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

TL;DR

This paper proves that the Euler method for random ODEs driven by semi-martingale noises achieves strong order 1 convergence in many typical cases, extending previous results and including various types of stochastic processes.

Contribution

It establishes strong order 1 convergence for the Euler method under semi-martingale noise, broadening the class of stochastic processes with optimal convergence results.

Findings

Euler method attains strong order 1 convergence for semi-martingale noises.

Numerical simulations confirm the optimality of the convergence order.

Example with fractional Brownian motion shows lower convergence order, highlighting the semi-martingale case's advantage.

Abstract

It is well known that the Euler method for 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. This order is known to increase to $1$ in some special cases. 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. In fact, 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. The result follows from estimating the global error as an iterated integral over both large and small mesh scales, and by switching the order of integration to move the critical regularity to the large scale. The work is complemented with numerical simulations showing the optimality of the strong order 1 convergence in those cases, and with an example 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 previous works.