Asymptotic error distribution for tamed Euler method with coupled monotonicity condition

This paper establishes the asymptotic error distribution of the tamed Euler method for stochastic differential equations (SDEs) with a coupled monotonicity condition, that is, the limit distribution of the corresponding normalized error process. Specifically, for SDEs driven by multiplicative noise, we first propose a tamed Euler method parameterized by $α\in (0, 1]$ and establish that its strong convergence rate is $α\wedge\frac{1}{2}$. Notably, $α$ can take arbitrary positive values by adjusting the regularization coefficient without altering the strong convergence rate. We then derive the asymptotic error distribution for this tamed Euler method. Further, we infer from the limit equation that among the tamed Euler method of strong order $\frac{1}{2}$, the one with $α= \frac{1}{2}$ yields the largest mean-square error after a long time, while those of $α>\frac{1}{2}$ share a unified asymptotic error distribution. In addition, our analysis is also extended to SDEs with additive noise and similar conclusions are obtained. Additional treatments are required to accommodate super-linearly growing coefficients, a feature that distinguishes our analysis on the asymptotic error distribution from established results.

💡 Research Summary

This paper investigates the asymptotic error distribution of a tamed Euler scheme applied to stochastic differential equations (SDEs) whose drift and diffusion coefficients may grow super‑linearly but satisfy a coupled monotonicity condition. The authors first introduce a family of tamed Euler methods indexed by a parameter α ∈ (0, 1] (later extended to any positive α) in which the drift and diffusion are regularized as

 f_α(x)=f(x)/(1+(Δt)^α|x|^{2l}), g_α(x)=g(x)/(1+(Δt)^α|x|^{2l}),

with Δt = T/N the time step and l the growth exponent appearing in the monotonicity assumptions. This regularization suppresses the super‑linear growth when the step size is small, while leaving the underlying dynamics unchanged in the limit Δt→0.

Under the coupled monotonicity assumptions (a one‑sided Lipschitz condition together with polynomial growth bounds for f and g), the SDE admits a unique strong solution with moments of all orders. The paper proves that the tamed Euler scheme is strongly convergent with order

 rate = α ∧ ½,

i.e. the minimum of α and ½. Remarkably, for any α > ½ the convergence order remains ½, so increasing α beyond ½ does not degrade the strong convergence. This extends earlier results where α was restricted to (0, ½].

The core contribution concerns the normalized error process

 E_N(t) = N^{½∧α}\bigl( X̂_{α,N}(t) – X(t) \bigr),

where X̂_{α,N} denotes the continuous‑time interpolation of the tamed Euler iterates. The authors decompose E_N into dominant terms (linear in the increments of the Brownian motion) and remainder terms. Because of the super‑linear coefficients, the usual approach of discarding the remainders is not justified. Instead, they establish convergence of both dominant and remainder parts in the space C(