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

TL;DR

The paper analyzes Euler discretizations for random ODEs driven by semi-martingale noises and proves strong order 1 convergence under a unified framework. It derives a global-error integral representation, leverages the semi-martingale decomposition, and applies FV/change-of-variables and martingale inequalities to bound the noise contributions, covering Itô diffusions, jump processes, transport-type noises, and time-changed Brownian motion. Numerical experiments across a wide range of models validate the theoretical rate, while showing that non-semi-martingale fractional Brownian motion can yield higher-order convergence only up to H<1/2, specifically p = H+1/2, and p = 1 for H≥1/2. The results provide sharp, broadly applicable convergence guarantees for RODEs and suggest potential extensions to pathwise convergence and random PDEs, with caveats about higher-order methods. Overall, the work delivers a unified, sharp understanding of the strong convergence behavior of the Euler method for RODEs under realistic noisy perturbations, informing both theory and computation in stochastic modeling.

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 $θ$-Hölder sample paths is estimated to be of strong order $θ$ 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 Itô 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.

Figures (23)

• Figure 1: (a) Estimated order of convergence of the strong error of the Euler method for the linear system \ref{['allnoisesRODEsystemintro']} as it depends on the noise, starting with all the noises combined (all) and continuing with the scalar equation, each with a different type of noise process, namely Wiener (W), Ornstein-Uhlenbeck (OU), geometric Brownian motion (gBm), a homogeneous linear Itô diffusion (hlp), compound Poisson process (cP), step point process (sP), Hawkes process (H), transport process (T), and also a fractional Brownian motion with Hurst parameter $1/2 < H < 1.$ (fBm); (b) Sample paths of all the noises used in the linear system \ref{['allnoisesRODEsystemintro']}.
• Figure 2: Estimated order of convergence of the strong error of the Euler method for (a) the population dynamics model \ref{['eqpopdyn']} and (b) the seismic model \ref{['eqearthquake']}.
• Figure 3: (a) Estimated order of convergence of the strong error of the Euler method for the toggle-switch model \ref{['toggleswitchsystemintro']}; (b) Euler approximations of the component $\{X_t\}_{t\in I}$ of a sample path.
• Figure 4: (a) Estimated order of convergence of the strong error of the Euler method for the insurance risk model \ref{['eqrisk']}; (b) Sample paths of the surplus $\{U_t\}_{t\in I},$ the Ornstein-Uhlenbeck process $\{O_t\}_{t\in I},$ the interest rate $\{I_t\}_{t\in I},$ and the insurance claims $\{C_t\}_{t\in I}.$
• Figure 5: (a) Estimated order of convergence of the strong error of the Euler method for the Fisher-KPP model \ref{['eqfisherkpp']}-\ref{['eqfisherkppbcs']}; Sample paths of (b) the Ornstein-Uhlenbeck and Hawkes processes, and of (c) the noise obtained as the product of these processes.

Theorems & Definitions (17)

• Remark 2.1
• Remark 2.2
• Lemma 2.1
• Lemma 2.2
• Remark 2.3
• Lemma 3.1
• Lemma 5.1
• proof
• Lemma 6.1: Discrete Gronwall Lemma
• Proposition 6.1: Basic strong bound