| The problem of stochastic integration with respect to fractional Brownian motion (fBm) with H < 1/2 and other 'rough path' Gaussian processes is considered. We use a Riemann sum approach to construct stochastic integrals. It is known, for example, that a Midpoint Riemann sum converges in probability to a stable integral for fBm with H > 1/4, but not in general if H ≤ 1/4. We consider four different types of Riemann sums and their associated critical values: Midpoint (2 types), Trapezoidal, and Simpson's rule. At the critical value (H = 1/4, 1/6, and 1/10, respectively), the sums converge only in distribution. Convergence in distribution is proved by means of theorems and techniques of Malliavin calculus. We consider asymptotic behavior of a specific stochastic integral with respect to fBm with H > 1/2. This result approximates an fBm version of Spitzer's theorem for planar Brownian motion. |
