WIAS Preprint No. 675, (2001)

The asymptotic behavior of semi-invariants for linear stochastic systems


  • Milstein, Grigori N.

2010 Mathematics Subject Classification

  • 60H10 93E15


  • Stochastic stability, moment Lyapunov exponent, analytic characteristic function, semi-invariants


The asymptotic behavior of semi-invariants of the random variable ln |X(t, 𝑥)|, where X(t,𝑥) is a solution of a linear system of stochastic differential equations, is connected with the moment Lyapunov exponent g(𝑝). Namely, it is obtained that the 𝑛-th semi-invariant is asymptotically proportional to the time 𝗍 with the coefficient of proportionallity g(n)(0). The proof is based on the concept of analytic characteristic functions. It is also shown that the asymptotic behavior of the analytic characteristic function of ln |X(t, 𝑥)| in a neighbourhood of the origin on the complex plane is controlled by the extension g(𝑖𝓏) of g(𝑝).

Appeared in

  • Stochatics and Dynamics, vol. 2 (2002), no.2, pp.281-294

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