Almost sure approximations in Hölder norms of a general stochastic process defined by a Young integral
Résumé
We focus on a stochastic process {Y (t)} t∈[0,v] defined by a pathwise Young integral of a general form. Thanks to the Haar basis, we connect the classical method of approximation of {Y (t)} t∈[0,v] through Euler scheme and Riemann-Stieltjes sums with a new approach consisting in the use of an appropriate series representation of {Y (t)} t∈[0,v]. This representation is obtained through a general compactly supported orthonormal wavelet basis. An advantage offered by the new approach with respect to the classical one is that a better almost sure rate of convergence in Hölder norms can be derived, under a general Wiener chaos condition. Also, this improved rate turns out to be optimal in some situations; typically, when the integrand and integrator associated to {Y (t)} t∈[0,v] are independent fractional Brownian motions with appropriate Hurst parameters.
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