  It is well known that the Euler method for approximating the solutions of a
random ordinary differential equation $\mathrm{d}X_t/\mathrm{d}t = f(t, X_t,
Y_t)$ driven by a stochastic process $\{Y_t\}_t$ with $\theta$-H\"older sample
paths is estimated to be of strong order $\theta$ with respect to the time
step, provided $f=f(t, x, y)$ is sufficiently regular and with suitable bounds.
Here, it is proved that, in many typical cases, further conditions on the noise
can be exploited so that the strong convergence is actually of order 1,
regardless of the H\"older regularity of the sample paths. This applies for
instance to additive or multiplicative It\^o process noises (such as Wiener,
Ornstein-Uhlenbeck, and geometric Brownian motion processes); to point-process
noises (such as Poisson point processes and Hawkes self-exciting processes,
which even have jump-type discontinuities); and to transport-type processes
with sample paths of bounded variation. The result is based on a novel
approach, estimating the global error as an iterated integral over both large
and small mesh scales, and switching the order of integration to move the
critical regularity to the large scale. The work is complemented with numerical
simulations illustrating the strong order 1 convergence in those cases, and
with an example with fractional Brownian motion noise with Hurst parameter $0 <
H < 1/2$ for which the order of convergence is $H + 1/2$, hence lower than the
attained order 1 in the examples above, but still higher than the order $H$ of
convergence expected from previous works.

Probability;Numerical Analysis;Numerical Analysis