Characterization of stochastic processes which stabilize linear companion form systems.
Consider the stochastic differential equation \[ dx^\varepsilon =Ax^\varepsilon d t +Ux^\varepsilon \circ dF^\varepsilon _t, \tag{1} \] (in Stratonovich sense), \(\varepsilon >0\), where \(A\) is a \(d\times d\) companion matrix to a linear differential equation of order \(d\), \[ U= \begin{pmatrix} 0 & \hdots & 0 &0\\ \\ u_1 & \hdots & u_{d-1} & u_d \end{pmatrix}, \] \(F^\varepsilon _t:= F_{t/\varepsilon }\) and the semimartingale \(F_t\) is defined by the equation \[ dF_t= f_0(\xi _t)d t +\sum _{k=1}^r f_k(\xi _t)\circ dW^k_t, \] where \(f_0,\dots ,f_r\) are real-valued \(C^\infty \)-functions on a connected smooth Riemann manifold \(M\), and \(\xi _t\) is a stationary ergodic process on \(M\). Stability of the trivial solution to the system (1) is investigated. The class of processes \(F\) is characterized for which the system (1) is stabilized and the Lyapunov spectrum is studied.
scientific article; zbMATH DE number 2098340
scientific article