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We propose a tamed-adaptive Euler-Maruyama approximation scheme for stochastic differential equations with locally Lipschitz continuous, polynomial growth drift, and locally Hölder continuous, polynomial growth diffusion coefficients. We consider the strong convergence and the stability of the new scheme. In particular, we show that under some sufficient conditions for the stability of the exact solution, the tamed-adaptive scheme converges strongly in both finite and infinite time intervals.

Tamed-adaptive Euler-Maruyama approximation for SDEs with locally Lipschitz continuous drift and locally Hölder continuous diffusion coefficients