The weak Stratonovich integral with respect to fractional Brownian motion with Hurst parameter 1/6

Ivan Nourdin (1) , Anthony Réveillac (2) , Jason Swanson (3)

1 LPMA - Laboratoire de Probabilités et Modèles Aléatoires
2 Institut für Mathematik [Berlin]
3 Department of Mathematics

Let B be a fractional Brownian motion with Hurst parameter H=1/6. It is known that the symmetric Stratonovich-style Riemann sums for $\int g(B(s))dB(s)$ do not, in general, converge in probability. We show, however, that they do converge in law in the Skorohod space of càdlàg functions. Moreover, we show that the resulting stochastic integral satisfies a change of variable formula with a correction term that is an ordinary Itô integral with respect to a Brownian motion that is independent of B.

Mots clés: Weak convergence; Stratonovich integral; Fractional Brownian motion; Malliavin calculus; Stochastic integration; Malliavin calculus.; Weak convergence

Domaines: Probabilités [math.PR]

Ivan Nourdin, Anthony Réveillac, Jason Swanson. The weak Stratonovich integral with respect to fractional Brownian motion with Hurst parameter 1/6. Electronic Journal of Probability, 2010, 15 (70), pp.2117-2162. ⟨hal-00493981⟩