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 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 $$d \in \mathbb N$$ be a natural number, let $$T \in (0, \infty)$$ be a positive real number, and let $$\mu \colon [0, T] \times \mathbb{R}^{d} \to \mathbb{R}^{d} \in C^{0, 1}([0, T] \times \mathbb{R}^{d})$$ be a function which is continuous on the time interval $$[0, T]$$ and continuously differentiable on the $$d$$-dimensional space $$\mathbb{R}^{d}$$. Let $$X \colon [0, T]^{2} \times \mathbb{R}^{d} \to \mathbb{R}^{d}$$, $$ (s, t, x) \mapsto X_{s, t}^{x}$$ be a continuous solution of the integral equation $$X_{s, t}^{x} = x + \int_{s}^{t} \mu(r, X_{s, r}^{x}) dr.$$ Furthermore, let $$Y \in C^{1}([0, T], \mathbb{R}^{d})$$ be continuously differentiable. We view $$Y$$ as the unperturbed function, and $$X$$ as the perturbed function. Then it holds that $$ X_{0, T}^{Y_{0}} - Y_{T} = \int_{0}^{T} \left( \frac{\partial}{\partial x} X_{r, T}^{Y_{s}} \right) \left( \mu(r, Y_{r}) - \frac{d}{dr} Y_{r} \right) dr. $$ The Alekseev–Gröbner formula allows to express the global error $$X_{0, T}^{Y_{0}} - Y_{T}$$ 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 is a generalization of the Alekseev–Gröbner formula which states in the deterministic case, that for a continuously differentiable function $$f \in C^{1}(\mathbb R^{k}, \mathbb R^{d})$$ it holds that $$ f(X_{0, T}^{Y_{0}}) - f(Y_{T}) = \int_{0}^{T} f'\left( \frac{\partial}{\partial x} X_{r, T}^{Y_{s}} \right) \frac{\partial}{\partial x} X_{s, T}^{Y_{s}}\left( \mu(r, Y_{r}) - \frac{d}{dr} Y_{r} \right) dr. $$