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

$$f(X_{0,T}^{Y_{0}})-f(Y_{T})=\int _{0}^{T}f'\left({\frac {\partial }{\partial x}}X_{r,T}^{Y_{s}}\right){\frac {\partial }{\partial x}}X_{s,T}^{Y_{s}}\left(\mu (r,Y_{r})-{\frac {d}{dr}}Y_{r}\right)dr.$$