Alekseev–Gröbner formula explained

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

Let

d \in N be a natural number, let T \in ( 0 , inf ty) be a positive real number, and let \mu \colon [ 0 , T] x R ^{d} \to R ^{d} \in C ^{0,} ([ 0 , T] x R ^{d} ) be a function which is continuous on the time interval [ 0 , T] and continuously differentiable on the d -dimensional space R ^{d} . Let X \colon [ 0 , T] ^{2} x R ^{d} \to R ^{d} , (s, t, x) \mapsto X ^{x} _{s, t} be a continuous solution of the integral equation$$X\_^\; =\; x\; +\; \backslash int\_^\; \backslash mu(r,\; X\_^)\; dr.$$Furthermore, let Y \in C ^{1} ([ 0 , T], R ^{d} ) be continuously differentiable. We view Y as the unperturbed function, and X as the perturbed function. Then it holds that$$X\_^\; -\; Y\_\; =\; \backslash int\_^\; \backslash left(\backslash frac\; X\_^\; \backslash right)\; \backslash left(\backslash mu(r,\; Y\_)\; -\; \backslash frac\; Y\_\; \backslash right)\; dr.$$The Alekseev–Gröbner formula allows to express the global error X ^{Y _{0}} _{0 , T} - Y _{T} in terms of the local error ( \mu(r, Y _{r} ) - \tfrac{d}{dr} Y _{r} ) .

The Itô–Alekseev–Gröbner formula

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

f \in C ^{1} ( R ^{k} , R ^{d} ) it holds that$$f(X\_^)\; -\; f(Y\_)\; =\backslash int\_^\; f\text{'}\backslash left(\backslash frac\; X\_^\; \backslash right)\; \backslash frac\; X\_^\backslash left(\backslash mu(r,\; Y\_)\; -\; \backslash frac\; Y\_\; \backslash right)\; dr.$$

Notes and References

1. Book: Gröbner . Wolfgang . Die Lie-Reihen und Ihre Anwendungen. . 1960 . VEB Deutscher Verlag der Wissenschaften . Berlin.
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. Book: Iserles . A. . A first course in the numerical analysis of differential equations . 2009 . Cambridge Texts in Applied Mathematics, Cambridge University Press . Cambridge . second.
4. Hudde . A. . Hutzenthaler . M. . Jentzen . A. . Mazzonetto . S. . On the Itô-Alekseev-Gröbner formula for stochastic differential equations . 2018 . math.PR . 1812.09857.