Convergence rate of Euler–Maruyama scheme for stochastic pantograph differential equations

• The convergence order of EM scheme for SPDE is 12.
• Under general conditions, the convergence order of EM scheme for SPDEwMJ is bounded.
• The convergence order for SPDE is completely different from the convergence order for SPDEwMJ.

This paper investigates the convergence rate of Euler–Maruyama scheme for a class of stochastic pantograph differential equations, which may not satisfy the general linear growth condition. We reveal that the convergence order for the equations driven by Brownian motion is 12. We also demonstrate that, along with the value of the exponent p⩾2p⩾2 increasing gradually, the convergence rate for the equations with Markovian jump is decreasing. This conclusion is entirely different from the case without Markovian jump.