Limits for weighted -variations and likewise functionals of fractional diffusions with drift

Let X t be the pathwise solution of a diffusion driven by a fractional Brownian motion B t H with Hurst constant H > 1 / 2 and diffusion coefficient σ ( t , x ) . Consider the successive increments of this solution, Δ X i = X i / n − X ( i − 1 ) / n . Using a cylinder approximation for the solution X t , our main result yields that if 1 / 2 < H < 3 / 4 then, if Z is a standard normal random variable which is independent of B H , the process 1 n ∑ i = 1 [ n t ] [ | Δ X i n H | p − σ p ( i / n , X i / n ) E ( | Z | p ) ] converges weakly to W ( C H , p ∫ 0 t σ p ( s , X s ) d s ) as n → ∞ where W is a Wiener process which is independent of B H and C H , p is a constant which depends on H and on p . In the place of p -variations we may consider functions that satisfy an almost multiplicative structure such as even polynomials or polynomials of absolute values. By considering second order increments of the discrete sample X i we obtain analogous results for the whole interval 1 / 2 < H < 1 . Finally, we show convergence is stable in the absence of drift and use this result to discuss weak convergence for weak solutions of the fractional diffusion equation.