6533b7ddfe1ef96bd1274ef1

RESEARCH PRODUCT

Interpolation and approximation in L2(γ)

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AbstractAssume 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 B2,qθ(γ)≔(L2(γ),D1,2(γ))θ,q, obtained via the real interpolation method, by the behavior of aX(f(X1);τ)≔∥f(W1)-PXτf(W1)∥L2, where τ=(ti)i=0n is a deterministic time net and PXτ:L2→L2 the orthogonal projection onto a subspace of ‘discrete’ stochastic integrals x0+∑i=1nvi-1(Xti-Xti-1) 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.