WIAS Preprint No. 377, (1997)

Moderate deviations for integral functionals of diffusion processes


  • Liptser, Robert
  • Spokoiny, Vladimir
    ORCID: 0000-0002-2040-3427

2010 Mathematics Subject Classification

  • 60F10


  • Large Deviations, Moderate Deviations, Diffusions


The moderate deviation principle (MDP) is established for a family of integral functionals $varepsilon^-kappaint_0^t g(xi_s^varepsilon)ds$, where $varepsilon$ is a small positive parameter, $xi_t^varepsilon$ is ``fast'' ergodic diffusions, and $g$ is a smooth unbounded function. It is shown that if $kappain (0,1/2)$ then the MDP for such a family is nothing but a large deviation principle for a family $varepsilon^1/2-kappasqrtgamma W_t$ with a proper parameter $gamma$ and a standard Wiener process $W_t$. An application of this result is a deviation principle of Freidlin-Wentzell's type for a family of random processes generated by the ordinary differential equation $ dotY^varepsilon,kappa_t=a(Y_t^varepsilon,kappa)+ varrho(Y_t^varepsilon,kappa)n^varepsilon(t) $ governed by, so called, wide-band noise $n^varepsilon(t)=varepsilon^-kappa g(xi_t^varepsilon)$. Another application is a large deviation type estimation for functionals $varepsilon^-kappaint_0^T^varepsilon Psi^varepsilon (X_s^varepsilon) g(xi_s^varepsilon)ds$, where $0< kappa<1/2$, $Psi^varepsilon $ is an unbounded in $varepsilon$ smooth mapping of ``slow'' diffusions $X_t^varepsilon$ governed by the ``fast'' ones, and $[0,T^varepsilon]$ is an unbounded in $varepsilonto 0$ time interval.

Appeared in

  • J. of Probability 4 (1999) No. 17, pp. 1-25. under the title: Moderate deviation type evaluation for integral functionals of diffusion processes.

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