6533b871fe1ef96bd12d2246

RESEARCH PRODUCT

On approximation of a class of stochastic integrals and interpolation

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Given a diffusion Y = (Y_{t})_{t \in [0,T]} we give different equivalent conditions so that a stochastic integral has an L 2-approximation rate of n −η, {\rm \eta \in (0,1/2],} if one approximates by integrals over piece-wise constant integrands where equidistant time nets of cardinality n + 1 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 n + 1 in case {\rm \eta > 0} , it turns out that one always obtains a rate of n^{ - 1/2}, 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 g(Y_{T}) \in D_{1,2} if and only if the equidistant time nets attain the optimal rate of convergence n^{ - 1/2}.