Interpolation and approximation in [formula omitted

Abstract

Assume a standard Brownian motion W = ( W t ) t ∈ [ 0 , 1 ] , a Borel function f : R → R such that f ( W 1 ) ∈ L 2 , and the standard Gaussian measure γ on the real line. We characterize that f belongs to the Besov space B 2 , q θ ( γ ) ≔ ( L 2 ( γ ) , D 1 , 2 ( γ ) ) θ , q , obtained via the real interpolation method, by the behavior of a X ( f ( X 1 ) ; τ ) ≔ ∥ f ( W 1 ) - P X τ f ( W 1 ) ∥ L 2 , where τ = ( t i ) i = 0 n is a deterministic time net and P X τ : L 2 → L 2 the orthogonal projection onto a subspace of ‘discrete’ stochastic integrals x 0 + ∑ i = 1 n v i - 1 ( X t i - X t i - 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 a X ( f ( X 1 ) ; τ ) can be used to describe the L 2 -error in discrete time simulations of the martingale generated by f ( W 1 ) and (in stochastic finance) to describe the minimal quadratic hedging error of certain discretely adjusted portfolios.