Improved error estimates for the order of convergence of the Euler method for random ordinary differential equations driven by semi-martingale noises

It is well known that the Euler method for approximating the solutions of a random ordinary differential equation (RODE) $\mathrm{d}X_t/\mathrm{d}t = f(t, X_t, Y_t)$ 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. This order was known to increase to $1$ in some special cases, such as that of an It\^o diffusion noise, in which case the Euler for RODE can be seen as a particular case of the Milstein method for an associated augmented system of stochastic differential equations. Here, it is proved that, in many more typical cases, further structures on the noise can be exploited so that the strong convergence is of order 1. More precisely, we prove so for any semi-martingale noise. This includes It\^o diffusion processes, point-process noises, transport-type processes with sample paths of bounded variation, and time-changed Brownian motion. This also yields a direct proof of order 1 convergence for It\^o process noises. The result follows from estimating the global error as an iterated integral over both large and small mesh scales, and then by 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 semi-martingale examples above, but still higher than the order $H$ of convergence expected from previous works.