Alekseev–Gröbner formula

Short description: Formula for expressing the global error of a perturbation

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 and Vladimir Mikhailovich Alekseev in 1961. 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.

Formulation

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

The Itô–Alekseev–Gröbner formula

The Itô–Alekseev–Gröbner formula is a generalization of the Alekseev–Gröbner formula which states in the deterministic case, that for a continuously differentiable function [math]\displaystyle{ f \in C^{1}(\mathbb R^{k}, \mathbb R^{d}) }[/math] it holds that [math]\displaystyle{ f(X_{0, T}^{Y_{0}}) - f(Y_{T}) = \int_{0}^{T} f'\left( \frac{\partial}{\partial x} X_{r, T}^{Y_{s}} \right) \frac{\partial}{\partial x} X_{s, T}^{Y_{s}}\left( \mu(r, Y_{r}) - \frac{d}{dr} Y_{r} \right) dr. }[/math]