Almost sure approximations in Hölder norms of a general stochastic process defined by a Young integral

We focus on a stochastic process $\{Y(t)\}_{t\in [0,v]}$ defined by a pathwise Young integral of a general form. Thanks to the Haar basis, we connect the classical method of approximation of $\{Y(t)\}_{t\in [0,v]}$ through Euler scheme and Riemann-Stieltjes sums with a new approach consisting in the use of an appropriate series representation of $\{Y(t)\}_{t\in [0,v]}$. 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 H\"older norms can be derived, under a general Gaussian condition. Also, this improved rate turns out to be optimal in some situations; typically, when the integrator associated to $\{Y(t)\}_{t\in [0,v]}$ is a fractional Brownian motion of an arbitrary Hurst parameter, and the integrand is a deterministic nowhere vanishing function with an invariant H\"older regularity large enough.