Stability theorems of stochastic difference equations
Let \(\{A_ n\); \(n\geq 0\}\) be an independent sequence of \(d\times d\) real-valued matrices and f(n,x) be an \({\mathbb{R}}^ d\) valued mapping defined on \({\mathbb{N}}\times {\mathbb{R}}^ d\). The author investigates various definitions of the stability of the solutions of the equation \[ x_{n+1}=A_ nx_ n+f(n,x_ n),\quad x_ 0\in {\mathbb{R}}^ d. \] A typical result is the following: Let us suppose that \[ {\mathbb{E}}\{\| A_{m-1}...A_ n\| \}\leq C\text{ for } m>n\geq 0, \] \[ {\mathbb{E}}\{\| f(n,x)\| \}\leq b_ n{\mathbb{E}}\{\| x\| \}\text{ with } \sum^{\infty}_{n=0}b_ n<\infty. \] Then for any positive \(\epsilon\) there exists a positive \(\delta\) (\(\epsilon\)) such that for any n one has \({\mathbb{E}}\{\| x_ n\| \}\leq \epsilon\) provided that \({\mathbb{E}}\{\| x_ 0\| \}\leq \delta (\epsilon).\) An other kind of result is the following: Let us suppose that \[ {\mathbb{E}}\{\| A_{m-1}...A_ n\| \}\leq C\delta^{m-n+1}\text{ for } m>n\geq 0\text{ and } \delta <1, \] \[ {\mathbb{E}}\{\| f(n,x)\| \}\leq b{\mathbb{E}}\{\| x\| \}\text{ with } b\quad sufficiently\quad small. \] Then \(\lim_{n\to \infty}{\mathbb{E}}\{\| x_ n\| \}=0.\) Analogous results hold for the \(L^ p\)-norm and various conditions on the behavior of the products of the matrices \(A_ n\) and contraction properties of the function f.
scientific article; zbMATH DE number 4166225
scientific article