Stochastic Integration for Some Rough Non‐adapted Processes

For the Wiener process W on the unit interval and 0 ⩽ θ ⩽ 1/4 consider the non‐adapted process u t =sgn(W t−θ )sgn(W 1 −W T+θ ), θ⩽t⩽1−θ. By giving explicit descriptions of their expansions in terms of generalized Hermite polynomials we show that they possess Skorohod integral processes. In the more interesting case θ = 0, a component due to the interaction of forwards and backwards Wiener process seems to dominate the non‐interaction part, whereas for θ > 0 interaction drops geometrically with respect to non‐interaction. We translate these purely analytic properties into more familiar terms of samples of stochastic processes. Firstly, we construct simple approximations of the integral process which are of the type of Riemann sums. We then show that their quadratic variations are the same as for the Wiener process on [θ, 1 − θ]. Finally, using the fact that W remains a semimartingale if its filtration is enlarged by W 1 , we compare the resulting Ito integral process to the Skorohod integral process of u in case θ = 0. We show that they differ by a local time. Consequently the Skorohod integral process of u is a semimartingale.