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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