\label{euler_moment}\end{aligned}$$ Then $$\lim_{n \rightarrow \infty}  \mathbf{E}  \max_{k=0, \ldots, n} |X_{t_k}- \widetilde{X}^n_{t_k} |^2=0.$$

From the results of the previous section $$\lim_{n \rightarrow \infty}  \max_{k=0, \ldots, n} |X_{t_k}- \widetilde{X}^n_{t_k} |= 0$$ hold, almost surely. However, assumption (\[euler\_moment\]) implies the uniform integrability of $$\max_{k=0, \ldots n} |X_{t_k}- \widetilde{X}_{t_k}^{n} |^2, \,\,\,\, n \in \mathbb{N}.$$ The assertion now follows, since uniform integrability allows integration to the limit.

Note that assumption is easily verified if the SDE coefficients have linear growth on $D$, i.e. $$|a(x)| + \sum_{j=1}^m |b_{j}(x)| \leq C \cdot (1+ |x|) , \qquad x \in D,$$ for some $C>0$. Turning back to the Cox-Ingersoll-Ross process this gives us strong convergence of the Euler schemes , and under the assumption $ 2 \kappa \lambda \geq \theta^2.$ Note that for the Euler schemes and strong convergence without a restriction on the parameter has been shown in [@delbaen] and [@hm] using a Yamada function technique. This technique has also been applied by Gyöngy & Rásonyi in [@Gyoengy_Rasonyi] to obtain the following result:

\[euler\_hoelder\] Let $a_1,a_2,b: \mathbb{R} \rightarrow \mathbb{R}$. Consider the one-dimensional SDE $$dX_t = (a_1(X_t) +a_2(X_t))\,dt + b(X_t)\, dW_t, \quad t \in [0,T], \qquad X_0=x_0 \in \mathbb{R}$$ and let ${ \overline{X}}^n$ be the corresponding Euler scheme with 