A new class of structure-preserving stochastic exponential Runge-Kutta integrators for stochastic differential equations

B-series for SDEs with application to exponential integrators for non-autonomous semi-linear problems

“…The following exponential midpoint rule was proposed by Yang et.al. [31]: t ] A ) = h 3 /48 we get Φ(τ ) = 3h 6 ∆W/768. Clearly Φ(τ ) ̸ = φ(τ ), so the order condition related to this particular tree is not satisfied, which could also not be expected, as the order of the tree is 6.5.…”

“…We will conclude the paper with an example from Yang et.al. [31], in which low order terms are found and mean square convergence of order 1 for Stratonovich SDEs is proved. We do not intend to reproduce their results, but we will only use this example as a demonstration on how our theory can be applied.…”

“…in [4,16]. This has been extended to exponential integrators in [2] and more recently in [31]. Consider a semi-linear non-autonomous SDE of the form…”

“…For these methods, construction of the random coefficients is discussed in [20]. Yang et al [31] derived B-series for the exact and numerical solution of this problem, given that the matrix function A(t) is commutative, by expanding exponentials of the form e tn+h tn A(s)ds . In the following, it will be demonstrated how these series can be derived significantly simpler by the theory derived above, and without the assumptions on commutativity of A(t).…”

“…A similar work has very recently been published by Yang et.al. [31], however, their results only apply to Stratonovich SDEs with a commutative linear part. The results which will be presented in the current paper do not have these limitations.…”