Alekseev–Gröbner formula

Template:Short description 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 $T\in (0,\mathrm{\infty })$ be a positive real number, and let $\mu :[0,T]\times {ℝ}^d\to {ℝ}^d\in C^{0,1}([0,T]\times {ℝ}^d)$ be a function which is continuous on the time interval $[0,T]$ and continuously differentiable on the $d$-dimensional space ${ℝ}^d$. Let $X:[0,T{]}^2\times {ℝ}^d\to {ℝ}^d$, $(s,t,x)\mapsto X_{s,t}^x$ be a continuous solution of the integral equation $$X_{s,t}^x=x+\underset{s}{\overset{t}{\int }}\mu (r,X_{s,r}^x)dr.$$ 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 $$X_{0,T}^{Y_0}-Y_T=\underset{0}{\overset{T}{\int }}\left(\frac{\partial }{\partial x}{X}_{r,T}^{Y_s}\right)(\mu (r,Y_r)-\frac{d}{dr}{Y}_r)dr.$$ The Alekseev–Gröbner formula allows to express the global error $X_{0,T}^{Y_0}-Y_T$ in terms of the local error $(\mu (r,Y_r)-\frac{d}{dr}{Y}_r)$.

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 $f\in C^1({ℝ}^k,{ℝ}^d)$ it holds that $$f(X_{0,T}^{Y_0})-f(Y_T)=\underset{0}{\overset{T}{\int }}{f}^{\prime }\left(\frac{\partial }{\partial x}{X}_{r,T}^{Y_s}\right)\frac{\partial }{\partial x}{X}_{s,T}^{Y_s}(\mu (r,Y_r)-\frac{d}{dr}{Y}_r)dr.$$