[Paper Review] Asymptotics for the normalized error of the Ninomiya-Victoir scheme

Gerbi, Anis Al, Benjamin Jourdain+1 more|arXiv (Cornell University)|Jan 20, 2016

Stochastic processes and financial applications14 references3 citations

TL;DR

This paper establishes the asymptotic distribution of the normalized error for the Ninomiya-Victoir scheme in SDEs driven by Brownian motion. It proves that $√{N}(X - X^{NV,\eta})$ converges stably in law to an affine SDE driven by Lie brackets of the Brownian vector fields and an independent Brownian motion, confirming the convergence rate is $1/2$ unless the vector fields commute, in which case the rate improves to $1$. This result supports the use of multilevel Monte Carlo methods with the scheme.

ABSTRACT

In a previous work, we proved strong convergence with order $1/2$ of the Ninomiya-Victoir scheme $X^{NV,η}$ with time step $T/N$ to the solution $X$ of the limiting SDE. In this paper we check that the normalized error defined by $\sqrt{N}\left(X - X^{NV,η} ight)$ converges to an affine SDE with source terms involving the Lie brackets between the Brownian vector fields. The limit does not depend on the Rademacher random variables $η$. This result can be seen as a first step to adapt to the Ninomiya-Victoir scheme the central limit theorem of Lindeberg Feller type, derived by M. Ben Alaya and A. Kebaier for the multilevel Monte Carlo estimator based on the Euler scheme. When the Brownian vector fields commute, the limit vanishes. This suggests that the rate of convergence is greater than $1/2$ in this case and we actually prove strong convergence with order $1$.

Motivation & Objective

• To analyze the asymptotic behavior of the normalized error $\sqrt{N}(X - X^{NV,\eta})$ for the Ninomiya-Victoir scheme.
• To establish the limiting distribution of the normalized error process in the context of strong convergence with order $1/2$.
• To determine whether the convergence rate can be improved when the Brownian vector fields commute.
• To provide a theoretical foundation for applying multilevel Monte Carlo methods with the Ninomiya-Victoir scheme.
• To extend central limit theorem-type results, as in Lindeberg-Feller for Euler schemes, to the Ninomiya-Victoir scheme.

Proposed method

• Use of stable convergence in law, as formalized by Rényi and extended by Jacod and Protter, to analyze the limiting behavior of the normalized error.
• Interpolation of the discrete-time Ninomiya-Victoir scheme to continuous time to derive the asymptotic error distribution.
• Derivation of the limit SDE involving Lie brackets $[\sigma^j, \sigma^m]$ between Brownian vector fields and an independent Brownian motion $B^{j,m}$.
• Application of Itô's formula and Itô-Taylor expansions to control the error between the scheme and the true solution.
• Use of Gronwall’s lemma to bound the moments of the error process and establish moment stability.
• Leverage Lipschitz continuity and regularity assumptions on the coefficients $b$ and $\sigma^j$ to derive moment estimates and convergence rates.

Experimental results

Research questions

• RQ1 Does the normalized error $\sqrt{N}(X - X^{NV,\eta})$ converge in law to a non-degenerate limit as $N \to \infty$?
• RQ2 What is the explicit form of the limiting SDE that governs the asymptotic distribution of the normalized error?
• RQ3 How do the Lie brackets between the Brownian vector fields influence the limiting error distribution?
• RQ4 Under what conditions does the limiting error vanish, and what does this imply for the convergence rate?
• RQ5 Can the central limit theorem for multilevel Monte Carlo estimators, previously derived for the Euler scheme, be adapted to the Ninomiya-Victoir scheme?

Key findings

• The normalized error $\sqrt{N}(X - X^{NV,\eta})$ converges stably in law to the solution of an affine SDE driven by the Lie brackets of the Brownian vector fields and an independent Brownian motion.
• The limit distribution does not depend on the Rademacher random variables $\eta$, indicating that the scheme's asymptotic error is robust to the choice of sign sequence.
• When the Brownian vector fields commute, the Lie bracket terms vanish, and the limiting error process is zero, implying a higher convergence rate.
• The paper proves strong convergence with order $1$ in the commutative case, confirming that the $1/2$ rate is not sharp when $[\sigma^j, \sigma^m] = 0$ for all $j,m$.
• The asymptotic distribution is characterized by an SDE with drift and diffusion coefficients involving $\partial b(X_s)$ and $\partial\sigma^j(X_s)$, respectively, and source terms from Lie brackets.
• The result confirms that the strong convergence rate of the Ninomiya-Victoir scheme is $1/2$ in the general non-commutative case, as the normalized error does not vanish in distribution.