This text surveys properties and applications of the exponential functional t 0 exp(−ξs)ds of real-valued Lévy processes ξ = (ξt, t ≥ 0).

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For every value of the Hurst index H ∈ (0, 1) we define a stochastic integral with respect to fractional Brownian motion of index H . We do so by approximating fractional Brownian motion by semi-martingales.Then, for H > 1/6, we establish an Itô's change of variables formula, which is more precise than Privault's Ito formula (1998) (established for every H > 0), since it only involves anticipating integrals with respect to a driving Brownian motion.

We show that the limiting free energy in Sherrington-Kirkpatrick's Spin Glass Model does not depend on the environment.