Strong convergence of the exponential Euler scheme for SDEs with superlinear growth coefficients and one-sided Lipschitz drift

TL;DR

This paper establishes the strong convergence of a semi-explicit exponential Euler scheme (Exp-EM) for one-dimensional SDEs with superlinear growth, where the diffusion is polynomially growing and may vanish at zero, and the drift is piecewise locally Lipschitz. It proves a strong rate of $1/2$ for continuous drift, while discontinuities in the drift incur an $\varepsilon$-penalty, with a robust analysis that uses occupation times and a change-of-time technique to control exponential moments and positivity. The authors also develop moment and local-error estimates for the Exp-EM scheme, demonstrate asymptotic stability relative to the continuous model, and provide extensive numerical experiments that illustrate the theoretical rates and stability properties across a range of coefficient regimes. Overall, the work broadens the applicability of exponential-type schemes to SDEs with non-Lipschitz and discontinuous drifts, offering practical guidance for adaptive timestepping and robust simulation in applications with superlinear dynamics.

Abstract

We consider the problem of the discrete-time approximation of the solution of a one-dimensional SDE with piecewise locally Lipschitz drift and continuous diffusion coefficients with polynomial growth. In this paper, we study the strong convergence of a (semi-explicit) exponential-Euler scheme previously introduced in Bossy et al. (2021). We show the usual 1/2 rate of convergence for the exponential-Euler scheme when the drift is continuous. When the drift is discontinuous, the convergence rate is penalised by a factor {$ε$} decreasing with the time-step. We examine the case of the diffusion coefficient vanishing at zero, which adds a positivity preservation condition and a convergence analysis that exploits the negative moments and exponential moments of the scheme with the help of change of time technique introduced in Berkaoui et al. (2008). Asymptotic behaviour and theoretical stability of the exponential scheme, as well as numerical experiments, are also presented.

Figures (5)

• Figure 1: Behaviour of function $\varphi_\Delta t(x) = B_1- (\frac{{\text{\scriptsize{$\Upsigma$}}}^2}{2}+B_2)x^{2(\alpha-1)}+\frac{b(0)}{x}+b(0)\left({\text{\scriptsize{$\Upsigma$}}}^2+B_2\right)x^{2\alpha-3} \Delta t$.
• Figure 2: Strong approximation error for the Exp-EM scheme applied to \ref{['eq:proto_case']}, with Cases 1 to 5 (in log-log scale), $\mathbb{E}^{1/2}\left[\sup_{0\leq t\leq T}|X_t - \overline{X}_t|^2\right]$ (left) and $\mathbb{E}^{1/2}\left[|X_T - \overline{X}_T|^2\right]$ (right). The strong error is compared with the reference slope of order 1/2 (in red).
• Figure 3: Strong approximation error (left) and stopped strong approximation error (right) for the Exp-EM scheme applied to \ref{['eq:proto_case']}, with Cases 6 to 9 (in log-log scale), the strong error is compared with the reference slope of order 1/2 (in red).
• Figure 4: Variance of the approximation error (stopped and non-stopped) for the Exp-EM scheme applied in Cases 6 and 8 (in log-log scale).
• Figure 5: Exp-EM approximation of the trajectories of the solution to $dX_t = (X_t - 6 X_t^3)dt + X_t^2 dW_t$, $X_0 = 1$, crossing infinite times the threshold $\xi^2=\tfrac{2}{13}$ (horizontal line in red) as shown in propositions \ref{['prop:asympt_sol']} and \ref{['prop:asympt_disc']}.

Theorems & Definitions (31)

• Definition 1.1
• Remark 1.2
• Proposition 1.3
• Remark 1.4
• Lemma 2.1
• proof
• Proposition 2.2
• Lemma 2.3
• proof
• Proposition 2.4