Efficient and Accurate Gradients for Neural SDEs

The paper’s Appendix D states and proves that the reversible Heun method (Definition D.1) converges strongly with order 1/2 under globally Lipschitz C^2 coefficients, and with order 1 in the additive-noise case (Theorems D.11–D.12 and D.16–D.17). The analysis proceeds via (i) a two-step reformulation to control the internal stage gap Y−Z with an L4 bound O(h^{1/2}) (Theorem D.6), (ii) Stratonovich–Taylor expansions of the numerical step and the exact solution to the second-iterated integral level (Theorems D.8–D.9), and (iii) a local L2 error recursion giving c4 h^2 (general) and c6 h^3 (additive), then a discrete Grönwall argument to globalize the bound. These steps match the candidate’s outline: the candidate’s local defect is attributed to the missing Lévy-area bracket term (equivalently reflected in the paper’s handling of W⊗2 versus W increments), and they arrive at the same global rates. Small differences are technical (paper uses an L4 control on Y−Z and works explicitly with the second iterated integral; the candidate uses an L2 control and speaks in terms of Lie brackets), but the logic and results are effectively the same. See Definition D.1 for the scheme, Theorem D.6 for the stage-gap control, Theorems D.8–D.9 for the expansions, Theorem D.11 for the local estimate (c4 h^2) and Theorem D.12 for strong order 1/2; Theorems D.13–D.17 give the additive-noise improvements to local c6 h^3 and global order 1 .