Stochastic Pantograph Differential Equations Research Articles

Convergence and stability of numerical methods with variable step size for stochastic pantograph differential equations

The stochastic pantograph equations (SPEs) are very special stochastic delay differential equations (SDDEs) with unbounded memory. When the numerical methods with a constant step size are applied to the pantograph equations, the most difficult problem is the limited computer memory. In this paper, we construct methods with variable step size to solve SPEs. The analysis is motivated by the example of a mean-square stable linear SPE for which the Euler–Maruyama (EM) method with variable step size fails to reproduce this behaviour for any nonzero timestep. Then we consider the Backward Euler (BE) method with variable step size and develop the fundamental numerical analysis concerning its strong convergence and mean-square linear stability. It is proved that the numerical solutions produced by the BE method with variable step size converge to the exact solution under the local Lipschitz condition and the Bounded condition. Furthermore, the order of convergence p=½ is given under the Lipschitz condition. The result of the mean-square linear stability is given. Some illustrative numerical examples are presented to demonstrate the order of strong convergence and the mean-square linear stability of the BE method.

A Taylor polynomial approach in approximations of solution to pantograph stochastic differential equations with Markovian switching

The subject of this paper are analytic approximate methods for pantograph stochastic differential equations with Markovian switching, as well as their counterparts without Markovian switching. Approximate equations are defined on equidistant partitions of the time interval, and their coefficients are Taylor approximations of the coefficients of the initial equation. In the case with Markovian switching we will present the approximate method based on Taylor approximation of coefficients in two arguments and show that the appropriate approximate solutions converge in the L p -norm to the solution of the initial equation. Then we will present the other approximate method which deals with Taylor approximation in the first argument. In both cases the closeness between the approximate solution and the solution of the initial equation depends on the number of degrees in Taylor approximations of coefficients, although the presence of the Markov chain affects it. These approximate methods are then adapted to the case without Markovian switching. The first method gives the possibility of proving L p convergence as well as a.s. convergence of the appropriate sequence of approximate solutions to the solution of the initial equation.