Interpolation and approximation in L2(γ)

Assume a standard Brownian motion W=(Wt)t∈[0,1], a Borel function f:R→R such that f(W1)∈L2, and the standard Gaussian measure γ on the real line. We characterize that f belongs to the Besov space , obtained via the real interpolation method, by the behavior of , where is a deterministic time net and the orthogonal projection onto a subspace of ‘discrete’ stochastic integrals with X being the Brownian motion or the geometric Brownian motion. By using Hermite polynomial expansions the problem is reduced to a deterministic one. The approximation numbers aX(f(X1);τ) can be used to describe the L2-error in discrete time simulations of the martingale generated by f(W1) and (in stochastic finance) to describe the minimal quadratic hedging error of certain discretely adjusted portfolios.