Interpolation and approximation in L2(γ)

  • Authors:

  • Stefan Geiss;Mika Hujo

  • Affiliations:

  • Department of Mathematics and Statistics, University of Jyvaeskylae, P.O. Box 35 (MAD), FIN-40014 Jyvaeskylae, Finland;Department of Mathematics and Statistics, University of Jyvaeskylae, P.O. Box 35 (MAD), FIN-40014 Jyvaeskylae, Finland

  • Venue:
  • Journal of Approximation Theory
  • Year:
  • 2007

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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 @c on the real line. We characterize that f belongs to the Besov space B"2","q^@q(@c)@?(L"2(@c),D"1","2(@c))"@q","q, obtained via the real interpolation method, by the behavior of a"X(f(X"1);@t)@?@?f(W"1)-P"X^@tf(W"1)@?"L"""2, where @t=(t"i)"i"="0^n is a deterministic time net and P"X^@t:L"2-L"2 the orthogonal projection onto a subspace of 'discrete' stochastic integrals x"0+@?"i"="1^nv"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);@t) 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.