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ARCANE: Scalable high-degree cubature formulae for simulating SDEs without Monte Carlo error

Peter Koepernik, Thomas Coxon, James Foster

TL;DR

ARCANE is presented, an algorithm that efficiently and automatically constructs cubature formulae of arbitrary degree that robustly achieve an error orders of magnitude smaller than Monte Carlo with the same number of paths.

Abstract

Monte Carlo sampling is the standard approach for estimating properties of solutions to stochastic differential equations (SDEs), but accurate estimates require huge sample sizes. Lyons and Victoir (2004) proposed replacing independently sampled Brownian driving paths with "cubature formulae", deterministic weighted sets of paths that match Brownian "signature moments" up to some degree $D$. They prove that cubature formulae exist for arbitrary $D$, but explicit constructions are difficult and have only reached $D=7$, too small for practical use. We present ARCANE, an algorithm that efficiently and automatically constructs cubature formulae of arbitrary degree. It reproduces the state of the art in seconds and reaches $\boldsymbol{D=19}$ within hours on modest hardware. In simulations across multiple different SDEs and error metrics, our cubature formulae robustly achieve an error orders of magnitude smaller than Monte Carlo with the same number of paths.

ARCANE: Scalable high-degree cubature formulae for simulating SDEs without Monte Carlo error

TL;DR

ARCANE is presented, an algorithm that efficiently and automatically constructs cubature formulae of arbitrary degree that robustly achieve an error orders of magnitude smaller than Monte Carlo with the same number of paths.

Abstract

Monte Carlo sampling is the standard approach for estimating properties of solutions to stochastic differential equations (SDEs), but accurate estimates require huge sample sizes. Lyons and Victoir (2004) proposed replacing independently sampled Brownian driving paths with "cubature formulae", deterministic weighted sets of paths that match Brownian "signature moments" up to some degree . They prove that cubature formulae exist for arbitrary , but explicit constructions are difficult and have only reached , too small for practical use. We present ARCANE, an algorithm that efficiently and automatically constructs cubature formulae of arbitrary degree. It reproduces the state of the art in seconds and reaches within hours on modest hardware. In simulations across multiple different SDEs and error metrics, our cubature formulae robustly achieve an error orders of magnitude smaller than Monte Carlo with the same number of paths.
Paper Structure (41 sections, 21 theorems, 109 equations, 15 figures, 3 tables)

This paper contains 41 sections, 21 theorems, 109 equations, 15 figures, 3 tables.

Table of Contents

  1. Keywords.
  2. Introduction
  3. Results
  4. Limitations
  5. Smoothness
  6. Long time horizons
  7. Methods
  8. Background
  9. Signatures and Cubature Formulae
  10. Size of Cubature Formulae
  11. Connection to rough path theory
  12. The ARCANE algorithm
  13. Parameter Ablation
  14. Theoretical results
  15. Conclusion
  16. ...and 26 more sections

Key Result

Theorem 1

Suppose $(\lambda_i^n, \omega_i^n)$ is an ARCANE cubature with degree $D_n \to \infty$, and dyadic depth $m_n \in \mathbb{N}$ such that for some $\varepsilon, C > 0$. Then goal:C holds.

Figures (15)

  • Figure 1: Monte Carlo estimation for the mean of a scalar SDE, in the ideal case where there exists a closed-form solution that can be evaluated on a discretised time domain.
  • Figure 2: An illustration of our ARCANE cubature formulae with degrees 17, 19, and 5 (dyadic depth 8). As the formulae contain 3194, 8362, and 3952 paths, respectively, we have added noise to help distinguish paths. Here, the thickness and opacity of each path is taken to be proportional to its associated weight (i.e. paths with larger weights are more visible); a histogram of the weights is to the right of each cubature formula.
  • Figure 3: Error plots showing the performance of our cubature formulae in comparison with plain Monte Carlo and QMC methods across a range of different SDEs and error metrics. Unlabelled cubature formulae in orange are dyadic cubature formulae, see Section \ref{['sec:methods']} as well as Appendix \ref{['app:plots']} for full-sized fully labelled versions of the same plots.
  • Figure 4: Error plot showing the performance of our cubature formulae in comparison with plain Monte Carlo and QMC methods measured in Call Price Relative Error (see \ref{['eq:loghestoncallprice']}) in the log--Heston model. Superscripts in cubature labels refer to the dyadic depth of the cubature, see Section \ref{['sec:methods']}.
  • Figure 5: Solution paths generated by the degree-19 cubature formula for the Wright–Fisher diffusion, simulated up to final times $T=0.1$, $T=1.0$, and $T=10.0$ (top row, left to right). Corresponding MVE plots are shown in the bottom row, with a shared y-axis scale.
  • ...and 10 more figures

Theorems & Definitions (43)

  • Theorem 1
  • Remark 1
  • Theorem 2
  • Remark 2
  • Definition 1
  • Theorem B.1: Extension Theorem
  • Definition 2
  • Remark 3
  • Theorem B.2: Universal Limit Theorem
  • Theorem \ref{thm:maindyadic}*
  • ...and 33 more