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It is well known that the Euler method for a random ordinary differentialequation $\mathrm{d}X_t/\mathrm{d}t = f(t, X_t, Y_t)$ driven by a stochasticprocess $\{Y_t\}_t$ with $\theta$-H\"older sample paths is estimated to be ofstrong order $\theta$ with respect to the time step, provided $f=f(t, x, y)$ issufficiently regular and with suitable bounds. This order is known to increaseto $1$ in some special cases. Here, it is proved that, in many more typicalcases, further structures on the noise can be exploited so that the strongconvergence is of order 1. More precisely, we prove so for any semi-martingalenoise. This includes It\^o diffusion processes, point-process noises,transport-type processes with sample paths of bounded variation, andtime-changed Brownian motion. The result follows from estimating the globalerror as an iterated integral over both large and small mesh scales, and byswitching the order of integration to move the critical regularity to the largescale. The work is complemented with numerical simulations showing theoptimality of the strong order 1 convergence in those cases, and with anexample with fractional Brownian motion noise with Hurst parameter $0 < H <1/2,$ which is not a semi-martingale and for which the order of convergence is$H + 1/2$, hence lower than the attained order 1 in the semi-martingale case,but still higher than the order $H$ of convergence expected from previousworks.

Strong order-one convergence of the Euler method for random ordinary differential equations driven by semi-martingale noises