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Single-Variate Strong Order 1.5 Schemes for Additive-Noise SDEs

Article Quant Q&A · Author: horchler
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Summary

The document examines whether an explicit, derivative-free stochastic integration method can attain strong order 1.5 using only one random draw per dimension and step when the system has diagonal additive noise. It compares a scheme attributed to Burrage and Burrage, which reportedly uses one variate, with later literature that labels an apparently identical method as only order 1.0 and with other order 1.5 Runge–Kutta schemes that use more variates.

The evidence presented is a literature discrepancy and informal numerical tests in which the one-variate scheme performs slightly better than Euler–Maruyama; no formal convergence study or equations are supplied. The question cites a theoretical limit on methods that use only Wiener increments, but leaves unresolved how that result applies to the stated noise case and whether the claimed order is valid. The document is therefore useful as a methodological question, while conclusions about accuracy require checking the original methods and conducting formal convergence tests.

Key ideas

  • The question concerns strong order 1.5 integration for diagonal additive-noise systems.
  • A cited scheme uses one random variate, but a later paper appears to assign it a lower order.
  • The reported comparison with Euler–Maruyama is informal and does not establish convergence order.
  • The document leaves the literature discrepancy unresolved and calls for clarification.

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# Order 1.5 strong SDE integration methods for systems with diagonal additive noise


# Order 1.5 strong SDE integration methods for systems with diagonal additive noise












I'm looking into simple-to-implement and efficient order 1.5 strong SDE integration schemes for my system. My noise is diagonal and additive (possibly time-varying). Thus methods designed for either Itô or Stratonovich are fine.

Are there any explicit, derivative-free, order 1.5 methods that only require a single random variate per dimension per time-step (or that reduce to this in the diagonal additive noise case)? Based on a proof by Rümelin 1982, acccording to Burrage & Burrage 1996 (p. 83):

> More general Runge-Kutta type schemes can be constructed but it is possible to show that a strong order of 1.5 cannot be surpassed if just the increments $\Delta W_n$ of the Wiener process are used.

In the same Burrage & Burrage paper, a method claimed to be order 1.5 strong is developed1 that indeed just uses one random variate. In simple tests, the Burrage & Burrage scheme performs slightly better than Euler-Maruyama (order 1.0 strong for additive noise), but I've yet to do anything formal. However, in a later paper (Burrage, et al. 2004), what looks to be the identical method (p. 388) is labelled as order 1.0 strong. Additionally, in my perusal of the recent literature, every explicit stochastic Runge-Kutta method of order 1.5 strong has required at least two random variates, e.g., those in Rößler 2010.

So, what's the deal? Is there such a scheme? And if so, can anyone point me to one? Or are there reasons why I might not be coming across such schemes in the literature (other than that they might be less exciting academically)? Perhaps order 1.5, being the limit for a single random variable, is hard to achieve in practice for real problems? Hence, why methods that claim to be order 1.5 generally use at least two variates in order achieve the desired order of convergence.

1 Let me know and I'll include the equations for the integration scheme if you think it necessary.

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.