Strong convergence of the tamed and the semi-tamed Euler schemes for stochastic differential equations with jumps under non-global Lipschitz condition
Provides a computationally efficient alternative to implicit methods for SDEs with jumps, but the result is incremental as it extends existing taming techniques to jump processes.
The paper proves that tamed and semi-tamed Euler schemes converge strongly with standard one-half order for SDEs with jumps under non-global Lipschitz conditions, addressing a gap where explicit Euler fails.
We consider the explicit numerical approximations of stochastic differential equations (SDEs) driven by Brownian process and Poisson jump. It is well known that under non-global Lipschitz condition, Euler Explicit method fails to converge strongly to the exact solution of such SDEs without jumps, while implicit Euler method converges but requires much computational efforts. We investigate the strong convergence, the linear and nonlinear exponential stabilities of tamed Euler and semi-tamed methods for stochastic differential equation driven by Brownian process and Poisson jumps, both in compensated and non compensated forms. We prove that under non-global Lipschitz condition and superlinearly growing drift term, these schemes converge strongly with the standard one-half order. Numerical simulations to substain the theoretical results are provided.