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.
{\displaystyle d\in \mathbb {N} }
be a natural number, let
∈ ( 0 , ∞ )
{\displaystyle T\in (0,\infty )}
be a positive real number, and let
μ : [ 0 ,
be a function which is continuous on the time interval
and continuously differentiable on the
-dimensional space
be a continuous solution of the integral equation
μ ( r ,
{\displaystyle X_{s,t}^{x}=x+\int _{s}^{t}\mu (r,X_{s,r}^{x})dr.}
be continuously differentiable.
We view
as the unperturbed function, and
as the perturbed function.
Then it holds that
μ ( r ,
{\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
in terms of the local error
( μ ( r ,
{\displaystyle (\mu (r,Y_{r})-{\tfrac {d}{dr}}Y_{r})}
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
it holds that
μ ( r ,
{\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.}