[Paper Review] Weak error expansion of the implicit Euler scheme

Omar Aboura|arXiv (Cornell University)|Apr 25, 2013

Advanced Numerical Methods in Computational Mathematics6 references3 citations

TL;DR

This paper establishes a weak error expansion for the implicit Euler scheme applied to It’s stochastic differential equations, extending the Talay-Tubaro framework to implicit schemes. It proves the scheme is of weak order 1 and derives a first-order expansion of the weak error in terms of a new function $\psi_i$, which differs from the explicit scheme’s $\psi_e$ by terms involving $b^2\Delta u$ and $\sigma^2\Delta(b\partial u)$, with the remainder bounded by $O(h^2)$. The result relies on the Kolmogorov equation, Itô and Clark-Ocone formulas, and interpolation techniques.

ABSTRACT

In this paper, we extend the Talay Tubaro theorem to the implicit Euler scheme.

Motivation & Objective

• To extend the Talay-Tubaro weak error expansion framework from explicit to implicit Euler schemes for SDEs.
• To analyze the weak convergence behavior of the implicit Euler scheme under smooth, bounded coefficients.
• To derive a precise first-order weak error expansion involving a new function $\psi_i$ that characterizes the leading-order error term.
• To provide a theoretical foundation for studying weak error in SPDEs, where implicit schemes are more natural.

Proposed method

• Derives the weak error expansion using the Kolmogorov backward equation associated with the SDE.
• Introduces a continuous interpolation of the discrete implicit Euler scheme to enable analysis of the weak error.
• Applies the Itô and Clark-Ocone formulas to decompose and estimate the error terms.
• Uses a function $S_h(x) = 1/(1 - h b'(x))$ to model the implicit scheme's stability and correction terms.
• Establishes a discrete-to-continuous approximation via a summation lemma, showing $h\sum E[v(t_{k+1}, X^{N}_{t_{k+1}}) w(t_k, X^{N}_{t_k})] \to E\int v w(t, X_t) dt$ with $O(h)$ error.
• Compares $\psi_{ih}$, the discrete approximation of $\psi_i$, to the continuous $\psi_i$, proving their difference is $O(h)$ in expectation.

Experimental results

Research questions

• RQ1 Can the Talay-Tubaro weak error expansion framework be extended to the implicit Euler scheme for SDEs?
• RQ2 What is the leading-order term in the weak error expansion of the implicit Euler scheme, and how does it differ from the explicit case?
• RQ3 How does the implicit scheme's weak error depend on the drift and diffusion coefficients beyond the standard $O(h)$ convergence?
• RQ4 What role does the function $S_h(x) = 1/(1 - h b'(x))$ play in correcting the weak error of the implicit scheme?
• RQ5 Is the weak error of the implicit Euler scheme still of order 1, and can a higher-order expansion be derived?

Key findings

• The implicit Euler scheme is of weak order 1, with $|\mathbb{E}[f(X_T^N)] - \mathbb{E}[f(X_T)]| \leq Ch$ for some constant $C$ independent of $N$.
• The weak error admits a first-order expansion: $\mathbb{E}[f(X_T^N)] - \mathbb{E}[f(X_T)] = h\mathbb{E}\int_0^T \psi_i(t, X_t) dt + O(h^2)$, where $\psi_i$ is explicitly defined in terms of $b$, $\sigma$, and derivatives of the solution $u$ to the Kolmogorov PDE.
• $\psi_i$ differs from the explicit scheme's $\psi_e$ by terms involving $b^2\Delta u$, $\sigma^2\Delta(b\partial u)$, and higher-order derivatives, reflecting the implicit nature of the scheme.
• The difference between the discrete approximation $\psi_{ih}$ and the continuous $\psi_i$ is bounded by $O(h)$, ensuring the expansion remains valid in the limit.
• The proof relies on the Kolmogorov equation, Itô and Clark-Ocone formulas, and a summation lemma that controls the error in discrete-continuous approximations.
• For $b=0$, $\psi_i = \psi_e = \frac{1}{8}\sigma^4 \partial^4 u - \frac{1}{8}\sigma^2 \Delta(\sigma^2 \Delta u)$, confirming consistency with the explicit case.