classical linear growth condition is replaced by polynomial growth conditions, under which there exists a unique global solution and the solution is almost surely exponentially stable. On the basis of a series of lemmas, the paper establishes a new criterion on convergence in probability of the Euler-Maruyama approximate solution. The criterion is very general so that many highly nonlinear stochastic pantograph equations can obey these conditions. A highly nonlinear example is provided to illustrate the main theory.
基金Supported by the NSF of the Higher Education Institutions of Jiangsu Province(10KJD110006)Supported by the grant of Jiangsu Institute of Education(Jsjy2009zd03)Supported by the Qing Lan Project of Jiangsu Province(2010)
文摘In this paper,we present the semi-implicit Euler（SIE）numerical solution for stochastic pantograph equations with jumps and prove that the SIE approximation solution converges to the exact solution in the mean-square sense under the Local Lipschitz condition.
文摘The paper introduces a new class of numerical schemes for the approximate solutions of stochastic pantograph equations. As an effective technique to implement implicit stochastic methods, strong predictor-corrector methods （PCMs） are designed to handle scenario simulation of solutions of stochastic pantograph equations. It is proved that the PCMs are strong convergent with order 1/2.Linear M^-stabiiity of stochastic pantograph equationsand the PCMs are researched in the paper. Sufficient conditions of MS-unstability of stochastic pantograph equations and MS-stability of the PCMs are obtained, respectively. Numerical experiments demonstrate these theoretical results.
文摘In this paper, a deterministic delay differential pantograph equation （DDPE） with an unbounded memory is stochastically perturbed by an Ito-type noise. The contribution of white noise to the oscillatory behaviour of the new stochastic delay differential pantograph equation （SDDPE） is investigated. It is established that under certain conditions and with a highly positive probability, the new stochastic delay differential pantograph equation has an oscillatory solution influenced by the presence of the noise. This is not possible with the original deterministic system which has a non-oscillatory solution due to the absence of noise.
文摘In this paper, we derive the stochastic maximum principle for optimal control problems of the forward-backward Markovian regime-switching system. The control system is described by an anticipated forward-backward stochastic pantograph equation and modulated by a continuous-time finite-state Markov chain. By virtue of classical variational approach, duality method, and convex analysis, we obtain a stochastic maximum principle for the optimal control.
