Convergence of the tamed Euler scheme for stochastic differential equations with Piecewise Continuous Arguments under non-Lipschitz continuous coefficients
Provides a convergence guarantee for a practical numerical method applied to a specific class of delay SDEs, which is an incremental extension of existing theory.
The paper proves that the tamed Euler method converges with strong order one half for stochastic differential equations with piecewise continuous arguments under superlinearly growing coefficients, extending previous results for standard SDEs.
Recently, Martin Hutzenthaler pointed out that the explicit Euler method fails to converge strongly to the exact solution of a stochastic differential equation (SDE) with superlinearly growing and globally one sided Lipschitz drift coefficient. Afterwards, he proposed an explicit and easily implementable Euler method, i.e tamed Euler method, for such an SDE and showed that this method converges strongly with order of one half. In this paper, we use the tamed Euler method to solve the stochastic differential equations with piecewise continuous arguments (SEPCAs) with superlinearly growing coefficients and prove that this method is convergent with strong order one half.