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Limit theorem for a differential equation with a long-range random coefficient. (Théorème limite pour une équation différentielle à coefficient aléatoire à mémoire longue.) (French) Zbl 1038.60033

The author considers a differential equation with a small parameter \(\varepsilon\) and a centered stationary long-range Gaussian process \(m\). For instance \(m\) may be a fractional white noise with Hurst index \(H\) (\(1/2<H<1\)). It is proved that the solution of this equation converges in distribution as \(\varepsilon\) goes to zero to the solution of a stochastic differential equation driven by a fractional Brownian motion: when \(m\) is an \(H\)-fractional white noise, this fractional Brownian motion has the same Hurst index. The proof is based on the theory of rough paths that has recently been put forward by T. Lyons.

MSC:

60G15 Gaussian processes
60H10 Stochastic ordinary differential equations (aspects of stochastic analysis)
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[1] Billingsley, P., Convergence of Probability Measures (1968), Wiley · Zbl 0172.21201
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