On approximation of a class of stochastic integrals and interpolation

Given a diffusion we give different equivalent conditions so that a stochastic integral has an L 2-approximation rate of n −η, if one approximates by integrals over piece-wise constant integrands where equidistant time nets of cardinality are used. In particular, we obtain assertions in terms of smoothness properties of g(Y T ) in the sense of Malliavin calculus. After optimizing over non-equidistant time-nets of cardinality in case , it turns out that one always obtains a rate of which is optimal. This applies to all functions g obtained in an appropriate way by the real interpolation method between the weighted Sobolev space D 1,2(μ) and L 2(μ), where μ is related to the law of Y T . Finally, we obtain the result that if and only if the equidistant time nets attain the optimal rate of convergence