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Large deviations of linear stochastic differential equations. (English) Zbl 0632.60021
Stochastic differential systems, Proc. IFIP-WG 7/1 Work. Conf., Eisenach/GDR 1986, Lect. Notes Control Inf. Sci. 96, 117-151 (1987).
[For the entire collection see Zbl 0619.00019.]
The authors study the linear stochastic differential system \[ dx(t)=A_ 0(\xi (t))x(t)+\sum^{m}_{i=1}A_ i(\xi (t))x(t)\circ dW_ i(t),\quad x(0)=x_ 0\in {\mathbb{R}}^ d, \] which is driven by both white and real noise, Ẇ(t) and \(\xi\) (t). Investigating the Lyapunov exponent of the p th moment of the solution, \[ g(p)=\lim_{t\to \infty}\log E| x(t,x_ 0)|^ p,\quad p\in {\mathbb{R}}, \] first, they prove a law of large numbers and a central limit theorem for \(\log | x(t;x_ 0)|\), \(t\to \infty\), by generalizing and uniforming the former separate treatment for white noise and real noise. Then, for \[ \{law[\log | x(t;x_ 0)| /| x_ 0|],\quad t\to \infty \} \] a large deviation principle is established, where the action functional or level-one entropy function is the Legendre-Fenchel transformation of g(p).
Comparison with deterministic periodic or vibrational noise yields conditions under which the system is stable for any real noise (i.e. any stationary Markov process). As an example the random oscillator is treated in detail with a universal stability diagram.
Reviewer: V.Wihstutz

60F10 Large deviations
60H10 Stochastic ordinary differential equations (aspects of stochastic analysis)
60F05 Central limit and other weak theorems
93E15 Stochastic stability in control theory
93D20 Asymptotic stability in control theory