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A high-order recombination algorithm for weak approximation of stochastic differential equations

Syoiti Ninomiya, Yuji Shinozaki

TL;DR

This work tackles the challenge of efficiently weakly approximating SDEs when high-order cubature methods suffer from exponential growth in the measure’s support. It introduces a refined high-order recombination framework that uses reduced measures and a recursive patch-division algorithm, incorporating patch weights into the error bound to preserve order and control computational cost. The approach is grounded in a free-Lie-algebra formulation and m-similar discretisations (e.g., Ninomiya–Victoir) under the UH/UFG structure, yielding an error bound that splits into discretisation and recombination components and remains of order $m$ with a targeted patching strategy. Numerical experiments, including Asian option pricing in the Heston model, demonstrate that the method maintains the expected convergence while suppressing exponential growth of the support to a polynomial rate, illustrating practical impact for high-order weak approximation in finance.

Abstract

This paper presents an algorithm for applying the high-order recombination method, originally introduced by Lyons and Litterer in ``High-order recombination and an application to cubature on Wiener space'' (Ann. Appl. Probab. 22(4):1301--1327, 2012), to practical problems in mathematical finance. A refined error analysis is provided, yielding a sharper condition for space partitioning. Based on this condition, a computationally feasible recursive partitioning algorithm is developed. Numerical examples are also included, demonstrating that the proposed algorithm effectively avoids the explosive growth in the cardinality of the support required to achieve high-order approximations.

A high-order recombination algorithm for weak approximation of stochastic differential equations

TL;DR

This work tackles the challenge of efficiently weakly approximating SDEs when high-order cubature methods suffer from exponential growth in the measure’s support. It introduces a refined high-order recombination framework that uses reduced measures and a recursive patch-division algorithm, incorporating patch weights into the error bound to preserve order and control computational cost. The approach is grounded in a free-Lie-algebra formulation and m-similar discretisations (e.g., Ninomiya–Victoir) under the UH/UFG structure, yielding an error bound that splits into discretisation and recombination components and remains of order with a targeted patching strategy. Numerical experiments, including Asian option pricing in the Heston model, demonstrate that the method maintains the expected convergence while suppressing exponential growth of the support to a polynomial rate, illustrating practical impact for high-order weak approximation in finance.

Abstract

This paper presents an algorithm for applying the high-order recombination method, originally introduced by Lyons and Litterer in ``High-order recombination and an application to cubature on Wiener space'' (Ann. Appl. Probab. 22(4):1301--1327, 2012), to practical problems in mathematical finance. A refined error analysis is provided, yielding a sharper condition for space partitioning. Based on this condition, a computationally feasible recursive partitioning algorithm is developed. Numerical examples are also included, demonstrating that the proposed algorithm effectively avoids the explosive growth in the cardinality of the support required to achieve high-order approximations.
Paper Structure (21 sections, 5 theorems, 88 equations, 8 figures, 1 table, 3 algorithms)

This paper contains 21 sections, 5 theorems, 88 equations, 8 figures, 1 table, 3 algorithms.

Table of Contents

  1. Introduction
  2. Weak approximation problem of SDEs
  3. High-order cubature on Wiener space method
  4. cub3: a cubature of degree $3$
  5. Tree-based simulation
  6. Order of discretisation
  7. High-order recombination
  8. Reduced measure
  9. High-order recombination measures and their construction procedures
  10. Contribution of this paper
  11. Preliminary
  12. Free Lie algebras
  13. High-order discretisation scheme
  14. N--V method
  15. High-order recombination: approximation theorem
  16. ...and 6 more sections

Key Result

Theorem 1

Assume that the vector fields $V_0, V_1, \dots, V_d$ satisfy the UFG condition and let $X$ be the diffusion process defined in eq:SDE. Let $\alpha_1, \alpha_2, \dots, \alpha_k \in A^\ast_1$. Then there exists a constant $C > 0$ such that

Figures (8)

  • Figure 1: Illustration of recombination method
  • Figure 2: Example of patch division
  • Figure 4: Total Error (Heston)
  • Figure 5: # supp(Heston)
  • Figure 6: Recombination Error (Heston)
  • ...and 3 more figures

Theorems & Definitions (20)

  • Definition 1
  • Definition 2
  • Definition 3: $\left\{P^X_t\right\}_{t \geq 0}$
  • Definition 4: UFG condition kusuoka2003malliavin
  • Theorem 1
  • Definition 5: $m$-similar operator
  • Theorem 2
  • Remark 1: The regularity assumption on $f$
  • Theorem 3: ninomiya2008weakkusuoka2013gaussian
  • Remark 2: The number of random variables involved in the N--V method
  • ...and 10 more