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Approximating the signature of Brownian motion for high order SDE simulation

James Foster

TL;DR

This work addresses the challenge of simulating the non-Gaussian iterated integrals that arise in the Brownian signature, which are central to high-order SDE solvers. It develops a framework that leverages Gaussian, depth-3 Brownian inputs ($W_{s,t}$, $H_{s,t}$, $K_{s,t}$) to construct accurate, efficiently generated approximations for Lévy areas and space-time Lévy areas, including a moment-matching weak Lévy-area estimator and refinements to better capture fourth moments in dimension four. The paper also extends these ideas to space-space-time and related quantities, providing conditional-expectation formulas and variance analyses that underpin high-order splitting methods and Strang-type schemes. Through extensive numerical experiments on Langevin dynamics, Heston, SABR, IGBM, and FHN models, the authors demonstrate tangible accuracy and efficiency gains, highlighting the practical impact on high-order SDE simulation and stochastic modeling in finance and physics.

Abstract

The signature is a collection of iterated integrals describing the "shape" of a path. It appears naturally in the Taylor expansions of controlled differential equations and, as a consequence, is arguably the central object within rough path theory. In this paper, we will consider the signature of Brownian motion with time, and present both new and recently developed approximations for some of its integrals. Since these integrals (or equivalent Lévy areas) are nonlinear functions of the Brownian path, they are not Gaussian and known to be challenging to simulate. To conclude the paper, we will present some applications of these approximations to the high order numerical simulation of stochastic differential equations (SDEs).

Approximating the signature of Brownian motion for high order SDE simulation

TL;DR

This work addresses the challenge of simulating the non-Gaussian iterated integrals that arise in the Brownian signature, which are central to high-order SDE solvers. It develops a framework that leverages Gaussian, depth-3 Brownian inputs (, , ) to construct accurate, efficiently generated approximations for Lévy areas and space-time Lévy areas, including a moment-matching weak Lévy-area estimator and refinements to better capture fourth moments in dimension four. The paper also extends these ideas to space-space-time and related quantities, providing conditional-expectation formulas and variance analyses that underpin high-order splitting methods and Strang-type schemes. Through extensive numerical experiments on Langevin dynamics, Heston, SABR, IGBM, and FHN models, the authors demonstrate tangible accuracy and efficiency gains, highlighting the practical impact on high-order SDE simulation and stochastic modeling in finance and physics.

Abstract

The signature is a collection of iterated integrals describing the "shape" of a path. It appears naturally in the Taylor expansions of controlled differential equations and, as a consequence, is arguably the central object within rough path theory. In this paper, we will consider the signature of Brownian motion with time, and present both new and recently developed approximations for some of its integrals. Since these integrals (or equivalent Lévy areas) are nonlinear functions of the Brownian path, they are not Gaussian and known to be challenging to simulate. To conclude the paper, we will present some applications of these approximations to the high order numerical simulation of stochastic differential equations (SDEs).
Paper Structure (21 sections, 20 theorems, 217 equations, 12 figures, 11 tables)

This paper contains 21 sections, 20 theorems, 217 equations, 12 figures, 11 tables.

Table of Contents

  1. Introduction
  2. Gaussian iterated integrals of Brownian motion
  3. Lévy area of multidimensional Brownian motion
  4. Conditional moments of two-dimensional Lévy area
  5. Conditional moments of multi-dimensional Lévy area
  6. Moment-matching approximation of Brownian Lévy area
  7. Space-space-time Lévy area of Brownian motion
  8. The shuffle product for calculations with iterated integrals
  9. Approximations of space-space-time Lévy area
  10. Approximation of $L_{s,t}$ for high order splitting methods
  11. Numerical examples
  12. Simulating Langevin dynamics using Gaussian integrals
  13. Space-space Lévy area examples
  14. Lévy area of a $d$-dimensional Brownian motion (with $d\leq 4$)
  15. Heston stochastic volatility model
  16. ...and 6 more sections

Key Result

Theorem 1

Consider the Itô SDE (eq:intro_sde) where the vector fields $f, g_i : \mathbb{R}^e \rightarrow\mathbb{R}^e$ are assumed to be sufficiently smooth and with linear growth (see condition (2.17) in milstein2004numerics). Then the solution $y$ of (eq:intro_sde) can be expressed over the interval $[s,t]$, where $h := t - s$, $W_{s,t}^{i} := W_t^i - W_s^i\space$ and the remainder term $R_{s,t}$ is given

Figures (12)

  • Figure 1: The (space-space) Lévy area is the chordal area enclosed by independent Brownian motions (diagram taken from foster2020thesis).
  • Figure 2: $H_{s,t}$ is the area enclosed by a Brownian path and its linear interpolant (diagram from foster2020poly).
  • Figure 3: Space-time-time Lévy area corresponds to how asymmetric in time the Brownian motion is.
  • Figure 4: Given $(W_{s,t}\space, H_{s,t})$, Brownian Lévy area always has more kurtosis than a normal distribution.
  • Figure 5: Space-time Lévy swing gives the side where the path has greater space-time Lévy area (diagram taken from foster2020thesis).
  • ...and 7 more figures

Theorems & Definitions (82)

  • Definition 1: Signature of a path hambly2010roughpathsmorrill2021nrdes
  • Remark 1
  • Theorem 1: Stochastic Itô-Taylor expansion milstein2004numerics
  • Remark 2
  • Definition 2: Lévy area
  • Definition 3: Path increment of Brownian motion
  • Definition 4: Space-time Lévy area
  • Remark 3
  • Definition 5: Space-time-time Lévy area
  • Remark 4
  • ...and 72 more