Alekseev–Gröbner formula

The Alekseev–Gröbner formula, or nonlinear variation-of-constants formula, is a generalization of the linear variation of constants formula which was proven independently by Wolfgang Gröbner in 1960[1] and Vladimir Mikhailovich Alekseev in 1961.[2] It expresses the global error of a perturbation in terms of the local error and has many applications for studying perturbations of ordinary differential equations.[3]

Formulation

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Let   be a natural number, let   be a positive real number, and let   be a function which is continuous on the time interval   and continuously differentiable on the  -dimensional space  . Let  ,   be a continuous solution of the integral equation   Furthermore, let   be continuously differentiable. We view   as the unperturbed function, and   as the perturbed function. Then it holds that   The Alekseev–Gröbner formula allows to express the global error   in terms of the local error  .

The Itô–Alekseev–Gröbner formula

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The Itô–Alekseev–Gröbner formula[4] is a generalization of the Alekseev–Gröbner formula which states in the deterministic case, that for a continuously differentiable function   it holds that  

References

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  1. ^ Gröbner, Wolfgang (1960). Die Lie-Reihen und Ihre Anwendungen. Berlin: VEB Deutscher Verlag der Wissenschaften.
  2. ^ Alekseev, V. "An estimate for the perturbations of the solution of ordinary differential equations (Russian)". Vestn. Mosk. Univ., Ser. I, Math. Meh. 2, 1961.
  3. ^ Iserles, A. (2009). A first course in the numerical analysis of differential equations (second ed.). Cambridge: Cambridge Texts in Applied Mathematics, Cambridge University Press.
  4. ^ Hudde, A.; Hutzenthaler, M.; Jentzen, A.; Mazzonetto, S. (2018). "On the Itô-Alekseev-Gröbner formula for stochastic differential equations". arXiv:1812.09857 [math.PR].