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

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

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 ${\displaystyle f\in C^{1}(\mathbb {R} ^{k},\mathbb {R} ^{d})}$ it holds that $${\displaystyle f(X_{0,T}^{Y_{0}})-f(Y_{T})=\int _{0}^{T}f'\left({\frac {\partial }{\partial x}}X_{r,T}^{Y_{s}}\right){\frac {\partial }{\partial x}}X_{s,T}^{Y_{s}}\left(\mu (r,Y_{r})-{\frac {d}{dr}}Y_{r}\right)dr.}$$