  In this paper, we use the truncated EM method to study the finite time strong
convergence for the SDEs with Poisson jumps under the Khasminskii-type
condition. We establish the finite time $ \mathcal L ^r (r \ge 2) $ convergence
rate when the drift and diffusion coefficients satisfy super-linear condition
and the jump coefficient satisfies the linear growth condition. The result
shows that the optimal $\mathcal L ^r$-convergence rate is close to $ 1/ (1 +
\gamma)$, where $\gamma$ is the super-linear growth constant. This is
significantly different from the result on SDEs without jumps. When all the
three coefficients of SDEs are allowing to grow super-linearly, the $ \mathcal
L^r (0<r<2)$ strong convergence results are also investigated and the optimal
strong convergence rate is shown to be not greater than $1/4$. Moreover, we
prove that the truncated EM method preserve nicely the mean square
exponentially stability and asymptotic boundedness of the underlying SDEs with
Piosson jumps. Several examples are given to illustrate our results.
